deconvolution of variable-rate reservoir-performance data using … · 2018. 11. 18. ·...

Deconvolution of Variable-RateReservoir-Performance Data

Using B-SplinesD. Ilk, P.P. Valkó, SPE, and T.A. Blasingame, SPE, Texas A&M U.

SummaryWe use B-splines for representing the derivative of the unknownunit-rate drawdown pressure and numerical inversion of theLaplace transform to formulate a new deconvolution algorithm.When significant errors and inconsistencies are present in the datafunctions, direct and indirect regularization methods are incorpo-rated. We provide examples of under- and over-regularization, andwe discuss procedures for ensuring proper regularization.

We validate our method using synthetic examples generatedwithout and with errors (up to 10%). Upon validation, we thendemonstrate our deconvolution method using a variety of fieldcases, including traditional well tests, permanent downhole gaugedata, and production data. Our work suggests that the new decon-volution method has broad applicability in variable rate/pressureproblems and can be implemented in typical well-test and produc-tion-data-analysis applications.

IntroductionThe constant-rate drawdown pressure behavior of a well/reservoirsystem is the primary signature used to classify/establish the char-acteristic reservoir model. Transient-well-test procedures typicallyare designed to create a pair of controlled flow periods (a pressure-drawdown/-buildup sequence) and to convert the last part of theresponse (the pressure buildup) to an equivalent constant-ratedrawdown by means of special time transforms. However, thepresence of wellbore storage, previous flow history, and rate varia-tions may mask or distort characteristic features in the pressure andrate responses.

With the ever-increasing ability to observe downhole rates, ithas long been recognized that variable-rate deconvolution shouldbe a viable option to traditional well-testing methods because de-convolution can provide an equivalent constant-rate response forthe entire time span of observation. This potential advantage ofvariable-rate deconvolution has become particularly obvious withthe appearance of permanent downhole instrumentation.

First and foremost, variable-rate deconvolution is mathemati-cally ill-conditioned; while numerous methods have been devel-oped and applied to deconvolve “ideal” data, very few deconvo-lution methods perform well in practice. The ill-conditioned natureof the deconvolution problem means that small changes in theinput data cause large variations in the deconvolved constant-ratepressures. Mathematically, we are attempting to solve a first-kindVolterra equation [see Lamm (2000)] that is ill-posed. However, inour case the kernel of the Volterra-type equation is the flow-ratefunction (i.e., the generating function); this function is not knownanalytically but, rather, is approximated from the observed flowrates. In practical terms, this issue adds to the complexity of theproblem (Stewart et al. 1983).

In the literature related to variable-rate deconvolution, we findthe development of two basic concepts. One concept is to incor-porate an a priori knowledge regarding the properties of the de-convolved constant-rate response. The observations of Coats et al.

(1964) on the strict monotonicity of the solution led Kuchuk et al.(1990) to impose a “nonpositive second derivative” constraint onpressure response. In some respects, this tradition is maintained inthe work given by von Schroeter et al. (2004), Levitan (2003), andGringarten et al. (2003) when they incorporate non-negativity inthe “encoding of the solution.” We note that in the examples given,this concept (non-negativity/monotonicity of the solution) requiresless-straightforward numerical methods (e.g., nonlinear least-squares minimization).

The second concept is to use a certain level of regularization(von Schroeter et al. 2004; Levitan 2003; Gringarten et al. 2003),where “regularization” is defined as the act or process of makinga system regular or standard (smoothing or eliminating nonstan-dard or irregular response features). Regularization can be per-formed indirectly, by representing the desired solution with a re-stricted number of “elements,” or directly, by penalizing the non-smoothness of the solution. In either case, the additional degree offreedom (the regularization parameter) has to be established,where this is facilitated by the discrepancy principle (effectivelytuning the regularization parameter to a maximum value while notcausing intolerable deviation between the model and the observa-tions). In some fashion, each deconvolution algorithm developedto date combines these two concepts (non-negativity/monotonicityof the solution or regularization).

In addition to these main features, individual deconvolutionalgorithms may have other distinctive features. From a numericalstandpoint, recent deconvolution methods tend to use advancedtechniques to solve the underlying system of linear equations [e.g.,singular value decomposition, the equivalent concept of pseudo-inverse, etc.—for example, see Cheney and Kincaid (2003)].

One class of approaches makes extensive use of transforma-tions (i.e., the Laplace transform, the Fourier transform, etc.). Sev-eral cases consider numerical Laplace transformation of observedtabulated data combined with numerical inversion (Roumboutsosand Stewart 1988; Onur and Reynolds 1998). In addition, Chenget al. (2003) consider the application of the Fourier transformationto solve the deconvolution problem.

Error-minimization methods, typically driven by some type ofleast-squares algorithm, include von Schroeter’s total least squaresmethod [von Schroeter et al. (2004); Gringarten et al. (2003);Levitan (2003)], as well as a group of methods that rely on the useof spline functions (in various steps of the deconvolution algo-rithm) [e.g., Guillot and Horne (1986) and Mendes et al. (1989)].

Formally, it would be easy to place our proposed method intoexisting categories. Our proposed approach uses B-splines; nu-merical inversion of the Laplace transform is also used for certaincomponents of the solution, and regularization is provided indi-rectly (i.e., by the number of knots used in the selected B-spline)and directly (by penalizing the nonsmoothness of the logarithmicderivative of the reconstructed constant-rate response). However,in detail our approach is radically different from any of the existingmethods—in contrast to previous methods, our approach does notfit B-splines to observed data. Rather, B-splines are used to representthe unknown response solution (i.e., as a linear combination of B-splines). With respect to the Laplace transform, our approach doesnot use the Laplace transform to transform the observed pressuredata series, nor does our approach use numerical inversion toprovide the objective response (i.e., the constant-rate response), asdo other spectral methods. In our approach, the application of

Copyright © 2006 Society of Petroleum Engineers

This paper (SPE 95571) was first presented at the 2005 SPE Annual Technical Conferenceand Exhibition, Dallas, 9–12 October, and revised for publication. Original manuscript re-ceived for review 17 July 2005. Revised manuscript received 10 July 2006. Paper peerapproved 18 July 2006.

582 October 2006 SPE Reservoir Evaluation & Engineering

numerical inversion of the Laplace transform is restricted to theconstruction of the sensitivity matrix for the least-squares problem.

