an investigation of recent deconvolution methods for well ... · pdf filean investigation of...

An Investigation of Recent DeconvolutionMethods for Well-Test Data Analysis

M. Onur, SPE, and M. Çınar,* SPE, Istanbul Technical University; D. Ilk, SPE, P.P. Valko, SPE, and T.A. Blasingame,SPE, Texas A&M University; and P.S. Hegeman, Schlumberger

Summary

In this work, we present an investigation of recent deconvolutionmethods proposed by von Schroeter et al. (2002, 2004), Levitan(2005) and Levitan et al. (2006), and Ilk et al. (2006a, b). Theseworks offer new solution methods to the long-standing deconvo-lution problem and make deconvolution a viable tool for well-testand production-data analysis. However, there exists no study pre-senting an independent assessment of all these methods, revealingand discussing specific features associated with the use of eachmethod in a unified manner. The algorithms used in this study forevaluating the von Schroeter et al. and Levitan methods representour independent implementations of their methods based on thematerial presented in their papers, not the original algorithmsimplemented by von Schroeter et al. and Levitan. Three syntheticcases and one field case are considered for the investigation.

Our results identify the key issues regarding the successful andpractical application of each method. In addition, we show thatwith proper care and attention in applying these methods, decon-volution can be an important tool for the analysis and interpreta-tion of variable rate/pressure reservoir performance data.

Introduction

Applying deconvolution for well-test and production data analysisis important because it provides the equivalent constant rate/pressure response of the well/reservoir system affected by variablerates/pressures (von Schroeter et al. 2002, 2004; Levitan 2005;Levitan et al. 2006; Ilk et al. 2006a, b; Kuchuk et al. 2005). Withthe implementation of permanent pressure and flow-rate measure-ment systems, the importance of deconvolution has increased be-cause it is now possible to process the well test/production datasimultaneously and obtain the underlying well/reservoir model (inthe form of a constant rate pressure response). New methods ofanalyzing well-test data in the form of a constant-rate drawdownsystem response and production data in the form of a constant-pressure rate system response have emerged with development ofrobust pressure/rate (von Schroeter et al. 2002, 2004; Levitan2005; Levitan et al. 2006; Ilk et al. 2006a, b) and rate/pressure(Kuchuk et al. 2005) deconvolution algorithms. In this work, wefocus on the pressure/rate deconvolution for analyzing well-test data.

For over a half century, pressure/rate deconvolution techniqueshave been applied to well-test pressure and rate data as a means toobtain the constant-rate behavior of the system (Hutchinson andSikora 1959; Coats et al. 1964; Jargon and van Poollen 1965;Kuchuk et al. 1990; Thompson and Reynolds 1986; Baygun et al.1997). A thorough review and list of the previous deconvolutionalgorithms can be found in von Schroeter et al. (2004). The pri-mary objective of applying pressure/rate deconvolution is to con-vert the pressure data response from a variable-rate test or produc-tion sequence into an equivalent pressure profile that would havebeen obtained if the well were produced at a constant rate for theentire duration of the production history.

If such an objective could be achieved with some success, then,as stated by Levitan, the deconvolved response would remove theconstraints of conventional analysis techniques (Earlougher 1977;Bourdet 2002) that have been built around the idea of applying aspecial time transformation [e.g., the logarithmic multirate super-position time (Agarwal 1980)] to the test pressure data so that thepressure behavior observed during individual flow periods wouldbe similar in some way to the constant-rate system response. Asalso stated by Levitan, the superposition-time transform does notcompletely remove all effects of previous rate variations and oftencomplicates test analysis because of residual superposition effects.

Unfortunately, deconvolution is an ill-posed inverse problemand will usually not have a unique solution even in the absence ofnoise in the data. Even if the solution is unique, it is quite sensitiveto noise in the data, meaning that small changes in input (measuredpressure and rate data) can lead to large changes in the output(deconvolved) result. Therefore, this ill-posed nature of the decon-volution problem combined with errors that are inherent in pres-sure and rate data makes the application of deconvolution a chal-lenge, particularly so in terms of developing robust deconvolutionalgorithms which are error-tolerant. Although there exists a varietyof different deconvolution algorithms proposed in the past, onlythose developed by von Schroeter et al., Levitan, and Ilk et al.appear to offer the necessary robustness to make deconvolution aviable tool for well-test and production data analysis. In this paper,our objectives are to conduct an investigation of these three de-convolution methods and to establish the advantages and limita-tions of each method.

As stated in the abstract, the algorithms used in this study forevaluating the von Schroeter et al. and Levitan methods representour independent implementations based on the material presentedin their papers; therefore, our implementations may not be identi-cal to their versions. However, as is shown later, validation con-ducted on the simulated (test) data sets (von Schroeter et al. 2004;Levitan 2005) sent to us directly by von Schroeter and Levitanshows that our implementations reproduce almost exactly the sameresults generated by their original algorithms for these simulateddata sets.

The paper is organized as follows: First, we describe the pres-sure/rate deconvolution model and error model considered in thiswork. Then, we provide the mathematical background of the vonSchroeter et al., Levitan, and Ilk et al. methods together with theirspecific features. We compare the performance of each methodby considering three synthetic and one field well-test data sets.Finally, we provide a discussion of our results obtained fromthis investigation.

Pressure/Rate Deconvolution ModelThe pressure/rate deconvolution model considered in this study [aswell as in von Schroeter et al. (2002, 2004), Levitan (2005), Levi-tan et al. (2006), and Ilk et al. (2006a, b)] is given by the well-known convolution integral (van Everdingen and Hurst 1949):

pm�t� = p0 − �0

tqm�t��

dpu�t − t��

dtdt�, . . . . . . . . . . . . . . . . . . . . (1)

where qm(t) and pm(t) are the measured flow rate and pressure atany place in the wellbore including the wellhead or sandface, andp0 is the initial pressure. In Eq. 1, pu is referred to as the drawdownpressure response of the well/reservoir system if the well wereproduced at a constant unit-rate. Thus, pu is the rate-normalized

* Currently pursuing PhD degree at Stanford University.

Copyright © 2008 Society of Petroleum Engineers

This paper (SPE 102575) was accepted for presentation at the 2006 SPE Annual TechnicalConference and Exhibition, San Antonio, Texas, 24–27 September, and revised for publi-cation. Original manuscript received for review 5 July 2006. Revised manuscript received 8December 2007. Paper peer approved 12 December 2007.

226 June 2008 SPE Journal

pressure response in psi/(STB/D), and any reference rate other thanunity can also be used for normalizing pu. Throughout this paper,pu is normalized by a constant rate of 1 STB/D.

Eq. 1 assumes that at the beginning of production, the wellbore/reservoir system is in equilibrium and pressure is uniform through-out the system at p0. In addition, Eq. 1 is an expression of theprinciple of superposition (van Everdingen and Hurst 1949) that isvalid only for linear systems (e.g., single-phase flow of slightlycompressible fluid in porous media).

It can be shown that Eq. 1 is valid if the wellbore-storagecoefficient and the skin factor are constant during the entire historyof pressure and rate measurements. If these constraints are satisfiedby the test data, then pu(t) generated from Eq. 1 will represent theconstant unit-rate drawdown pressure response of the system in-cluding the effects of the skin and wellbore-storage, Cw, which iscaused by the wellbore volume below the point at which the rateqm is measured (Kuchuk 1990). By considering simulated test datathat have different wellbore-storage coefficients during differentflow periods, Levitan demonstrates that Eq. 1 fails to produce aphysically meaningful response function pu(t), particularly duringthe late-time portion of the response, when Eq. 1 is applied to thepressures for the entire test sequence. Levitan (2005) and Levitanet al. (2006) also discuss other cases (e.g., interference effects andcommingled reservoirs) and note that Eq. 1 is not the correctsuperposition equation for such cases.

When dealing with inconsistent data sets, which are usually thecase in real test data, Eq. 1 should be used with caution (vonSchroeter et al. 2004; Levitan 2005). Here, inconsistent data maybe simply defined as the data (acquired during the test sequence)that cannot be described by the deconvolution model of Eq. 1 (e.g.,the data acquired under the situations where wellbore-storage co-efficient and/or skin change, or multiphase flow effects occur nearthe wellbore during the test sequence). For inconsistent data sets,Levitan and Levitan et al. suggest that a deconvolution algorithmbased on Eq. 1 should be applied by using the pressure data froman individual flow period(s) and by keeping rates and initial pres-sure fixed in each run. Typically, the individual flow period(s) isa pressure buildup portion of the test sequence. Levitan shows thatwhen used in this mode, the algorithm produces reliable results forpu(t), and hence deconvolution of test pressure data performed oneflow period at a time provides for a consistency check. Based onour understanding of his approach, Levitan seems to assume thatinconsistency (or quality) of rate data could be determined bycomparison of the constant unit-rate drawdown derivative curvespertaining to each buildup period. His basic assumption is that ifthe curves lay on top of each other, particularly at late times, thenthe rates are consistent, unless the model changes during the test.Should the curves not overlay, then one can think that there is aproblem with the rates, which has to be resolved before proceedingwith deconvolution. However, it is not clear how such an approachwould work when both initial pressure and rate history could con-tain errors because Levitan et al. also show that initial pressure hasa profound effect on the late portion of deconvolved responses.They recommend a trial-and-error procedure, wherein the initialpressure is treated as a constraint parameter, to determine initialpressure from at least two buildup periods. An application of Levi-tan’s methodology to a synthetic test for which both initial and ratedata are uncertain can be found in the meeting version of this work[see Synthetic Example 3 (Çinar et al. 2006)].