Our new technique represents the derivative of the unknownconstant-rate drawdown pressure response as a weighted sum ofB-splines, using logarithmically distributed knots. For the discreteflow-rate function, we use piecewise constant, piecewise linear, orany other appropriate representation for which the Laplace trans-form can be obtained easily. Taking the Laplace transform of theB-splines, we then apply the convolution theorem in the Laplacedomain, and we calculate the sensitivities of the observed pressureresponse with respect to the B-spline weights by numerical inver-sion of the Laplace transform.

Finally, the sensitivity matrix is used in conjunction with aleast-squares criterion to yield the sought B-spline weights and theunit-rate response (the spline formulation), which then yields thewell-testing derivative functions in the real domain. The combinedapproach of B-splines, Laplace-domain convolution, and the least-squares procedure are innovative and robust and should find nu-merous applications in the petroleum industry (production- andwell-test-data analysis, inversion of water influx behavior, etc.).

This new deconvolution approach is made possible by our abil-ity to solve a certain class of convolution problems with any de-sired accuracy, in particular, by the availability of a robust nu-merical inversion procedure for the Laplace transform [i.e., theGaver-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 deconvolution method based on

B-spline representations of the derivative of unknown constant-ratedrawdown pressure response (i.e., the undistorted pressure response).

• Create a practical and robust deconvolution tool that cantolerate relatively large (random) errors in the input rate and pres-sure functions. The proposed process also should be capable oftolerating small systematic errors in the input functions (by meansof calibrated regularization).

• Apply this new method to traditional variable-rate/pressureproblems, such as wellbore-storage distortion, long-term produc-tion data, permanent downhole (pressure) gauge data, and welltests having multiple flow sequences.

Statement of the ProblemDuhamel’s principle states that the observed pressure drop is theconvolution of the input-rate function and the derivative of theconstant-rate pressure response—at t�0, the system is assumed tobe in equilibrium [i.e., p(r,t�0)�pi]. For reference, the convolu-tion integral is defined as:

�0

t

q�t − ��p�u��d� = �p�t�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

The goal of variable-rate deconvolution is to estimate the unit (i.e.,constant) -rate reservoir response with observed measurements ofpressure drop and flow rate. This objective leads us to the inverseproblem, in which the time-dependent “input signal” (constant-ratepressure response) must be extracted from the output signal (pres-sure-drop response distorted by the specified variable-rate profile).To reconstruct the unknown function (i.e., the derivative of theunit-rate reservoir pressure response [K(t)�p�u(t)], we require thesolution of the Volterra integral equation of the first kind (i.e., theconvolution integral), where the role of the convolution kernel isrepresented by the sandface rate.

Lamm (2000) shows that for q(t–�)�=t�0—that is, when therate does not jump instantaneously at the first moment—the leftside of the equation is “smoothing at least of order two,” and,hence, the problem is severely ill-conditioned. In addition, thesandface flow rate is also observed in discrete points and is alwayscorrupted in practice by some form of error; therefore, the resultswill further vary depending on the way in which we make the tran-sition from discrete observations into the convolution kernel: q(t–�).

As noted by Lamm, the convolution operation has a naturalsmoothing effect; therefore, significantly different K(t) functions

can (and may) reconstruct essentially the same �p(t) response fora given rate history. Ideally, K(t)�0 and decreases monotonically[Coats et al. (1964) showed that derivatives of a higher order mustremain monotonic]. Unfortunately, even using downhole measure-ments, measured (observed) flow rates contain effects that maycause deviation from monotonic behavior (e.g., wellbore storage orother variable-rate conditions). If the response contains a skineffect and no wellbore storage, then the impulse response, K(t),will contain a Dirac delta component (van Everdingen and Hurst1949). Therefore, the conceptual representation of the K(t) func-tion is not trivial, a point often established by numerous failed at-tempts to represent this function. In particular, many algorithms sub-stitute the convolution integral in Eq. 1 by the trapezoidal rule (orsome similar approach), which is difficult to justify near the origin.

Development of a New B-Spline-BasedDeconvolution MethodSplines Over Logarithmically Distributed Knots. Spline func-tions (Cheney and Kincaid 2003) are piecewise polynomial func-tions that are defined on subintervals connected by points calledknots (ti) and have the additional property of continuity (of thefunction and its derivatives). In this work, we represent the un-known K(t) function as a second-order spline with logarithmicallyspaced knots. To reveal characteristic reservoir behavior, the num-ber of knots should be on the order of at least 2 to 6 knots per logcycle [for typical kernels of interest (i.e., input flow-rate functionsobserved in reservoir engineering). Therefore, we never use moreknots, but we may be forced to reduce the number of knots in casesin which the data quality does not justify the pursuit of sophisti-cated (reservoir pressure) signatures. A spline function can berepresented effectively with a linear combination of basic splinefunctions called B-splines (Cheney and Kincaid 2003). Once theknots (or continuity points) are set, generation of B-splines is easybecause of their intrinsic recurrence relation. Distributing the knotslogarithmically, we have ti�bi, b>1 i�0, ±1, ±2, . . . ; with asuitable selected b>1 basis, the B-spline of degree 0 is defined by

Bi0�t� = �1 ti � t � ti+1

0 otherwise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

Higher-degree B-splines are generated recursively using

Bik�t� = � t − ti

ti+k − ti�Bi

k−1�t� + � ti+k+1 − t

ti+k+1 − ti+1�Bi+1

k−1�t�, . . . . . . . . . . . (3)

where k�1,2, . . . . The ith B-spline has a nonzero value (support)between ti and ti+k+1, as is illustrated in Appendix B. Any kth-degree spline function (over the given system of knots) can berepresented as a linear combination of kth-order B-splines.