On the other hand, when applying the deconvolution model ofEq. 1 to inconsistent data sets, von Schroeter et al. (2004) suggesta strategy based on applying deconvolution successively to in-creasing portions of the signal while monitoring rate and pressurematches. In this strategy, rate history and/or initial pressure aretreated as unknown in deconvolution. In our view, this strategyassumes that we have an a priori knowledge of expected errormargins in pressure and rate data, and most importantly, that sucha prior knowledge is correct. If so, then one can detect any incon-sistency in rates as well as inconsistency between the data set andthe deconvolution model by using a statistical evaluation of theresults. As will be shown with the example applications later, thisstrategy also proves useful for detecting inconsistencies in decon-

volution in cases for which the expected error margins for rate andpressure data are known or can be estimated.

As to be discussed later, the three deconvolution methods eachhave built-in flexibility to process the pressure for any subset offlow and buildup periods in a test sequence by accounting for theentire flow rate history from the start of the test to the end of theflow or buildup period to be deconvolved. Note that only thedeconvolution methods of von Schroeter et al. and Levitan have abuilt-in flexibility to allow the user to treat the rate and/or initialpressure as unknown in deconvolution; Ilk et al. does not have thisflexibility. Later, we will discuss the conditions under which es-timating rate or correcting measured rates and initial pressure bydeconvolution of measured pressure data alone could be feasible.

Error Model. In this work, we compare the performance of thethree deconvolution methods based on Eq. 1 using simulated pres-sure and rate data sets corrupted by noise. As in the von Schroeteret al. work, we consider independent, identically distributed nor-mal random errors with zero mean and specified standard devia-tions for pressure and rate data computed from, respectively,

�p =�p��p�2

�M, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

and

�q =�q�q�2

�N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

Here, �p represents the M-dimensional vector of true pressurechanges generated from a particular well/reservoir model, whileq represents the N-dimensional vector of true input flow rates.(Note: throughout, a boldface lower case letter denotes a columnvector, while a boldface capital letter denotes a matrix.)�x�2(�√x1

2+x22+ � � �) is the l2-norm. The error levels as a fraction

of the pressure change and rate are represented by �p and �q,respectively. For example, �p��q�0.05 corresponds to an errorlevel of 5%. Given true pressure change and rate data, along withspecified �p and �q, Eqs. 2 and 3 are used to compute � and �, theM-dimensional and N-dimensional vectors of normal errors gen-erated for pressure change and rate data, respectively. These sets ofrandom errors are added to the true pressure and rate data togenerate the corrupted (or “measured”) pressure pm and rate qm

which will be used for deconvolution. That is,

pm = p + �, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

and

qm = q + �. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

This error generation process will provide the same standard de-viation of errors in both the absolute pressure pm and the pressurechange �pm (�p0–pm). However, the specified error level (�p) forthe pressure change data may not correspond to the same errorlevel in the pressure data pm because magnitudes of the pressureand pressure change data may differ significantly.

One may wish to question the validity of using independent,identically distributed normal random errors with zero mean andspecified standard deviations to model errors in both the pressureand rate data. However, this is the most commonly used model tointerpret the errors in actual pressure and rate measurements. Also,the assumption of randomly distributed errors forms the basis ofthe all three methods considered in this study. However, we notethat the model may not be realistic in all cases and can lead tomisleading results in deconvolution—particularly for tests wherethe rates are allocated based on a few measurements. It is possible,but beyond the scope of this paper, to use more general errormodels; for example, systematic and/or normal errors with zeromean and variable variance and with correlations (i.e., non-diagonal covariance matrix for errors).

227June 2008 SPE Journal

Description of DeconvolutionMethods ConsideredHere, we present a description of each of the three pressure/ratedeconvolution methods.

Von Schroeter et al. Deconvolution Method. The von Schroeteret al. method is based on three novel ideas: First, Eq. 1 is solvednot for the constant-unit-rate response pu(t), but for the z (or re-sponse) function based on its derivate with respect to the naturallogarithm of time as defined by

z�� = ln�dpu�t�

d ln�t�� = ln�dpu��

d� �, . . . . . . . . . . . . . . . . . . . . . . . . (6)

where ��ln(t). The selection of z(�) as a new solution variableensures that the dpu(t)/d lnt is positive, a necessary condition thata constant-unit-rate response of the system should satisfy. This useof z(�) converts Eq. 1 to a nonlinear convolution equation given by

pm�t� = p0 − �−�

lntqm�t − e�� ez��d�. . . . . . . . . . . . . . . . . . . . . . . . (7)

Second, a regularization based on the curvature of z(�) is im-posed to achieve some degree of smoothness of the solution z(�)and to improve the conditioning of the deconvolution algorithm.

Third, their formulation accounts for errors in both the rate andpressure data by considering a special least-squares formulationknown as the Total Least-Squares (TLS) problem (von Schroeteret al. 2002, 2004; Golub et al. 1999; Björck 1996). Within the TLSconcept, the deconvolution problem can be posed as an uncon-strained nonlinear minimization for which the objective function isdefined as (von Schroeter et al. 2002, 2004)

E�p0, z, y� = �p0e − pm − C�z�y�22 + ��y − qm�2

2 + ��Dz − k�22

. . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

The arguments given in the parentheses of the objective function Erepresent the unknowns to be determined by minimizing E. Aswith von Schroeter et al., we minimize the objective function ofEq. 8 with the variable projection algorithm given by Björck(1996). This algorithm is based on the separable least-squaresproblem and requires minimization of two (one linear, other non-linear) least-squares objective functions. For the solution of theseleast-squares problems, we implemented the singular value decom-position algorithm called SVDCMP found in Press et al. (1986).When updating the solution vector in Eq. 8, we use the line searchalgorithm called LINMIN found in Press et al. (1986). We use thesame strategies as von Schroeter et al. for choosing grid nodes,starting values of p0, z, and y in the nonlinear minimization as wellas the same termination criterion.

In Eq. 8, y represents the N-dimensional column vector ofcomputed (or true, unobserved) rates, and C is the M×N dimen-sional matrix of response coefficients, D is the (L−1)×L constantmatrix in curvature measure. z is the L-dimensional vector of re-sponse coefficients, and k is the (L−1)-dimensional vector in cur-vature measure as given by kT�(1, 0, . . . , 0). The analyticalexpressions for the elements of the M×N matrix C are derived byapproximating the flow rate qm(t) with stepwise constant-rate func-tions [as prescribed in von Schroeter et al. (2002, 2004)]. Theelements of the (L−1)×L constant matrix D are given by Eqs. B-6aand B-6b in von Schroeter et al. (2004). The solution zi in Eq. 8 isapproximated as a piecewise-linear function on a grid of points �i,i�1, 2, . . . , L, usually with a uniform constant grid increment ��[see Eq. 63 of von Schroeter et al. (2004)]. As suggested by vonSchroeter et al., we use 40 grid nodes and choose the first node as10−3 hours and the last node as the total test duration, unlessotherwise stated. Furthermore, as consistent with the material pre-sented by von Schroeter et al., L−1 curvature constraint equationsare used in Eq. 8, and the wellbore-storage unit-slope line assump-tion at/before the first node is implemented.

The � and � appearing in the right side of Eq. 8 are the relativeerror weight and regularization parameter (von Schroeter et al.2002, 2004), respectively. Providing a comprehensive error analy-sis based on Gaussian statistics, von Schroeter et al. recommendthat the default values of these two parameters be computed from

�def =N

M

��pm�22

�qm�22

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

and

�def =��pm�2

2

M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

In cases where the initial pressure p0 is known, �pm is formed withthis p0. Otherwise, it is formed with the maximal pressure samplein place of p0, as suggested by von Schroeter et al. In our appli-cations of the von Schroeter et al. method, we find that the defaultvalue given by Eq. 9 for the relative error weight � to be used inEq. 8 works fine in most cases.

On the other hand, the regularization parameter � has a stronginfluence on the z function. We found that the default value com-puted from Eq. 10 divided by the square of the Frobenius norm ofthe curvature matrix D [see Eq. B-7 in von Schroeter et al. (2004)for the definition of the Frobenius norm] provides a good initialguess for determining an “optimal” level of regularization. This isequivalent to the same level of regularization as in von Schroeteret al., where instead the curvature measure �Dz−k�2 in Eq. 8 isnormalized to unit Frobenius norm. Throughout this paper, wereport �def as the � value computed from Eq. 10 divided by thesquare of the Frobenius norm of D, i.e.,

�def =1

M

��pm�22

�D�F2

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

whereas von Schroeter et al. report �def as the one computed fromEq. 10. Thus, the difference is only in reporting.

Similar to von Schroeter et al.’s suggestion, we start by usinga � value equal to the value of �def computed from Eq. 11. As vonSchroeter et al. (2004) state, determining the optimal value of � inthis way requires a trial-and-error procedure and is subjective.Similar to von Schroeter et al., we find that for a fixed number ofnodes, the optimal value of � depends on well/reservoir model andon the error levels in the pressure and rate data.

A disadvantage that may be noted about the von Schroeter et al.method is their assumption regarding the behavior of the z functionbefore the first node. They assume that the wellbore-storage unit-slope line is valid for the behavior of the z response before/at thefirst grid node. As is shown in “Synthetic Example 1” section, incases where this assumption is violated, the deconvolved responseat early times can be in error. As discussed in the next two sec-tions, the Levitan and Ilk et al. methods remove this restriction.

The deconvolution method of von Schroeter et al. gives the useran option to treat not only the z-response as unknown in decon-volution, but also treat the rate history and/or initial pressure asunknown. (The same option is also true for the Levitan method,but not for the Ilk et al. method.)

Unlike Levitan and Ilk et al., von Schroeter et al. (2002, 2004)provide a concrete error analysis based on Gaussian statistics andshow how to compute such statistics as a posteriori estimates ofstandard deviation of pressure and rate errors, and 95% confidenceintervals for the estimates. Inspecting such statistics with the avail-able knowledge of expected error margins in pressure and rate dataare indeed very useful in evaluating the validity of the resultsobtained from deconvolution. In our implementation of the vonSchroeter et al. method (and Levitan method), we also computesuch statistics and show their utility in some of the synthetic andfield data cases to be considered later.

We also note that in the work of von Schroeter et al. the focusis the reconstruction of the derivative of the constant-unit-ratepressure (i.e., z(t)) with respect to the logarithm of time (see Eq. 6).They do not comment how to reconstruct the constant-unit-ratepressure response pu(t) from the estimated z response.

Our results show that their method can be used to reconstructpu(t) from the estimated z response by use of the discretized ver-sion of Eq. 7 (with all qms replaced by unity; see Eqs. 12 and 13)in cases where wellbore-storage effects exist at early-time portionsof the data. In cases where there is minimal or no wellbore-storage


effects, our results indicate that we can still reconstruct pu(t), butin such cases, the z value estimated for the first grid node (i.e., z1

at t1) interestingly contains a “Dirac delta” component which rep-resents the skin effect. In other words, z has a discontinuity at thefirst node in the von Schroeter et al. method; estimated ez1 is equalto the correct value of the pu(t) function at t1, but not to the correctvalue of the dpu(t)/d lnt function at t1, and estimated ez at the othernodes at the very early times suffer from the “left-end” effects forapproximately a log-cycle due to the unit-slope line assumptionbefore/at the first grid node. This point will be further examined inthe “Synthetic Example 1” section.

An equation to construct the pu(t) response for the von Schroe-ter et al. method can be obtained by setting qm(t–e�)�1 STB/Dand replacing p0–pm(t) by pu(t) in Eq. 7 to represent the unit-ratepressure change. The result is

pu�t� = �−�

lnt

ez��d�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

Assuming piecewise linear representation of the z function, and awellbore-storage unit-slope trend at/before the first node, we canderive the following discrete form of Eq. 12:

pu�ti� = ez1 + �j=2

i�ezj − ezj−1�

�zj − zj−1�ln� tj

tj−1�; for i = 1, 2, . . . , L,

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

where ti represent the times at which the z function is computed,and L represents the number of nodes.

Thus, after the z function is estimated by minimizing the ob-jective function given by Eq. 8, Eq. 13 can be used to construct thepu(t) function. If the assumption of a wellbore-storage unit-slopetrend at/before the first node is satisfied, then it can be shown that

ez1 =dpu�t1�

d lnt= pu�t1�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

However, as mentioned previously, the estimated ez1 from the vonSchroeter et al. encoding for the data sets with minimal or nowellbore effect represents pu(t1), the unit-rate pressure drop at thefirst node including the skin effect, if it exists, but not the value ofdpu(t1)/d lnt.

Levitan Deconvolution Method. Levitan’s method is based onthe same concepts as proposed by von Schroeter et al. However,Levitan’s method differs in two respects.

First, Levitan uses a deconvolution equation which removes therestriction of the von Schroeter et al. assumption that the wellbore-storage unit-slope line is valid before the first node. Levitan’sencoding for pressure/rate deconvolution equation assumes that thetime corresponding to the first grid node �1 (or equivalently t1) issufficiently small so that Eq. 7 can be accurately approximated by

pm�t� = p0 − qm�t�pu�t1� − ��1

lnt

qm�t − e�� ez��d�. . . . . . . . . . . . . (15)

Second, Levitan uses an unconstrained, nonlinear weightedleast-squares objective function involving the sum of three mis-match terms for pressure, rate, and curvature:

E�p0, pu�t1�, z, y� =1

2 �p0e − pm − pu�t1�y − C�z�y

p�

2

2

+1

2 �y − qm

q�

2

2

+1

2 �Dz

c�

2

2

. . . . . . . . . . . . . . . . (16)

where C is the M×N dimensional matrix of response coefficientsfor which analytical expressions can be derived by approximatingthe flow rate qm(t) using stepwise constant-rate functions and aregiven in von Schroeter et al. (2002, 2004). Note that the matrix Cin the Levitan method is not identical to the matrix C in the von

Schroeter et al. method because the integration limits in Eq. 7 andEq. 15 are different. y in Eq. 16 is an M-dimensional column vectorof flow rates at the times where the pressure measurements, pm, aremade. The elements of y are also included in the elements of theunknown flow-rate vector of y. p, q, and c represent the errorbounds (or scale parameters) for the pressure, rate and curvatureconstraints (Levitan 2005).

Like Levitan, we use the algorithm for unconstrained minimi-zation given by Dennis and Schnabel (1983) to minimize the ob-jective function of Eq. 16. The Levitan method considers pu(t1) asan unknown, in addition to the unknowns p0, y, and z to be esti-mated by minimization of Eq. 16. We use the same strategiessuggested by Levitan for choosing starting values of pu(t1), p0, z,and y in nonlinear minimization of Eq. 16, and use 70 uniformlyspaced nodes, as he suggested.

However, there are several points that are not stated or revealedby Levitan, regarding specific implementation of his method basedon Eqs. 15 and 16. We believe that our observations given herewill be useful to those readers interested in implementing the Levi-tan method.

The first point is related to the selection of the time for the firstnode, t1. In his paper, Levitan does not describe how to choose thevalue of t1 to be used for constructing the z-response. He onlystates that if t1 is sufficiently small, the model pressure can berepresented as Eq. 15. In our applications, we usually choose t1 tobe less or equal to the minimum elapsed time (relative to the startof its flow period) of any point in the pressure signal. In the“Synthetic Example 1” section, we also examine the consequencesof choosing small and large values of t1.

Secondly, it is not clear from Levitan’s paper how the last term(or curvature constraint part) in the right side of the objectivefunction given by Eq. 16 is treated, particularly for the first node.In his paper, Levitan states that he uses L−1 curvature constraints,where L is the number of nodes of the grid �i, and he states that heuses the model prediction of a curvature constraint as given by vonSchroeter et al. (2004) [see Eq. 9 of Levitan (2005)]. For thepurpose of discussion, here we recall Levitan’s Eq. 9:

ci�z� =1

��zi−1 − 2zi + zi+1�. . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

When i�1, we have z0 in this equation, and there is no such z0 inhis formulation. Therefore, if he is treating the curvature con-straints as the same as von Schroeter et al. (2004), as stated in hispaper, then when i�1, we would expect that he uses [see also Eq.B-6a of von Schroeter et al. (2004)]

c1�z� =1

��z2 − z1�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

In this paper, unless otherwise stated, we will present the resultsbased on the implementation of Eq. 18 for i�1, and Eq. 17 fori�2, 3, . . . , L−1 because we believe this is consistent with thematerial presented in Levitan’s paper. There are other alternativesthat one can use for the treatment of the curvature constraint termin Eq. 16. For example, one can use Eq. 17 with L−2 curvatureconstraints by starting with i�2, 3, . . . , L−1. As is shown later,we often find that the latter option provides a slightly better rep-resentation of the reconstructed z-response at early times than theformer option.

In addition, Levitan does not specifically show how he con-structs the pu(t) function. He only states that pu(t) can be recon-structed from the estimated z by integrating Eq. 6 with the value ofpu(t1), estimated by minimizing the objective function given byEq. 16. There are several ways to perform the integration of Eq. 6with the values of pu(t1), z1, z2, . . . , zL estimated from nonlinearregression (e.g., composite trapezoidal rule of integration, and therectangular rule of integration). Our point is that Levitan does notspecifically show how he reconstructs pu(t) from Eq. 6.

In our implementation of his method, we reconstruct the pu(t)function by setting qm(t)�qm(t-e�)�1 STB/D in Eq. 15, and re-arranging the resulting equation to obtain:


pu�t� = pu�t1� + ��1

lnt

ez��d�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)

Assuming piecewise linear representation of the z function, we canderive the following discrete form of Eq. 19:

pu�ti� = pu�t1� + �j=2

i�ezj − ezj−1�

�zj − zj−1�ln� tj

tj−1�; for i = 1, 2, . . . , L.

. . . . . . . . . . . . . . . . . . . . . . . . . . (20)

Comparison of Eqs. 13 and 20 indicates that the ez1 used toconstruct the pu(t) function in the von Schroeter et al. method (Eq.13) can be thought as the pu(t1) used to construct pu(t) function inLevitan’s method (Eq. 20). However, recall that in Levitan’smethod, pu(t1) is an unknown model parameter to be determinedby minimizing the objective function given by Eq. 16, while ez1 iscomputed after z1 is determined by minimizing the objective func-tion given by Eq. 16 or Eq. 8. It is guaranteed that ez1 to be usedin Eq. 13 will always be positive. However, there is no guaranteefor pu(t1) to be positive in Levitan’s method because it is treated asan unknown model parameter to be estimated by unconstrainedminimization of the objective function given by Eq. 16. Thus, it ispossible that in some cases, pu(t1) could converge to a negativevalue, as happens for our Synthetic Example 2 and field examplecases to be discussed later.