S�t� = �i=l

u

ciBik�t�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

where the number of B-splines involved is u–l+1. For second-ordersplines having support in an interval (ti1, ti2), we obtain l�i1–2,u�i2–1, and the number of B-splines involved is n�i2–i1+2. Bylinearity, the Laplace transform of Eq. 4 is

S�s� = �i=l

u

ciBik�s�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

We will represent K(t) as a weighted sum of B-splines of degree 2,defined over logarithmically evenly spaced knots:

K�t� = �i=l

u

ciBi2�t�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

Substituting Eq. 6 into Eq. 1, we have:

�p�t� = �0

t

�i=l

u

ciBi2�� q�t − �� d�. . . . . . . . . . . . . . . . . . . . . . . . (7)

583October 2006 SPE Reservoir Evaluation & Engineering

If we have m drawdown observations collected in a vector �p thatwere observed at times (t1, t2, . . . tm), then the problem can bewritten as an overdetermined system of linear equations:

Xc = �p, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

where X is the m×n sensitivity matrix (or “design” matrix.) Beforesolving Eq. 8 in the least-squares sense, each row of the overde-termined system is multiplied by a weight factor. Typically, we usethe reciprocal of the observed value; in other words, we assume theresponses are corrupted by a constant level of relative errors. Also,additional rows are added; we will discuss this operation in thesection on regularization.

Theoretically, the elements of the sensitivity matrix are defined as

Xj,i = �0

t j

Bi2��q�tj − �� d�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

In contrast to other methods, we do not assume that the rate functionis given in a stepwise manner. Rather, we allow for a general ratefunction with a Laplace transform of q(s). We calculate the elementsof the X matrix using numerical inverse Laplace transformation:

Xj,i = L−1�q�s�Bi2�s��tj�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

Here, Bi2(s), the Laplace transform of the ith B-spline, is known in

closed form (which is given in Appendix A). We exclude B-splineswith indices i for which bi>tj because those splines “start” laterthan the time of the given observation.

Rate Representation. To obtain the Laplace transform of the ratefunction, we have various options. The one we prefer is to dissectthe rate into nq segments with starting times t k

q, k�1, . . . , nq. Ineach segment,we describe the incremental contribution by an ex-ponential term and,hence,the total rate is obtained in the form

q�t� = �k=1

nq �akq + bk

q exp�−t − t k

q

ckq �� t − t k

q�, . . . . . . . . . . . (11)

where �( ) is the Heaviside (unit-step) function. The parameters foreach segment are found by linear least-squares fit combined bydirect search on the nonlinear parameter ck

q. A reasonable range forthis time constant is between t1 and tm. The fewer segments weuse, the more we smooth the observations. A shut-in period isalways represented as an individual segment. The selection of flowsegments is completely independent from the location of the pres-sure observations.

With the special form of the rate given by Eq. 11, the elementsof the sensitivity matrix are calculated as

Xj,i = �k=1

nq

L−1��qk�s�Bi2�s��tj − t k

q�, . . . . . . . . . . . . . . . . . . . . . . (12)

where

�qk�s� =ak

q

s+

bkqck

q

1 + ckq s

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

For strongly varying flow rates, we can use a piecewise constantrepresentation of the flow rates with many segments, as given inEq. 14.

q�t� = �k=1

nq

�qk − qk−1��t − t kq�. . . . . . . . . . . . . . . . . . . . . . . . . . (14)

In such a case, however, there is no need for inverse Laplacetransformation because Eq. 9 can be calculated by use of closedformulas in which the analytical integral of the second-order B-spline is available (see Appendix A).

Numerical Laplace Inversion. The success or failure of our pro-posed approach depends primarily on the numerical Laplace-transform inversion. For the previous three decades, fixed preci-sion computing has defined the status of numerical inversion [see

Davies and Martin (1979) and Narayanan and Beskos (1982)].Recently, several new algorithms were introduced using multi-precision methods (Abate and Valkó 2004).

In this work, we use the publicly available GWR algorithm.With this algorithm, it is possible to invert a large class of Laplacetransforms with essentially any desired accuracy. For more detailson the particular algorithm and Laplace transform inversion ingeneral, see Appendix B.

Decoupling the Unit Response and Its Derivative. As we alreadymentioned, in general, the earliest part of the unknown K(t) func-tion is critical, especially if the observed variable-rate responsecontains a skin effect [but minimal (or no) wellbore-storage ef-fects]. In such cases, the K(t) function contains a Dirac delta com-ponent. To reconcile this condition, we add additional “anchor”B-splines to the left of the first observed time point. After consid-erable experimentation, we established that four additional B-splines are sufficient for virtually any scenario.

These anchor B-splines do not affect the reconstructed pud(t)function (that is, the logarithmic derivative at the observationpoints) directly, but they do affect the reconstructed pu(t) (con-stant-rate) response at all observation points [i.e., the integral ofthe pud(t) function]. In fact, these additional B-splines are a con-venient vehicle to represent the Dirac delta component of the func-tion K(t).

Regularization. We found that using the concept of a pseudoin-verse (a cutoff for small singular values) does not provide a sufficientregularization as the noise level in the data increases. Therefore,we require an additional regularization that makes sense in a physi-cal context and ensures the relevance of the spline representation.

Appended to the overdetermined system (Eq. 8) to be solved byleast squares are the following two conditions for each spline interval:

��t �i=l

u

ciBi�t��t=bk

−�t �i=l

u

ciBi�t��t=bk+

1

2

� = 0

and ��t �i=l

u

ciBi�t��t=bk+

1

2

−�t �i=l

u

ciBi�t��t=bk+1

� = 0 . . . . . . (15)

In other words, we require that the value of the logarithmic de-rivative of the constant-rate response differ only slightly betweenthe knot and the middle location between knots. Because the entiresystem is overdetermined, these equations will not be satisfiedexactly, and their influence on the solution will depend on howlarge � is selected. When ��0, there is no regularization, and withgenerated data with practically no error, ��0 results in the re-quired smooth solution for the constant-rate response and its loga-rithmic derivative. In the presence of random noise and/or otherinconsistencies, a positive � is selected on the basis of an informalinterpretation of the discrepancy principle—that is, we increase thevalue of the regularization parameter until the calculated (model)pressure difference begins to deviate from the observed pressuredifference in a specific manner. The mean and standard deviationof the arithmetic difference of the computed and input pressurefunctions are also computed, but algorithmic rules (e.g., L-curvemethod) for selecting � are not recommended, for reasons dis-cussed by von Schroeter et al. (2004) and Gringarten et al. (2003).