In cases where pressures pertaining to only buildup portions arematched to reconstruct the constant unit-rate drawdown responses,it can be shown that in the Levitan encoding of Eq. 15, the sen-sitivity of the model pressure with respect to pu(t1) depends ononly the flow rate value at the instant of shut-in. To show this, wedifferentiate Eq. 15 with respect to pu(t1) to obtain:

dpm�t�

dpu�t1�= −qm�t�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)

Therefore, if we apply deconvolution to only buildup pressuresacquired at t 1

b, t 2b, � � � , tNb

b , where t 1b represent the time for the

instant of shut-in, then qm(t) in Eq. 21 will be all zero, except at thetime for the instant of shut-in. As a result, the sensitivity of pm topu(t1) is only nonzero at the instant of shut-in, i.e.,

dpm�t�

dpu�t1�= �−qm�t 1

b�, for t = t 1b

0, for t � t 1b

. . . . . . . . . . . . . . . . . . . . . . . (22)

This shows that the sensitivity of model pressure to pu(t1) dependson only a single flow-rate information at the instant of shut-in,when processing the pressures for a buildup period.

Our investigation indicates that an improper selection of t1 andof curvature constraint c, and/or magnitude of errors in pressure/rate data and time synchronization errors in rate and pressure dataare possible causes of driving an unconstraint minimization algo-rithm to a negative value for pu(t1). A possible way to avoid thisis to use a constraint when minimizing Eq. 16 in the form of aone-sided inequality constraint, such as pu(t1)>0. For example, theimaging procedure suggested by Carvalho et al. (1996) can be usedto achieve this.

Regarding the error bound (or scale) parameters p, q, and c

in Eq. 16, Levitan states that these are not free parameters that canbe manipulated at will to control least-squares minimization, andthat these parameters are based on prior knowledge of data quality.Based on his experience, he suggests the use of p�0.01 psi(which is a typical value for the resolution of quartz gauges) andc�0.05 in Eq. 16, but does not provide any default value to beused for the rate error bound q. Levitan states that choosing thevalues of the error bounds for rates is not straightforward andrecommends that larger values of q be assigned to the flow peri-ods where we expect large uncertainty in flow rate data.

We agree that p and q should be determined based on knowl-edge of data quality; however, we also note that it is usuallydifficult to have a correct prior knowledge of error levels in pres-sure and particularly in rate data. This difficulty is because error

level in data is not solely dependent on the sensor/tool resolutionand measurement-error characteristics; there could be other typesof errors that cause the spread (variance or standard deviation) ofpressure and rate data to be larger than that caused by sensor/toolresolution or error characteristics alone. Therefore, a single choicefor c (�0.05) together with the gauge resolution for p (�0.01psi) could not be expected to work for all data sets, and it may bedesirable to use different values of p, q, and c than those sug-gested by Levitan, depending on the specific data quality and toolcharacteristics.

Finally, we make a comment regarding the relationship of p,q, and c in the Levitan method and � and � in the von Schroeteret al. method. Ignoring the differences of encoding (see Eqs. 7 and15), and assuming that errors in pressures and rates are indepen-dently, identically distributed normal random variables with zeromean and variances equal to p

2 and q2, respectively, it can be

shown that the objective functions are equivalent if we define��(p/q)2 and ��(p/c)

2. Minimizing E given by Eq. 16 is thesame as minimizing any constant multiple of it, so the necessaryparameters are p, p/q, and p/c; hence, the number of param-eters is exactly the same in both methods. Our results indicate thatboth objective functions yield almost identical results if these ra-tios are kept the same in both objective functions.

We can make the following observations based on the expres-sions ��(p/q)2 and ��(p/c)

2. For example, in cases where wedo not have a prior knowledge of data quality, the default valuesof � and � computed from the equations presented by vonSchoreter et al. (see Eqs. 9 and 11) can be used in the expressions��(p/q)2 and ��(p/c)

2 to determine the default values for p

and q to be used in the Levitan objective function (Eq. 16) witha presumed value of c. For example, if we compute ��0.01(psi.d/stb)2, and ��0.04 psi2 from Eqs. 9 and 11 for a given dataset, then for a presumed value of c�0.05, the expressions willgive p�0.01 psi and q�0.1 stb/d. (Note that the Levitan’s sug-gested values of c�0.05 and p�0.01 psi give ��0.04 psi.2) Onthe other hand, in cases where we have a prior knowledge ofstandard deviations of errors in rate and pressure data (i.e., weknow the values of p and q a priori), then the expressions ��(p/q)2 and ��(p/c)

2 with a presumed value of c and with theknown values of p and q can be used to estimate the defaultvalues of � and � to be used in the von Schroeter et al. objectivefunction (Eq. 8).

Validation of Our Algorithms Based on the von Schroeter et al.and Levitan Methods. Here, we validate our implementations ofthese authors’ methods using the synthetic data provided to thefirst author of the paper by von Schroeter and Levitan, throughpersonal communication.

First, we compare the results generated from our implementa-tion with those provided by von Schroeter. The data set used forthis comparison is the “Simulated Example” of von Schroeter et al.(2004) for the case where pressure and rate data contain 5% and10% errors (i.e., �p=0.05 and �q=0.10), respectively. The simu-lated model is for a vertical well with wellbore-storage and skineffects, near a sealing fault, and all quantities in this example aredimensionless (von Schroeter et al. 2004). In generating the resultsshown in Table 1 for estimated rates and Fig. 1 for estimatedunit-rate responses, we used his parameters supplied, i.e., t1�2,��44.8676, and ��def�1.21351 (which is equal 424.868 com-puted from Eq. 10 divided by the Frobenius norm, which is350.063 for this example). The number of nodes is 21, and initialpressure is fixed at its true value of 60. We have an excellentagreement (correct up to three significant figures) between datagenerated from our algorithm and data provided by von Schroeter.The pu(t) curve shown in Fig. 1 represents our estimated valuesgenerated by Eq. 13 because von Schroeter did not provide us thepu(t) data. For this data set, the deconvolved derivative response(Fig. 1) deviates considerably from the true behavior [see Fig. 2of von Schroeter et al. (2004)] at early and late times. The reasonfor the odd behaviors seen at the deconvolved derivative responseat early and late times is mainly caused by use of the default val-ues of the parameters � and � in Eq. 8, which appear to be not


appropriate for reconstructing the true derivative response for thisdata set.

Next, we compare the results generated from our implementa-tion of Levitan method with those provided by Levitan. The dataset used for this comparison is “Example 1” of Levitan (2005). Thesimulated model is for a vertical well with wellbore-storage andskin effects, located in the middle of a channel with two parallelno-flow boundaries. Pressure and rate data are free of errors. Thedata sent to us only included unit-rate pressure, not the unit-ratelogarithmic derivative responses. He also did not provide us withthe values of p, q, c, and t1 (time value for the first node), but histabulated values of unit-rate pressures started from a time value of0.01 h. We used t1�0.01 h, p�0.01 psi, c�0.05, and 70 nodes.We kept rates and initial pressure fixed, and used all pressure datafor the entire test sequence to generate the unit-rate responses fromour algorithm. A comparison of the results is shown in Fig. 2. Theconstant unit-rate drawdown derivate response, dpu/d lnt, shown assolid circular data points in Fig. 2 for Levitan, was generated by

use of a numerical derivative routine based on the Bourdet method(1989) without smoothing. In Fig. 2, we have also reported thevalue of pu(t1) obtained from our implementation and Levitan’svalue. As can be seen, our results are in excellent agreement withLevitan for his simulated “Example 1.”

Based on all the results presented here, we conclude that ourindependent implementations of the von Schroeter et al. and Levi-tan methods reproduce almost exactly the same results provided tous by von Schroeter and Levitan. We do not claim that our imple-mentations of the methods are identical to those of the authors;however, on the basis of the material presented in their papers andthe data sets provided to us, we believe we have validated ourimplementations of the algorithms.

Ilk et al. Deconvolution Method. The Ilk et al. (2006a) methodconsiders the following form of the convolution integral:

�p�t� = �0

t

q�t − t��dpu�t��

dtdt� . . . . . . . . . . . . . . . . . . . . . . . . . (23)

Fig. 1—Comparison of unit-rate responses from our implemen-tation of von Schroeter et al. (2004) method with those gener-ated by von Schroeter for “Simulated Example” considered invon Schroeter et al. (2004) for the case with �p=0.05 and�q=0.10.

Fig. 2—Comparison of unit-rate responses from our implemen-tation of Levitan’s method (2005) with those generated by Levi-tan for his “Example 1” in his paper (2005).


and uses quadratic (second-order) B-splines, denoted by B2

throughout, with logarithmically distributed knots to represent theunknown dpu(t)/dt, simply denoted by p�u(t),

p�u�t� = �i=l

K

ciBi2�t�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (24)

Bi2(t) is the second order B-spline being nonzero only in the inter-

val bi<t<bi+3 for i�0, ±1, ±2, . . . , where b is the basis; typicallyIlk et al. use b�1.6. If l is the index of the first knot and K is theindex of the last knot, then there are K−l+1 B-splines involved inthe interpolation of the unknown p�u(t) response function. InsertingEq. 24 in Eq. 23, the following equation is obtained:

�0

t ��i=l

K

ciBi2�t��q�t − t��dt� = �p�t�, . . . . . . . . . . . . . . . . . . . (25)

or, equivalently,

�i=l

K

ci��0

t

Bi2�t��q�t − t��dt�� = �p�t�, . . . . . . . . . . . . . . . . . . (26)

where c is the (K−l+1)-dimensional vector of unknown coefficients.Selecting the most appropriate basis for logarithmic distribu-

tion of B-splines is very influential on the deconvolved responsefunctions (Ilk 2005). In other words by selecting the basis, thenumber of B-splines is set—the number of B-splines has a directeffect on the resolution of the desired function. Fewer B-splinescan result in over-smoothing and introduce bias; greater B-splinescan result in oscillations in the deconvolved response functions,especially in well-test derivative functions. In addition, Ilk et al.note that the earliest part of the unit-rate derivative response iscritical, especially if the observed variable-rate response containsa skin effect (but minimal or no wellbore-storage effects) becausein such cases the unit-rate response contains a Dirac delta compo-nent. To reconcile this condition, they add four additional “anchor”B-splines to the left of the first observed time point in the signal(Ilk 2006a).

If there are M pressure-change observations collected in theM-dimensional vector �pm that were observed at times (t1, t2, . . . ,tM), then the problem can be translated into an over-determinedsystem (i.e., the number of measurements are greater than thenumber of sought B-spline coefficients, Mkn) of linear equations as:

Xc = �pm, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (27)

where X is the M×n sensitivity matrix (or “design” matrix). [Notethat n (�K–l+1) represents the number of B-splines.] Its elementsare defined as:

Xj,i = �0

tj

Bi2�t��q�tj − t��dt�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (28)

As mentioned before, Eq. 27 is an over-determined linear systemof equations and can be solved in least-squares sense. It is alsopossible to solve the system of equations in weighted least-squaressense. For this case, by introducing a weight matrix W, we modifyEq. 27 as:

WXc = W�pm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (29)

Ilk et al. usually prefer a diagonal weighting matrix, W, with thediagonal elements defined by:

wj = 1��pm,j. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (30)

In other words, the reciprocal of measured pressure drop is used—it is assumed that the responses are corrupted by a constant levelof relative error. In this case, the goal of B-spline deconvolution isto find c that minimizes the following weighted least-squares ob-jective function:

E�c� = �Xc − �pm�TWTW�Xc − �pm�, . . . . . . . . . . . . . . . . . . . (31)

where W is the M×M diagonal weighting matrix with diagonalelements equal to wj, for j�1, 2, . . . , M, and E(c) is the objectivefunction. The solution of Eq. 31 can be given by:

c = �XTWTWX�−1XTWTW�pm. . . . . . . . . . . . . . . . . . . . . . . . . . (32)

Here, a few remarks are in order regarding the weighted least-squares objective function of Ilk et al., where the weights aredefined by Eq. 30. If we view the least-squares only as a purecurve-fitting method, it may be reasonable to define the weights asthe reciprocal of the observed pressure drop values. In this case,the solution may even provide reasonable estimates of the constantunit-rate responses. However, when the objective function isviewed within the concept of the maximum likelihood method ofestimation based on Gaussian statistics, then the statistics of theestimated parameters will be correct and the estimates would beoptimal only if the weights are defined as the inverse of standarddeviation of errors in the data [see, for example, Bard (1974)].Therefore, we do not expect that using a weighting matrix based onEq. 30 as proposed by Ilk et al. should yield correct statistics (e.g.95% confidence intervals for the unit-rate responses), even thoughthe resulting estimates of the unit-rate response may be reasonable.However, we were not able to verify this issue because the versionof the algorithm that we used in this study does not provide astatistical analysis of the results.

The solution c from Eq. 32 does not provide sufficient regu-larization as the noise level increases in the data. Hence, for theirregularization approach, Ilk et al. consider the following two con-ditions per each spline interval:

�� t �i=l

K

ciBi2�t��

t=bk

− � t �i=l

K

ciBi2�t��

t=bk+1/2

� = 0 . . . . . . . (33)

and

�� t �i=l

K

ciBi2�t��

t=bk+1/2

− � t �i=l

K

ciBi2�t��

t=bk+1

� = 0, . . . . . (34)

for k�l,l+1,. . . ,K–1. The regularization conditions (Eqs. 33 and34) require that the value of the logarithmic derivative of theconstant-rate response differ only “slightly” between the knot andthe middle location and can be represented as

Xrc = 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (35)

where Xr is the (2K–2l)×n regularization matrix. Then, by intro-ducing the regularization parameter �, Ilk et al. append the systemof regularization equations (Eq. 35) to the system of equationsgiven by Eq. 29 to obtain a new system of equations which may bewritten as

X*c = �p*m, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (36)

where X* is the (M+2K–2l)×n augmented matrix obtained by join-ing the M×n matrix(1−�) WX and the (2K−2l)×n matrix� Xr, and�p*m is a M+(2K−2l)-dimensional vector obtained by joining theM-dimensional vector (1−�)W�pm and the (2K−2l)-dimensional(zero) vector 0.

Solving Eq. 36 in a least-squares sense is equivalent to mini-mizing the following objective function:

E�c� = �1 − ��2E�c� + �2cTXrTXrc, . . . . . . . . . . . . . . . . . . . . . . (37)

where E(c) is the objective function given by Eq. 31. The solutionof Eq. 37 can be given by:

c = Xa+�pm, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (38)

where the n×M matrix Xa+ represents the pseudoinverse of the

augmented matrix and is given by

Xa+ = �1 − ��2��1 − ��2XTWTWX + �2Xr

TXr�−1XTWTW. . . . . . (39)

The over-determined system (Eq. 38) can be solved after comput-ing the pseudoinverse Xa

+ by singular value decomposition with adefault cut-off for the smallest singular value. Further details canbe found in Ilk (2005).


We note that the solution c given by Eq. 38 will not satisfy Eqs.33 and 34 exactly because the system of equations (Eq. 36) isover-determined. The influence of Eqs. 33 and 34 on the solutionwill depend on the size of the regularization parameter �. If � is setto zero, there is no regularization, and the solution (Eq. 38) reducesto the solution given by Eq. 32 for the no-regularization case. Inthe presence of random noise and/or other inconsistencies, a posi-tive � is selected based on an informal interpretation of the dis-crepancy principle—i.e., the value of the regularization parameteris increased until the calculated (model) pressure difference beginsto deviate from the observed pressure difference in a specific man-ner. The mean and standard deviation of the arithmetic differenceof the computed and input pressure functions are also computed.Increasing � to the point where the problem is over-regularizedwill yield a solution that is oversmooth and dominated by bias.

By using the c coefficients obtained by minimizing Eq. 37 (orequivalently solving Eq. 38), the constant-unit-rate pressure and itslogarithmic derivative responses are reconstructed by the follow-ing equations, respectively:

pu�t� = �i=l

K

ciBi,int2 �t�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (40)

and

dpu�t�

d lnt= t �

i=l

K

ciBi2�t�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (41)

where ci, for i�l, l+1, . . . , K, represents the coefficients estimatedby minimization, and the integral of the second order B-spline,Bi,int

2 (t), is obtained analytically.In contrast to the von Schroeter et al. and Levitan methods, the

Ilk et al. method does not assume that the rate function is given ina constant step-wise manner. Instead, Ilk et al. allow for a generalrate function whose Laplace transform, q(s), is analytically de-fined. The rate profile is not required to be an analytic function, butsegments of the rate profile are represented by individual functionswhich can be easily transformed into the Laplace domain.

The elements of the matrix X (Eq. 28) are calculated by us-ing numerical inverse Laplace transformation (here denotedby L−1[F(s)][t]):

Xj,i = L−1�q�s�Bi2�s�� tj�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (42)

Here, Bi2(s) is the Laplace transform of the ith B-spline and it is

known in closed form. The B-splines with indices i for which bi>tjare excluded, because those splines start later than the time of thegiven observation. The success of this approach relies heavily onthe accuracy of the numerical Laplace transform inversion. Assuch, the multi-precision GWR algorithm (Abate and Valkó 2004)is used to obtain the elements of the matrix X.

To obtain the Laplace transform of the rate, the Ilk et al.method considers various options. The one that they prefer is todissect the rate into N segments with starting times t k

q, k�1, . . . ,N. In each segment they describe the incremental contribution byan exponential term and hence the rate is obtained in the form

q�t� = �k=1

N �a kq + bk

q exp�−t − t k

q

ckq �� t − t k

q�, . . . . . . . . . . . (43)

where (t) is the Heaviside (unit-step) function. The parameters foreach segment are found by a linear least-squares fit combined with“direct search” on the nonlinear parameter c k

q. [Here, “directsearch” is related with the optimization algorithm to find the op-timum value of the function which is called the “golden sectionsearch” [see Cheney and Kincaid (2003)]. This is a sequentialsearch method which makes use of previous information to locatesubsequent experiments. A reasonable range for this time constantis between t1 and tM. The fewer segments used the more smooththe resulting rate approximations will be. A shut-in period shouldbe always included as an individual segment, and it is possible torepresent the shut-in period with exponentials. In simple terms, theIlk et al. rate approximation is the superposition of exponential

functions. However, the rates can also be modeled in a constantstep-wise fashion. In this case, the rate function is given by

q�t� = �k=1

N

�qk − qk−1� �t − t kq�, . . . . . . . . . . . . . . . . . . . . . . . . . (44)

and can be solved in the real-time domain without Laplace trans-formation using the superposition principle and the analytic inte-gral of quadratic B-splines. However, for “smoothly” changingrates Ilk et al. prefer Eq. 43.

Application of the Ilk et al. method requires not only the mea-sured rate and pressure data, but also specification of the startingtimes of the rate segments, t k

q, the basis for the spline-knots, b, theregularization parameter, �, and an estimate of the initial reservoirpressure p0.

An advantage of the method is that it can treat variable ratewithout using step-wise approximation. This can be useful, par-ticularly when dealing with continuously measured bottomholeflow rates. When a profile of piecewise constant surface rates likein a well-test sequence is considered, there is little benefit fromusing the Ilk et al. method, and such data can be equally wellprocessed by the von Schroeter et al. and Levitan methods.