Solution Using Pseudoinverse. Once the augmented sensitivitymatrix Xa has been constructed, the estimate of the unknown pa-rameter vector (c) is obtained from

c = Xa+ �p, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)

where the superscript plus sign denotes pseudoinverse calculatedthrough singular value decomposition. (For simplicity, here weshow the case in which all the weights are unity.) To create thepseudoinverse, small singular values are removed from the calcu-lations; in our computations, the automatic cutoff selection ofMathematica (2005) was used. Using the c coefficient estimates


obtained from this process, we reconstruct the constant-rate re-sponse as follows:

pu�t� = �i=l

u

c Bi,int2 �t�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

The logarithmic derivative of the constant-rate response is given as

pud�t� = t �i=l

u

c Bi2�t�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

where the integral of the second-order B-spline is easily obtainedin symbolic form (see Appendix A). Hence, the unit response andits logarithmic derivative can be calculated anywhere between thestart and the end of the test sequence.

ValidationIn this section, we provide validation of the new method by com-paring the results from the deconvolution procedure with the exactresponses obtained from a known reservoir model. As a base case(specifically, the case of a reservoir model with strong character-istic behavior), we selected a dual-porosity reservoir with circularboundary (��1×10−5, ��5×10−2, and reD�1,600; see Table 1for details). To generate the synthetic pressure histories required

for the validation phase, we convolve the known unit-rate pressuresolution with the flow history, which includes two variable-rate-production and two pressure-buildup (PBU) periods.

In our validation, we use the following variable-rate profile:

q�t� = 405.23 − 736.79 e−t�5 + 368.39 et�20 − 3.684 t . . .

0 � t 60 hours

0 . . . . . . . . . . . . . . . . . . . . . . . . . . 60 � t � 100 hours

300 e�100−t��100 . . . . . . . . . . . . . . . . 100 t 200 hours

0 . . . . . . . . . . . . . . . . . . . . . . . . . . 200 � t � 300 hours. . . . . . . . . . . . . . . . . . . . . . . . . . (19)

Because we ultimately intend to process permanent gauge data,this selected rate profile was used to mimic typical pressure-drawdown/-buildup test sequences. In the validation process, weincrease the complexity of the reservoir model step by step toassess what influence the underlying reservoir model may have onthe performance of our deconvolution procedure. We first considera model without wellbore storage and skin effects (Fig. 1); we thenincorporate a skin factor of 5 (Fig. 2), and, finally, we use adimensionless wellbore-storage coefficient of 100 and a skin factorof 5 (Fig. 3).

In the first example, we perform deconvolution with the inputdata shown in Fig. 1. Fig. 4 presents our deconvolution results forthis case, and we note that when the deconvolution results arecompared with the exact input response, one hardly sees any dif-ference. All the characteristic features of a dual-porosity reservoirmodel are present in the results, including the boundary-dominatedflow regime.

Fig. 1—Input data for Validation Case 1 [no wellbore-storageeffects, skin factor (sx)=0].

Fig. 2—Input data for Validation Case 2 [no wellbore-storageeffects, skin factor (sx)=+5].

Fig. 3—Input data for Validation Case 3 [dimensionless well-bore-storage coefficient (CD)=100, and skin factor (sx)=+5].


In Fig. 5, we present the deconvolution results for the case inwhich a skin factor of 5 is included in the input pressure response;we note an additional increase in the unit-rate drawdown pressureresponse (from the positive skin factor), and we see that the well-testing derivative function remains essentially identical to thetrends observed in Fig. 4 (as would be expected).

In Fig. 6, we present the deconvolution results when wellbore-storage and skin effects are incorporated into the base model.

We note that the deconvolution shown in Fig. 6 is performedusing “surface” rates, which do not reflect wellbore-storage con-ditions (although wellbore storage effects are, obviously, includedin the observed pressure data).

Fig. 6 is of particular practical interest because we clearly mustrecognize that although deconvolution to remove wellbore-storageeffects is a worthwhile goal, in current practice, it is very unlikelythat we would be able to obtain the downhole flow-rate data withsufficient accuracy such that we could remove wellbore-storageeffects. However, given a general variable-rate/variable-pressuredata sequence, we can use deconvolution to reconstruct the con-

stant-rate drawdown response, which would include wellbore-storage effects (as shown quite effectively in Fig. 6).

In the validation cases considered to this point, only ideal datahave been used as input (i.e., noiseless/consistent data), and regular-ization was not necessary. As noted, the deconvolved results areessentially the same as the (input) ideal model responses. Regular-ization is not required for synthetic cases, but for most (if not all) fieldcases (which usually have inconsistencies and significant measure-ment errors), we do require some sort of regularization procedure.

We provide a synthetic case in which we have added randomand systematic data errors (Fig. 7); we note that we have used thesame production history and reservoir model with wellbore-storage and skin effects [base performance (no error) is shown inFigs. 3 and 6)]. In this particular example, we have added 10%random error to both the pressure and rate data, and we also seededthe data with a small systematic error in pressure (see Fig. 7 for asummary of the input data).

We can still perform deconvolution on these data, but in thiscase, we must use regularization to address the combined effects ofrandom and systematic errors in the data.

The value of the regularization parameter has a profound effecton the solution for this case. Figs. 8 through 10 illustrate thediscrepancy principle by which the regularization parameter, �, ischosen. When regularization is not used (Fig. 8), there is a randomerror between the observed response (�p) and its reconstructed

Fig. 5—B-spline deconvolution results for Validation Case 2 (alldata used); no wellbore-storage effects, skin factor (sx)=+5.

Fig. 6—B-spline deconvolution results for Validation Case 3 (alldata used); dimensionless wellbore-storage coefficient(CD)=100, and skin factor (sx)=+5.

Fig. 7—Validation Case 4: input data with systematic error andnoise; dimensionless wellbore-storage coefficient (CD)=100,and skin factor (sx)=+5.

Fig. 4—B-spline deconvolution results for Validation Case 1 (alldata used); no wellbore-storage or skin effects.


value (Xc). If a higher value of regularization parameter is used,oversmoothing is obvious from the one-sided deviations (Fig. 10).The optimum value of the regularization parameter (in this case,��0.0007) is obtained as the largest value still not causing sys-tematic discrepancy between observed and reconstructed re-sponses (Fig. 9).

Fig. 11 presents the results of deconvolution when errors andinconsistencies are present in the data. Clearly, some artifactscaused by inconsistent data are seen (particularly in the well-testing derivative), but deconvolution reconstructs the characteris-tic features of the reservoir model for the duration of the entire testsequence. (From another point of view, deconvolution without regu-larization can effectively reveal inconsistencies present in the data.)