One of the disadvantages is that a variable rate profile must bedissected into continuous segments. Also, the method as presentedhere does not allow for the estimation of initial pressure and rates.In addition, unlike the deconvolution methods of von Schroeteret al. and Levitan, the method cannot ensure positivity of thelogarithmic derivative of the constant-unit-rate pressure responseof the system. Using regularization based on Eqs. 33 and 34 is anindirect attempt to satisfy the positivity condition as well as pro-vide smoothness in the response function p�u(t). In certain cases, weobserved negative values of p�u(t) (Eq. 24) and hence, the logarith-mic derivative values (Eq. 41) were negative.

Comparisons of Methods With SimulatedTest DataTo test the proposed deconvolution methods, we consider threesynthetic (simulated) test examples. The examples are defined asfollows:

1. A simulated well-test example for an infinite-conductivityfracture well without wellbore-storage effects in a reservoirbounded by two parallel (no-flow) boundaries.

2. A simulated well-test example [Example 2 from Levitan(2005)], where a different wellbore-storage coefficient is modeledfor the final pressure buildup sequence.

3. A simulated well-test example for which we have con-tinuous measurements of flow rates.

Synthetic Examples 1 and 3 were generated by the Saphir soft-ware (2005).

Synthetic Example 1. As noted previously, the von Schroeteret al. method is based on the assumption that the wellbore-storageunit-slope line is valid for the behavior of the z response before thefirst grid node. In our experience, the wellbore-storage unit-slopetrend at the start of a transient is rare. Either the storage is poorlydefined (because of small storage, or data acquisition not being fastenough), or the storage is changing. Therefore, here, we investi-gate the consequences of using the von Schroeter et al. method ondata with no unit-slope trend at the very early time, in comparisonwith the use of the Levitan and Ilk et al. methods, which removethe unit-slope line assumption. For this investigation, we considera simulated well-test example (see Fig. 3 for pressure/rate data) foran infinite-conductivity fracture well without wellbore-storage ef-fects in a reservoir bounded by two parallel (no-flow) boundaries(see Table 2 for input model parameters). Pressure data weregenerated by assuming a gauge with a resolution of 0.01 psi, andrate history does not contain any errors. The data set is consistentwith the deconvolution model of Eq. 1.

The constant unit-rate drawdown responses generated based onmatching the entire pressure history for all periods from our imple-mentations of the von Schroeter et al. and Levitan methods areshown in Fig. 4a. The parameters used for the von Schroeter et al.


method are ��1 (psi.d/stb)2 and ��0.04 psi2, and for Levitan arep�0.01 psi, q�0.1 stb/d, and c�0.05. The minimum elapsedtime of any point in the pressure signal is 10−3 h. Hence, for bothmethods, we have selected this value of elapsed time as the valueof time for the first node (i.e., t1�10−3 h), and 70 nodes were used.

Fig. 4a shows that our implementation based on the von Schroe-ter et al. method does not correctly reconstruct the logarithmicderivative of the constant unit-rate drawdown response for the firstlog-cycle. However, the effect of assuming a wellbore-storageunit-slope trend at/before the first node for the von Schroeter et al.method disappears fast, and this assumption does not cause anyartifacts in the late-time portion of the derivative response. Al-though it may not be readily apparent from Fig. 4a, our algorithmfor the von Schroeter et al. method yields the same values for bothpu(t) and dpu/d lnt functions at the first node [i.e., dpu(t1)/dlnt�pu(t1)�ez1], as expected. Most interestingly, the estimated ez1

is equal to the correct value of the constant unit-rate drawdownpressure drop including the skin effect at t1 [i.e., pu(t1)], but notequal to the correct value of the logarithmic derivative of theunit-rate response at t1 [i.e., dpu(t1)/d lnt]. Therefore, there is a jump(or discontinuity) at the first node for the von Schroeter et al. method.

Fig. 4a also shows two implementations for the Levitanmethod; the first with L−1 curvature constraints (based on Eq. 18for the first node and Eq. 17 for the remaining nodes), and thesecond with L−2 constraints (based on Eq. 17 for all nodes). Bothcases reconstruct the unit-rate responses quite accurately for alltimes, though the implementation based on L−2 curvature con-straints gives slighlty better results.

Fig. 4b shows the results obtained from the Ilk et al. method.For this method, four additional “anchor” B-splines were added tothe left of the first observed time point in the signal. In Fig. 4b, wealso show the results for the case if the “anchor” B-splines are notadded, although Ilk et al. suggest that the anchor splines be addedfor all data sets with/without storage effects (Ilk et al. 2006a). Thetotal number of B-spline knots is 37 for the case without “anchor”B-splines. No regularization was considered (i.e., ��0). Fig. 4bindicates that the Ilk et al. method with “anchor” B-splines alsoconstructs the unit-rate responses quite accurately for all times.

Next, we investigate the effect of the value of t1 on deconvo-lution for our implementations of the von Schroeter et al. andLevitan methods. We consider two different values of t1; Fig. 5shows the results for t1�1×10−4 h, and Fig. 6 shows results fort1�1×10−2 h. The unit-rate responses shown for the von Schroeteret al. and Levitan method were generated by using the same valuesof the “weight” parameters that were used to generate the results inFig. 4a. From the results of Figs. 5 and 6, we conclude that t1 hasan effect on the early time portion of the unit-rate derivative re-sponses generated from both methods. For this example, the effectis more pronounced (approximately a log-cycle) on the unit-ratederivative response generated from the von Schroeter et al.method. Levitan’s method with L−2 curvature constraints pro-duces the correct unit-rate responses for all times if t1 is chosensmaller than the minimum elapsed time in the pressure signal.

Thus, we conclude that the effect of t1 is only on the very earlydata, approximately a log-cycle for both methods and recommend

Fig. 4—Comparison of unit-rate responses generated (witht1=10−3 h) from our implementations for (a) the von Schroeteret al. and Levitan methods, and (b) for the Ilk et al. method forSynthetic Example 1.

Fig. 3—Pressure and rate data for Synthetic Example 1.


that t1 be chosen equal or less than the minimum elapsed time inthe signal.

Synthetic Example 2. Pressure and rate data for this simulated testexample are shown in Fig. 7. The test data were provided byLevitan (through a personal communication) and do not containany errors. The simulation used two different values of the well-bore-storage coefficient; Cw�0.05 RB/psi for the first three peri-ods and 0.001 RB/psi for the last buildup period. The reservoir isconstrained by two parallel no-flow boundaries (Levitan 2005).The data set constitutes an example of inconsistent test data withthe deconvolution model of Eq. 1—the individual flow periods inthe example are consistent with the deconvolution model of Eq. 1,but jointly they are not. Our objective is to test the performancesof the three algorithms for this inconsistent data set and also testthe ability of the algorithms when applied to process pressure datapertaining to only individual flow periods.

Figs. 8 and 9 present comparisons of the results of deconvo-lution obtained from the first buildup (50 to 100 hr) and the secondbuildup (200 to 250 hr) periods by processing them separately withthe three deconvolution algorithms. For each algorithm, the flowrates and initial pressure are fixed at their given true values. There-fore, this actually represents the strategy proposed by Levitan et al.in deconvolution. The recommended default values of the requiredparameters for each algorithm are used. The total number of nodesused for the von Schroeter et al., and Levitan algorithms is 40 and

70, respectively, while the total number of B-spline knots used forthe Ilk et al. algorithm is 21. The same t1 value of 10−2 h, whichis equal to the minimal elapsed time in the pressure signal, wasused for the three methods. Initial pressure and rates were fixed attheir true values in deconvolution. The default values for � and �computed from Eqs. 9 and 11 for the von Schroeter et al. objectivefunction (Eq. 8) are �def�0.0085 (psi.d/stb)2 and �def�2.0643psi2 for the first buildup period, and �def�0.0046 (psi.d/bbl)2 and�def�1.3576 psi2 for the second buildup period. The default val-ues of p�0.01 psi and c�0.05 recommended by Levitan wereused in Eq. 16. Regularization was not used for the Ilk et al.algorithm (i.e., ��0 in Eq. 37) because pressure and rate data donot contain any noise.

As shown in Figs. 8 and 9, each algorithm gives almost iden-tical results for the pu(t) function and its logarithmic derivative forboth buildup periods. All methods work well with their defaultparameters (because we applied them to the periods where data areconsistent with the deconvolution model of Eq. 1) and are free oferrors. Although it is not shown, when we applied the deconvolu-tion algorithms to process the pressure data for the entire testsequence at once (with keeping initial pressure and rates at theirtrue values in deconvolution), all of the algorithms failed to pro-duce the correct solution, and we obtained a very similar responsefunction to that of Levitan [as shown in Fig. 5 of Levitan (2005)].

Fig. 6—Comparison of unit-rate responses generated witht1=10−2 h for Synthetic Example 1.

Fig. 7—Pressure and rate data for Synthetic Example 2; Ex-ample 2 of Levitan (2005).

Fig. 8—Comparison of constant-unit-rate pressure responsesobtained by deconvolution of the first buildup period; SyntheticExample 2.

Fig. 5—Comparison of unit-rate responses generated witht1=10−4 h for Synthetic Example 1.


These results verify that when all three algorithms are applied tosubsets of the pressure/rate data that are consistent with the de-convolution model of Eq. 1, the algorithms produce meaningfulresults. In addition, the results show that for this test example, theLevitan et al. strategy detects the inconsistency between data andthe deconvolution model because of changing wellbore-storagecoefficient.

We also applied the von Schroeter et al. and Levitan algorithmssuccessively to increasing portions of the signal, but treating initialpressure and rates as unknown and monitoring pressure and ratematches. Note that this is the strategy proposed by von Schroeteret al. (i.e., jointly estimate initial pressure and rates together withthe unit-rate derivative response). In this application, we use thevon Schroeter et al. algorithm with � and � at their default values(not reported here) computed from Eqs. 9 and 11 by using themaximal pressure value of 4,992.419 psi. This value of initialpressure was also taken as the initial guess for the unknown initialpressure; the true (or given) flow rates were taken as initial guessfor the flow rates.

Fig. 10 shows derivative responses deconvolved over increas-ing portions of the data set from the start to the end of the test.

With the application of this strategy, we are also able to detect thatthere is indeed inconsistency between the data for the secondbuildup and the deconvolution model because all deconvolvedderivative responses, except the one from the start to the end of thetest (i.e., from 0.01 to 250 h), show the same well/reservoir be-havior (Fig. 10). The pressure match obtained by deconvolution ofall pressure data from the start to the end of the test is not accept-able (Fig. 11). The pressure matches obtained by deconvolution ofother portions of the pressure data were perfect (not shown). Es-timated initial pressures are 4,999.816 psi by deconvolution of datafrom 0.01 to 50 h; 4,999.986 psi by deconvolution of data from0.01 to 100 h; 4,999.991 psi by deconvolution of data from 0.01 to200 h; and 5,044.511 psi by deconvolution of data from 0.01 to250 h, compared to the true initial pressure of 5,000 psi. Estimatedflow rates for the first and second flow periods are almost the sameas the input values of 1,000 STB/D by deconvolution of all por-tions of the pressure data, except by deconvolution of pressure datafrom 0.01 to 250 h. For that case, the estimated flow rates for thefirst and second flow periods are 1143.68 and 793.46 STB/D,respectively. Thus, the estimated initial pressure and estimatedflow rates differ significantly from the true values if the pressuredata for the second buildup are included in deconvolution. In sum-mary, these results show that for this example, the von Schroeter et al.strategy also detects the inconsistency between data and the decon-volution model by checking the pressure and rate matches as well asthe unit-rate responses obtained for different subsets of the data.

Synthetic Example 3. The true pressure and rate data as well astheir noisy versions with different error levels for this simulatedtest example are shown in Fig. 12. The model is for an unfracturedvertical well in a double-porosity reservoir with a no-flow circularboundary and including wellbore-storage and skin effects. Theinput parameters are given in Table 3. The true production rate asa function of time was produced with the following formula:

q�t� = �350 − 300e−t�20� �t� + �−350 + 350e−�t−100��50� �t − 100�

+ �300e−t�20 − 350e−�t−100��50� �t − 150�. . . . . . . . . . . . . . (45)

We consider continuous measurements of flow rate data in thisexample so that we can investigate the performance of the decon-volution algorithms based on constant stepwise approximation offlow rates. In addition, we consider different levels of normallydistributed errors generated from Eqs. 2 and 3 and added to flowrate and pressure data to investigate the robustness of the algo-rithms. The standard deviations of errors in pressure data with0.5%, 2%, and 5% error levels are �p�10.42, 41.88, and 101.69

Fig. 9—Comparison of constant-unit-rate pressure responsesobtained by deconvolution of the second buildup period; Syn-thetic Example 2.

Fig. 10—Comparison of constant-unit-rate derivative responsesdeconvolved (using the von Schroeter et al. algorithm) overincreasing portions of the data set; Synthetic Example 2.

Fig. 11—Pressure match obtained by deconvolution of datafrom 0.01 to 250 h, using the von Schroeter et al. algorithm;Synthetic Example 2.


psi, respectively, while the standard deviations of errors inrate data with 1% and 5% error levels are �q�2.44 and 11.24STB/D, respectively.

Fig. 13 presents a comparison of the true flow rate history andthe constant stepwise approximations used to represent these datafor application in the von Schroeter et al. and Levitan algorithms.Although it may not be clear from Fig. 13, we have approximatedthe continuous flow rate history by 125 piecewise constant steps.Noisy rate data containing 1% and 5% random errors were simi-larly approximated for the algorithms. In the Ilk et al. algorithm,piecewise exponential rates (Eq. 43) were used, and the parametersfor each segment were found by linear least-squares fit, as ex-plained previously. For example, for rate data containing 5% ran-dom noise, we use the following formula:

q�t� = �349.7 − 300.8e−t�20� �t�

+ �−339.7 + 336.8e−�t−100��48� �t − 100�

+ �−10 − 118.7e−�t−150��48� �t − 150�, . . . . . . . . . . . . . . . . (46)

for 0<t�300. Fig. 14 shows a comparison of the rate data con-taining 5% error level with the rate data approximated from Eq. 46.Although not reported here, our piecewise constant and exponen-tial representations of the continuous flow rate with or withoutnoise do honor the cumulative production during the test periodreasonably well.

We apply the deconvolution algorithms to process the pressuredata for the whole test sequence in one pass. In our applicationswith the von Schroeter et al. and Levitan methods, we only fix theinitial pressure at its true value of 8,000 psi. For the Ilk et al.algorithm, in addition to fixing the initial pressure, we fix the ratehistory approximated by fitting piecewise exponentials to the noisyrate data because flow rate cannot be treated as unknown. We sett1�10−3 h, the minimal elapsed time in the pressure signal, for thevon Schroeter et al. and Levitan methods; we start to constructB-splines with four “anchors” added to the left of this t1 in the Ilket al. method.

Our objective is to investigate the robustness of the three meth-ods by considering pressure and rate data sets that are contami-nated with different levels of Gaussian errors. In Figs. 15 through 17,we show the deconvolution results generated for three differentcombinations of error imposed on the input rate and pressure data.For the von Schroeter et al. (Fig. 15) and the Levitan (Fig. 16)methods, piecewise constant rates with errors are used and treatedas unknowns in the deconvolution process. For the Ilk et al.method (Fig. 17), the rate history is fixed and fitted using piece-wise exponential functions (e.g., see Eq. 46 for 5% error in ratedata, and Fig. 14).

For each method we have developed a “best fit” using theregularization mechanism for a particular algorithm. The controland regularization parameters are presented in Table 4. The �-parameters given in Table 4 for the von Schroeter et al. methodrepresent the default values computed from Eq. 9. The �-

Fig. 14—Comparison of flow-rate data with 5% error level withits piecewise exponential approximation given by Eq. 46, Syn-thetic Example 3.

Fig. 12—Pressure and rate data for Synthetic Example 3; bothtrue and corrupted data sets of pressure and rate.

Fig. 13—Comparison of true flow rate with its piecewise con-stant approximation, Synthetic Example 3.


parameters represent 10�def (with �def computed from Eq. 11)values. Thus, we actually use the strategy recommended by vonSchroeter et al. (i.e., for fixed �, start with a positive default level�def and multiply it by powers of 10). In Fig. 15 we present ourresults only for 10�def because this value provides the best resultsfor this strategy.

The results shown in Fig. 16 for the Levitan method weregenerated by setting p and q equal to the standard deviations oferrors in pressure and rate data for each data set considered, andadjusting the value of c to find its “optimal” value giving the bestunit-rate response curve. We note that the value of c�0.05 sug-gested by Levitan together with the p and q values given in Table 4did not produce smooth unit-rate responses for any of the threedata sets considered in the experiments. In addition, for data set 3(5% error levels in p and q), using p�102 psi, q�11.2 STB/D,and c�0.05 yielded a negative pu(t1) value (not shown here). Wehave not varied the values of p and q in these experiments be-cause in view of the maximum likelihood concept (Bard 1974),these parameters in the weighted least-squares objective functionconsidered by Levitan (Eq. 16) should represent the standard de-viations of error in pressure and rate data, respectively. We cancertainly compute the standard deviation of errors in pressure and

data and use these values for p and q because this is a syntheticexample. However, how this could be achieved in practice is animportant question.

The results of Figs. 15 through 17 indicate that the von Schroe-ter et al. and Levitan methods perform better than the Ilk et al.method for this simulated example, particularly when the level oferror in rate/pressure data increases. Fortunately, each method ap-pears, from this synthetic example to be competent (i.e., robust) forpractical applications in well-test and production data.

Fig. 18 shows reconstructed pressure signals obtained from allthree methods together with the given “measured” pressure fordata set 3 (i.e., 5% error levels in p and q) and the true (clean)pressure signal. Fig. 19 shows reconstructed rate signals obtainedfrom the von Schroeter et al. and Levitan methods (recall in the Ilket al. method rate is fixed, so it is not shown in Fig. 19) togetherwith the “measured” rate for this data set, and the true (clean) ratesignal. As can be seen, reconstructed pressures from all threemethods agree well with the true (clean) pressures. The averageroot-mean-square (rms) errors for the pressure match for the vonSchroeter et al., Levitan, and Ilk et al. methods are 94.1, 93.1, and100.2 psi, respectively, which are close to the true standard devia-tion of errors �p�101.7 psi for “measured” pressure data contain-ing 5% error level. Similarly, the average rms errors for rate matchfor the von Schroeter et al. and Levitan methods are 7.7 and 8.2STB/D, respectively, which are close to the true standard deviationof errors (�q�11.24 STB/D) for “measured” rate data containing5% error level.

Field Example ApplicationIn this section, we consider one field well-test example to illustrateapplication of the deconvolution methods under investigation.