From our validation experiments, we can conclude that veryaccurate (essentially exact) results can be obtained using any majorevent in the test sequence (i.e., a production test, a shut-in, or thecomplete history). We also note that when the input data haveerrors or inconsistencies, we need some form of regularization toprovide meaningful results. Later, we show that our deconvolutionapproach can address relatively high levels of noise and otherdata inconsistencies.

In the deconvolution process, the initial pressure is required tocompute the observed pressure drop response (�p). Therefore, theinitial reservoir pressure is an input parameter for the B-splinedeconvolution algorithm, and an accurate estimate of the initialpressure is a critical component of deconvolution. Using an erro-neous estimate of initial reservoir pressure will cause a systematicerror in the deconvolved, equivalent constant-rate pressure re-sponse. In addition, there is disagreement between the base pres-

sure data and the reconstructed pressures. For example, using aninitial pressure larger than the actual initial pressure will result inan apparent pseudosteady-state flow regime at the end of the de-convolved test sequence or production data set. Thus, we note thatan apparent pseudosteady-state flow regime is a possible artifact ofusing an incorrect estimate of the initial reservoir pressure.

It is possible to recover the initial reservoir pressure usingdeconvolution [see von Schroeter et al. (2004) and Levitan et al.(2004)]. We also are able to recover the initial reservoir pressureusing an iterative process in which we start with an initial estimateto minimize the error between base pressure data and reconstructedpressures. This is not an automated process; user intervention is re-quired to realize the discrepancy between reconstructed pressuresand the base pressure data. Nevertheless, we conclude that thisprocedure is an effective way to estimate initial reservoir pressure.

Validation 2: Test of the New DeconvolutionMethod Using Only the PBU Data Portion of theTest SequenceIn this section, we will test the methodology using individual partsof the well-test sequence. For our purposes, we have generated twosynthetic cases in which we select a simple homogeneous reservoirmodel with wellbore-storage effects for simplicity. The propertiesfor the base reservoir model are provided in Table 2.

Fig. 9—Validation Case 4: noisy pressure observations and theB-spline pwf model response [optimal regularization (�=0.0007)].

Fig. 10—Validation Case 4: noisy pressure observations andthe B-spline pwf model response [“over-regularization” case(�=0.005)].

Fig. 11—Validation Case 4: deconvolved pressure response[optimal regularization (�=0.0007)].

Fig. 8—Validation Case 4: noisy pressure observations and theB-spline pwf model response (no regularization).


In the first case, a constant-rate production of 50 STB/D is heldfor 5 hours and followed by 500 hours shut-in. We specifically setthe duration of the buildup to be 100 times larger than the durationof the drawdown to establish the difference between deconvolutionand conventional PBU analysis when the durations of productionsequences are not consistent.

We recognize that this is an extreme test of the algorithm (i.e.,to use only the data from the PBU portion, plus the productionhistory, to yield the equivalent constant-rate drawdown responseby deconvolution). However, in practice, the primary utility of ouralgorithm is likely to be the resolution of PBU data. While obvi-ous, it is worth restating that an accurate production history (i.e.,accurate flow rates) is absolutely critical for deconvolution.

Fig. 12a presents the input data for deconvolution and PBUanalysis. In Fig. 12b, we immediately note artifacts in the very-late-time pressure derivative of the PBU case (conventional PBUplot), where these artifacts are caused by the fact that the reservoirpressure has completely built up to the limiting value of the aver-age reservoir pressure.

However, our deconvolution method is not affected by thisbehavior. Using only the PBU data for this case, we performdeconvolution, and we then construct the constant-rate pressure-drawdown response for the entire history. We present the results ofthe deconvolution as well as the conventional PBU data functionsin Fig. 12b, and we compared these data functions to the correctmodel responses for each case.

As shown in Fig. 13a, for our second case, we add anotherproduction and shut-in period to the well-test sequence. The du-ration ratio between both drawdown and buildup sequences ispreserved (shut-in duration is 100 times larger than the duration ofthe drawdown).

In this case, we use only the data from the final PBU portion forthe deconvolution process. As noted previously, we can constructthe constant-rate pressure-drawdown response for the duration ofthe buildup using deconvolution.

In our work, we set the weights of the input pressure data tozero, except for the data from the buildup portion [or any indi-vidual portion of a test in which data are consistent; PBU data(zero rate case) are the most “consistent” data because they areunaffected by flow rate during the test]. As such, we use the datafrom the second (final) PBU portion and set the weights of priordata to zero. The results for this case are presented in Fig. 13b.

It is worth noting that using only the PBU data from the sec-ond (final) PBU test, we can recover the equivalent constant-ratepressure-drawdown response for the entire well-test sequence us-ing deconvolution.

Applications of the NewDeconvolution TechniqueHaving verified our new deconvolution technique, we can nowapply the same procedure to field data. For convenience, we clas-sify our field examples into four groups:

• Wellbore-storage distorted pressure data.• Well-test data, including multiple flow sequences.• Permanent downhole gauge data (several flow sequences).• Long-term production data (rates and pressures).

These examples illustrate the use of B-spline deconvolution foranalyzing various events (production/shut-in sequences) andshould be considered typical of cases that could be encountered infield operations.

Field Case 1: B-Spline Deconvolution of Pressure-TransientData Distorted by Wellbore Storage. The most straightforwardapplication of deconvolution is to deconvolve PBU data to elimi-nate the effects of wellbore storage. Fetkovich and Vienot (1984)published data on an oil well that was hydraulically fractured atinitial completion. The early-time pressure response is distorted bywellbore-storage effects, but sandface rates are available. Usingthis input, we perform deconvolution, the results of which areshown in Fig. 14.

We compare the deconvolved response along with reservoir-model responses (simulation) for consistency. Fig. 14 presents thedistorted pressure data, the deconvolved constant-rate pressure

Fig.12—(a) Case 1: input data for the first case, where the du-ration of the buildup is 100 times larger than the duration ofdrawdown. (b) Case 1: B-spline deconvolution results for thefirst case with conventional PBU analysis and exact results(only PBU data are used for deconvolution).


drawdown and its logarithmic derivative, and the model matchesfor the deconvolved response. We used two models to match thedeconvolved pressure response: the uniform flux fracture (Fetkov-ich and Vienot used this model in their analysis) and the finite-conductivity vertical fracture model. Our deconvolution resultscompare extraordinarily well with these model responses and serveas confirmation of previous interpretations.

In this case, we have used deconvolution to eliminate wellbore-storage effects, and we suggest that this case serves to verify ourcontention that our method is sufficiently accurate for use in elimi-nating wellbore-storage effects (when sandface rates are available).

Field Case 2: B-Spline Deconvolution of Conventional WellTests Including Multiple Flow Periods (Complex Gas Reser-voir). In Fig. 15a, we present the rate and pressure data obtainedduring the four-point test of a gas well, which was followed by anextended shut-in.

In this case, the PBU data seem to be of sufficient duration fora conventional PBU analysis; therefore, we simply could proceedand analyze the individual PBU portion of these data. However,our intention is not to replace PBU analysis by deconvolution but,rather, to implement our deconvolution algorithm as an option forwell-test analysis.

We choose to analyze all the available data, including the pro-duction periods. First, the plot of the base pressure data and the

pressures generated by the B-spline model indicate that pressuredata and reconstructed pressures by deconvolution are consistent.Therefore, we note that deconvolution using the entire test se-quence is successful in this case (Fig. 15b). We proceed and con-struct the unit-rate pressure and well-testing derivative functionsin Fig. 16, together with model matches of the deconvolved pres-sure drawdown.

Fig. 15—(a) Field Case 2: moderate-pressure gas well in a com-plex reservoir. (b) Field Case 2: pressure base data and theB-spline pwf model response.

Fig. 13—(a) Case 2: input data for the second case, where thedurations of the buildups are 100 times larger than the dura-tions of drawdowns. (b) Case 2: B-spline deconvolution resultsfor the second case with conventional PBU analysis and exactresults (only data from the final PBU are used for deconvolution).

Fig. 14—Field Case 1: wellbore-storage distorted pressure re-sponse, deconvolved response, and model match; data fromFetkovich and Vienot (1984).


Because the flowing fluid is gas, we must perform thepseudopressure transform to adhere to theoretical considerations(we note that pseudotime should also be considered, but to betheoretically rigorous, pseudotime requires the average reservoirpressure history, which is never available in practice). The decon-volved pressure response suggests that a complex, channel-typereservoir geometry is possible.

Therefore, we proceed and match the deconvolved responsewith two models to verify this possibility. The models consideredinclude a well in a homogeneous reservoir bounded by parallelfaults and a well in a homogeneous reservoir with closed rectan-gular boundaries. The results of our model matches validate thecomplexity of the reservoir. We do observe the familiar “half-slope” trend of the pressure and pressure-derivative functions onthe log-log plot, which indicates a channel-type reservoir geometry.

On the other hand, we also note that the signature of a closedreservoir is evident in the pseudopressure drop and pseudopres-sure-drop derivative functions at late times.

In this application, we have used deconvolution to extract in-formation from a time interval larger than just the PBU test se-quence. Analysis of the entire testing sequence using deconvolu-tion may reveal reservoir characteristics not “seen” by a singleevent in the sequence.

Field Case 3: B-Spline Deconvolution of Conventional WellTests Including Multiple Flow Periods (Low-Permeability OilReservoir). In our third example, we perform deconvolution usingthe data from a selected flow period during a well-test sequence, asshown for the selected oil well in Fig. 17a. This test sequenceincludes 170 hours of data, where we have a 50-hour constant-rateproduction (at 200 STB/D), followed by a shut-in for 16 hours;then, production resumes at a constant rate of 160 STB/D for 24hours and is then shut in for 80 hours.

Inconsistencies are observed; in particular, we suspect that inthe first production period, the given rate is not correct (we notefluctuations in the drawdown pressure profile). We have two op-tions in this case: first, we can deconvolve the entire sequence, butwe have to use a high value of regularization parameter to over-come the effects of inconsistencies that most likely will introducebias. We also can use the data from the final buildup portion toperform deconvolution but take the preceding flow history intoaccount. We choose the second option and perform deconvolutionusing only the PBU data.

In Fig. 17b, reconstructed pressures obtained from the B-splinemodel and base pressure data are shown together. It is obvious thatthe data are not consistent. We note that for construction of Fig. 16b,we set the weights of pressure data to zero except for the data fromthe last buildup; as expected, we have a very good match of the

reconstructed pressures for both sets of PBU data that are assumedconsistent. These results suggest that, most probably, reportedrates are wrong, and performing deconvolution using the entire setof test data will not produce meaningful results. Nevertheless, thedeconvolved pressure response (obtained using the final PBU data,which are considered consistent with the given rates) and a modelmatch are shown in Fig. 18. The deconvolved well-testing pres-sure-derivative function suggests that the well has a finite-

Fig. 16—Field Case 2: deconvolved response and model match(entire history used in this deconvolution).

Fig. 17 (a)—Field Case 3: input data for fractured oil well. (b)Field Case 3: pressure base data and the B-spline pwf modelresponse.

Fig. 18—Field Case 3: deconvolved response and model match(only final PBU data used in this deconvolution).


conductivity vertical fracture and that this portion of the reservoiris bounded by parallel faults. We confirm this supposed reservoirmodel by matching the deconvolved pressures with just such a model.

Field Case 4: B-Spline Deconvolution of Permanent DownholeGauge (PDG) Data. Our next example is taken from an oil wellequipped with a permanent downhole measurement system, whichis used to provide continuous (downhole) pressure and rate data. Inthis case, a tremendous amount of data is available for a very shorttime—10 hours (as shown in Fig. 19).

Nevertheless, PBU analysis or other traditional-type analysis isnot a realistic option because the buildups are very short and, inaddition, other phenomena (most likely temperature change and/orphase resegregation) cause nonintuitive behavior (e.g., between 2 and4 hours, the well is shut in, but the observed pressure is decreasing).

Fig. 20 presents the results of deconvolution for this case. Theapparent oscillations in the deconvolved pressure-derivative func-tion indicate that the data are somewhat inconsistent, but we can pro-ceed and match the deconvolved pressure with simple models [a wellin an infinite-acting homogeneous reservoir with wellbore storageand skin, and a well in a homogeneous reservoir with a singlesealing fault (also including wellbore-storage and skin effects)].

Using analytic simulation, we reproduce the pressure responserelative to the observed rate history for the entire test sequence, asshown in Fig. 21. The pressures generated by the model fail tohonor the observed data at some points (especially in the intervalsin which we have already established that inconsistencies exist inthe data); however, these simulations do honor the observed pres-sures at later times, in agreement with our expectations.

Field Case 5: B-Spline Deconvolution Applied to ProductionData (Tight Gas Reservoir). Our deconvolution method is notlimited to analyzing well tests; we can also apply this methodologyto analyze and interpret traditional (long-term) production data.Production data are regarded as “low-frequency” and “low-resolution” data. For these reasons, rigorous variable-rate/variable-pressure analysis of production data is rarely attempted. Our ob-jective is to convert the entire production sequence (includingpoor-quality data) into an equivalent constant flow-rate pressureresponse. We corroborate our results with material-balance decon-volution (Johnston and Lee 1991), which includes plotting �pp/qg

(rate-normalized pseudopressure) vs. material-balance pseudotimeand comparing our deconvolution results and the material-balancedeconvolution results with a reservoir model established from thedeconvolved data.

This example considers a tight gas reservoir (Pratikno et al.2003). The data are of good (to excellent) quality, and we note theeffect of shut-ins as well as daily production fluctuations (Fig. 22a).

Fig. 22b presents the reconstructed pressures by B-spline de-convolution compared with base pressure data. As seen, pressuresobtained using the B-spline model honor the measured data over aconsiderable portion of the production history. We can conclude that

production data are consistent and that our B-spline deconvolutionbased on the entire long-term production history was successful.

In Fig. 23, we present the deconvolved responses (our methodand material-balance time) compared with an appropriate reservoirmodel (a well with a finite-conductivity vertical fracture in abounded reservoir); we note good agreement for this case.

We note some anomalies in the well-testing pressure-derivativefunction at early times, where these anomalies were presumablycaused by inconsistencies in the pressure data (or simply a scarcityof data at early times). From the derivative trend, we find thatboundary-dominated flow is in transition to full development, andthere does exist a slight difference in our interpretation of the dataand model selected for this portion of the analysis.

Field Case 6: B-Spline Deconvolution Applied to ProductionData (Low-Productivity Gas Reservoir). The second field ex-ample is also a gas field case (Palacio et al. 1993). The dailyproduction data are quite erratic (Fig. 24a) because of liquid load-ing; in spite of these features, the B-spline deconvolution methodperforms reasonably well (see Fig. 24b).

In this case, the margin of error between data and reconstruc-tion is acceptable, and we expect that errors will result in the formof artifacts, especially in the deconvolved well-testing derivativefunction. In Fig. 25, we again compare our deconvolution results

Fig. 19—Field Case 4: permanent downhole gauge data for10 hours.

Fig. 20—Field Case 4: deconvolved response and model match(entire data sequence used).

Fig. 21—Pressure history match using the models shown inFig. 19.


with traditional production analysis based on material-balancetime deconvolution.

Artifacts caused by inconsistencies are seen at early times inthe well-testing pressure-derivative function, again (we believe)because of sparse/erratic data. The late-time well-testing pressure-derivative function trend confirms that the boundary-dominatedflow regime has been achieved. The solution for a homogeneousreservoir with a closed circular boundary matches all of the de-convolved pressure-response functions reasonably well, where thisobservation also confirms our analysis.

We believe that B-spline deconvolution can be applied success-fully to production data (regularly measured flow-rate and pres-sure data). Processing low-quality production data with deconvo-lution can provide additional insight into reservoir performance-based analysis.

It is worth noting that numerous other field cases of long-termflow-rate and (surface) pressure data have been analyzed and in-terpreted successfully using the proposed B-spline deconvolutiontechnique. In fact, we believe that B-spline deconvolution may beused more often for the analysis of production data, compared toits potential use for well-test data. This hypothesis arises fromthe fact that while often of poor quality, production data are gen-erally available.

ConclusionsOur proposed deconvolution technique is robust because of theconstruction of the coefficient (sensitivity) matrix of the underly-ing least-squares problem using B-splines. Because of the (semi-logarithmic) B-spline representation of the unknown impulse re-sponse and the high reliability of the numerical Laplace transforminversion technique used, the elements of the coefficient matrix arecomputed accurately, and the resulting deconvolution process isstable. On the other hand, modeling of the rates requires a signifi-

cant effort for accurate computation of the sensitivity matrix. Ratesare required to be in functional form, and if the rate history indi-cates discontinuities, it should be decomposed into smooth seg-ments. An increasing number of segments will increase the com-putational time. With the appropriate description of rates, the pro-posed technique is robust and reliable.

Fig. 22—(a) Field Case 5: production data history plot for eastTexas gas well. (b) Field Case 5: pressure base data and theB-spline pwf model response.

Fig. 24—(a) Field Case 6: production-data history plot for a mid-continent gas well. (b) Field Case 6: pressure base data and theB-spline pwf model response.

Fig. 23—Field Case 5: deconvolution response functions,model match, and material-balance time-normalized data.


For the deconvolution of data in practice, it is necessary toincorporate a physically sound regularization scheme into our pro-posed deconvolution method. Regularization might be seen as an“add-on” to the proposed algorithm, but in a practical sense (i.e.,for field applications), our proposed deconvolution scheme islikely to perform poorly without the use of regularization methods.

We have successfully demonstrated the used of B-spline de-convolution for the analysis of a wide range of field data, fromtraditional PBU data, to permanent downhole gauge data, to tra-ditional production data.

Nomenclatureak

q � rate function fitting parameterb � base of logarithmically evenly distributed knots

bkq � rate function fitting parameterB � formation volume factor, RB/STB

Bik(t) � kth degree B-spline starting at bi

Bi,intk (t) � integral of the kth degree B-spline

c � vector of unknown coefficientsck

q � rate function fitting parameterct � total system compressibility, psi−1

CD � dimensionless wellbore-storage coefficientFc � fracture conductivity, md-ftGp � cumulative gas production, Mscf

h � pay thickness, ftk � permeability, md

K(t) � impulse response (�pud / t)p(t) � pressure, psi

pu � constant (unit) -rate pressure response, psipud � well-testing (log) derivative of pu, psi

pi � initial reservoir pressure, psiq(t) � rate, STB/D or Mscf/D

re � reservoir outer-boundary radius, ftrw � wellbore radius, ft

reD � outer reservoir-boundary radius, dimensionlesss � Laplace transform variable

sx � skin factorS(t) � spline function

t � flowing time, hourst kq � segment starting time (in rate function)

tsh � shut-in time, hoursX � sensitivity matrix� � regularization parameter

�p � drawdown (with respect to pi)� � interporosity flow parameter � viscosity, cp� � dummy variable� � porosity, fraction� � storativity parameter

Superscripts

^ � estimated (from least squares)∼ � observed

— � Laplace transform+ � pseudoinverseT � transpose of a matrix

Special Functions

K0(t) � modified Bessel function of the second kind, zero order

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Appendix A—B-Splines Over LogarithmicallyDistributed Knots and Their Laplace TransformsThe 0th quadratic B-spline (defined over logarithmically spacedknots with basis b>1) is given by Eq. A-1, where �(x) denotes theHeaviside function. The Laplace transform of the above B-splinecan be written in a compact form (Eq. A-2). In Fig. A-1, we show

two B-splines (of order 2, basis b�2 and indices 0 and 1); we notethat the 0th index B-spline is nonzero only in the interval (1;8). Aninteresting property is that the Heaviside function can be repre-sented as the sum of all B-splines with indices from negative topositive infinity (this is the “partition of unity” property). In ad-dition, an analytical integral of a second-order B-spline is availableand given by Eq. A-3.

B02�x� =

�x − 1�� x − 1��x − 1� − ��x − b��

b − 1 �b2 − 1

+�x − 1�� b2 − x��x − b� − ��x − b2��

b2 − b�

b2 − 1

+�b3 − x�� x − b��x − b� − ��x − b2��

b2 − b�

b3 − b

+�b3 − x�� b3 − x��x − b2� − ��x − b3��

b3 − b2 �b3 − b

, . . . . . (A-1)

B02�s� =

2e�−1−b−b2−b3�s�b3eb�1+b+b2�s + �1 + b + b2�eb�1+b+b2�s�

�b − 1�2b2�b + 1�s2

−2e�−1−b−b2−b3�s�e�1+b+b2�s + b�1 + b + b2�e�1+b+b2�s�

�b − 1�2b2�b + 1�s2.

. . . . . . . . . . . . . . . . . . . . . . . . (A-2)

Integrals of second-order B-spline functions are calculated by thegiven formula below (Cheney and Kincaid 2003):

Bi,int2 �t� = �

−�

t

Bi2�x�dx = bi�b3 − 1

3 ��j=i

�

Bj3�t�. . . . . . . . . . . . . (A-3)

Appendix B—Comparison of LaplaceInversion AlgorithmsThe GWR algorithm for numerical inversion of Laplace trans-forms is publicly available from http://library.wolfram.com/database/MathSource/4738/. Because the algorithm relies on mul-tiprecision computing, it can be realized only in software systemssupporting variable precision methods for arithmetic operationsand for special functions. The significance of built-in automaticcontrol of variable precision cannot be overemphasized [see Steh-fest (1970) and Valkó and Vajda (2002)]. Here, we show a com-parison of the GWR algorithm with the standard Stehfest (1970)algorithm realized in double precision. Such an approach has beenused to generate the overwhelming majority of results used inpetroleum engineering well-testing applications so far. For illus-trative purposes, we use the Theis function (i.e., the constant-ratepressure solution for a single, unfractured well in an infinite-actinghomogeneous reservoir).

It can be seen from Table B-1 that for this simple inversionproblem, the standard double-precision Stehfest algorithm givesreasonable accuracy (five digits) if the Stehfest parameter n isselected optimally. Unfortunately, the use of this procedure doesnot provide any systematic way to estimate the number of signifi-cant digits or the optimum value of n. This fact has been the originof considerable confusion in the past 30 years. The results givenabove also show that the GWR algorithm performs somewhatbetter even when the Stehfest algorithm is realized in an idealinfinite-precision computational environment. This is caused bythe nonlinear sequence acceleration used in GWR. The particularrealization of the GWR algorithm used here automatically selectsthe variable precision needed for the internal calculations; there-

Fig. A-1—B-splines over logarithmically equidistant knots,B0

2(t) and B12(t) (with base b=2).


fore, the user can increase the parameter M and stop when therequired significant digits are already stabilized. For instance, inthe example given above, if we require 10 significant digits, thena calculation with M�16 is satisfactory, and a check with M�20proves this fact, without ever knowing the true value of the function.

SI Metric Conversion Factorsbbl × 1.589 873 E−01 � m3

cp × 1.0 E−03 � Pa·sft × 3.048* E−01 � m

psi × 6.894 757 E+00 � kPa

*Conversion factor is exact.

Dilhan Ilk is a PhD student in the Dept. of Petroleum Engineer-ing at Texas A&M U. e-mail: [email protected]. His researchinterest includes well-test/production-data analysis, numerical

analysis, and signal processing. He holds a BS degree fromIstanbul Technical U., Turkey, and an MS degree from TexasA&M U., both in petroleum engineering. Peter P. Valkó is anassociate professor of petroleum engineering at Texas A&M U.e-mail: [email protected]. Previously, he taught at aca-demic institutions in Austria and Hungary and worked for theHungarian Oil Co. (MOL). His research interests include designand evaluation of hydraulic fracture stimulation treatments,rheology of fracturing fluids, and performance of advancedand stimulated wells. Valkó has coauthored three books andpublished several dozen research papers. He holds BS and MSdegrees from Veszprém U., Hungary, and a PhD degree fromthe Inst. of Catalysis, Novosibirsk, Russia. Tom Blasingame is aprofessor of petroleum engineering at Texas A&M U., where heworks in the areas of reservoir engineering and analysis of res-ervoir performance. e-mail: [email protected]. He holdsBS, MS, and PhD degrees from Texas A&M U., all in petroleumengineering. Blasingame is a Distinguished Member of SPE andreceived the 2005 SPE Distinguished Service Award. He haschaired SPE Forums, Applied Technology Workshops (ATWs),and committees.


deconvolution of variable-rate reservoir-performance data using … · 2018. 11. 18. ·...

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