The pressure and rate for this example are presented in Fig. 20.The test consists of four distinct step-rate changes and a shut-inacquired over a 16-hr period from a geothermal well producingfrom a liquid-dominated, highly permeable fractured/faulted res-ervoir system. The flow in the reservoir is single-phase water witha salinity of approximately 5000 ppm, and a dissolved CO2 contentapproximately 0.4% by weight [see Onur et al. (2007) for details].In the field, the reservoir thickness is variable and not known verywell, and the permeability-thickness product (kh) ranges from 100Darcy-m to 1900 Darcy-m (Onur et al. 2007). The well is vertical.The pressure was measured with a downhole quartz gauge, posi-tioned at the middle of the open interval, with a specified resolu-tion of 0.01 psi. The rate measurements were taken at the surfaceusing a weir, and were accounted for the steam phase at the sur-face. The temperature was also recorded downhole and was nearlyconstant at 107.8°C throughout the test. In the well, we had alsorun a static survey and a flowing survey (conducted with a rate

Fig. 16—Deconvolved constant-unit-rate pressure responsesgenerated by the Levitan method for Synthetic Example 3.

Fig. 17—Deconvolved constant-unit-rate pressure responsesgenerated by the Ilk et al. method for Synthetic Example 3.

Fig. 15—Deconvolved constant-unit-rate pressure responsesgenerated by the von Schroeter et al. method for Synthetic Ex-ample 3.


equal to the rate for the fourth flow period shown in Fig. 20) ofpressure/temperature vs. depth, and these surveys indicated thatthere was no gas or steam phase at the gauge location.

The measured initial pressure is approximately 280 psi, butthere is an uncertainty in the value. The conventional (multirate)test analysis of each flow period (Fig. 21) gives an indication of aninconsistent test example with changing wellbore-storage effects(owing to nonisothermal and multiphase flow effects in the well-bore) within each period as well as non-Darcy flow (i.e., a rate-dependent skin profile). The non-Darcy flow is typical of the geo-thermal system under consideration and results from a high-permeability fracture network intersecting the wells. Based on theresults of Fig. 21, we also suspect that measured rates for someperiods may not be accurate (e.g., the derivative responses at latetimes for the third and fourth flow periods do not overlay with thederivative responses for the second flow and buildup periods).Otherwise, we would expect rate-normalized derivatives for allperiods to overlay because they should be independent of rate-dependent skin after storage effects die out. Although the samelevel of smoothing was applied for each of the derivative responses

shown in Fig. 21, derivative data for the flow periods are muchnoisier than those for the buildup period, and derivative data forsome of the flow periods are not interpretable.

Given the issues stated previously and considering that we haveonly one buildup period, first we process the pressures for theentire test sequence in one pass to estimate the initial pressure andrates jointly with the unit-rate derivative response functions. In asense, this is the strategy proposed by von Schroeter et al. How-ever, we must realize that in this approach, we may end up ad-justing the rates incorrectly because of rate-dependent skin for thesake of matching (or honoring) the entire measured pressure data.For this application, we use our algorithm based on the von Schroe-ter et al. method. The maximal pressure in the test sequence of279.872 psi is used as an estimate of initial reservoir pressure. Theminimal elapsed time in the test sequence of 2.78×10−3 h is usedfor the first node (t1) to reconstruct constant unit-rate drawdownresponses. For this value of initial pressure and t1, the defaultvalues of � and � are �def�3.16×10−8 (psi.d/stb)2 and�def�1.92×10−3 psi.2

Fig. 18—Comparison of reconstructed pressure together with“measured” (containing 5% error level) and true (clean) rate;Synthetic Example 3.

Fig. 19—Comparison of reconstructed rate together with “mea-sured” (containing 5% error level) and true (clean) rate; Syn-thetic Example 3.


Fig. 22 shows pressure derivative estimates obtained for fivedifferent sets of � and �—in four of the sets, � is increased from�def by successive powers of 10, leaving � at its default value,while in one of the sets, we increase � from �def by factor of 10,leaving � at its default value. (The conventional rate-normalizedbuildup derivative based on the superposition of measured rates isalso shown in the same figure for comparison.) For the sets where� is fixed at its default value and ��10n×�def for n�0, 1, 2, 3,estimates for initial pressure are almost identical, ranging from280.081 to 280.087 psi. Also, rms errors for the pressure and ratematches are almost identical; rms for pressure matches rangingfrom 0.071 from 0.080 psi, and rms for rate matches ranging from2,684 to 2,688 STB/D, as n increases from 0 to 3. For the set with��101×�def, and ��def, the estimated initial pressure is 280.40psi, and rms errors for pressure and rate matches are 0.084 psi and2004 STB/D, respectively. These results indicate that increasing �with � fixed at its default has little effect on the pressure and ratematches. The derivative estimate for the set with �=�def and��def is not interpretable because of its oscillations, and also thederivative estimate for the set with ��101×�def, and ��def

shows an odd behavior after 1 h. The derivative estimate for �=�def

and ��101×�def best matches the conventional buildup derivativefor times until approximately 3 h, though both are based on dif-ferent rate histories. (We find this good agreement to be puzzling

because the derivative responses are based on different rate histo-ries—the conventional buildup derivate is based on measuredrates, while the derivative estimate for �=�def and ��101×�def isbased on the adjusted rates shown in Fig. 23. For now, we post-pone our discussion of this issue.)

Thus, we choose the derivative estimate for �=�def and��101×�def for further analysis. The rate and pressure matchesobtained with this choice are shown in Figs. 23 and 24, respec-tively. The pressure match has an rms error of 0.072 psi and looksexcellent; however, the rms error is higher than the gauge resolu-tion of 0.01 psi. The estimated initial pressure is 280.081 psi witha 95% confidence interval of ±0.049 psi, which indicates smalluncertainty in initial pressure and that the true value may be insidethe interval [2.80.032, 280.130 psi]. We do not have a prior knowl-edge of error level in rate data, so it is difficult for us to judgewhether the rate match shown in Fig. 23 (with an average rms errorof 2691.2 STB/D) is acceptable: The rates for the first two flowperiods are changed by approximately 30%, which is quite large,while the rates for the third and fourth flow periods are changed byapproximately 11%. The 95% confidence intervals for the esti-mated rates are quite narrow, indicating small uncertainty in theestimated rates. However, none of the measured rates lie within the

Fig. 21—Conventional rate-normalized pressure change and de-rivatives for each flow period in the test sequence; field ex-ample.

Fig. 22—Comparison of reconstructed unit-rate derivative re-sponses by the von Schroeter et al. method, by processing thepressures for the entire test sequence in one pass with initialpressure and rates as unknown for various values of � and �;field example.

Fig. 20—Pressure and rate data; field example.

Fig. 23—Rate match for �=�def and �=101×�def and 95% confi-dence intervals for the estimated rates; field example.


95% confidence intervals. This indicates the rate errors cannot becharacterized as normally distributed random variables, which isone of the basic assumptions of the deconvolution model. Based onthese results and the fact that rate-dependent skin and changingwellbore-storage effects exist in the test, the data set cannot beconsistent with the deconvolution model of Eq. 1.

Next, we consider deconvolutions by using the pressure datafrom an individual flow period(s) and by keeping rates and initialpressure fixed in each run. For this purpose, we perform decon-volutions by using the pressure data from the second and third flowand buildup periods separately. In a sense, this strategy can beviewed as the strategy proposed by Levitan for inconsistent datasets, though we process pressure data for the flow periods too.Here, we will use the rate history (Fig. 23) and initial pressure of280.081 psi estimated previously by processing pressures for theentire test sequence in deconvolution with �=�def and ��101×�def

to examine whether the inconsistency detected by the conventionalmultirate analysis (Fig. 21) was caused by the incorrect measuredrates or the model changes. From now on, estimated rates arereferred to those shown as gray lines in Fig. 23. If deconvolutionsbased on these individual periods with the estimated rates produceunit-rate drawdown derivative responses that are consistent witheach other at least at late times, then the data and deconvolutionmodel may be said to be consistent, and the inconsistency observedin Fig. 21 is caused by incorrect measured rate history. Otherwise,the test data and the deconvolution model are inconsistent mainlybecause of the model changes in each period. For each deconvo-lution, we use our algorithm based on the von Schroeter et al.method with �=�def and ��101×�def, where �def and �def werecomputed by using the pressure data and rate history for the periodunder consideration—their values change depending on the periodconsidered in deconvolution. (We also performed deconvolutionswith �=�def and ��def, but the results were similar to thoseobtained with �=�def and ��101×�def.) Fig. 25 shows a compari-son of the deconvolution results for the three periods. The esti-mated derivatives for the second and third flow periods are verydifferent from that for the buildup period. Hence, the same decon-volution model cannot be valid for these flow periods and thebuildup period. This is not surprising to us because we havenonisothermal and multiphase effects (steam and noncondensablegas phases above its flashing point) in the wellbore reflected as acomplicated wellbore-storage phenomena, as well as rate-dependent skin. Hence, we do not expect the pressure transientbehavior of a flow period to be consistent with that of a buildupperiod even if the initial pressure and rate history were known withcertainty or estimated correctly. In summary, for this example,

deconvolution based on individual flow periods helps us to detectthe inconsistency. However, deconvolution does not help us toprove or disprove whether the initial pressure and particularly flowrates estimated simultaneously with the constant unit-rate draw-down derivative response by deconvolution of all measured pres-sures are correct.

Fig. 26 presents further interesting results for this field ex-ample. In Fig. 26, we compare the estimated derivative responsesobtained from deconvolutions of buildup pressures alone withmeasured and estimated rates. For all cases, in addition to rates,initial pressure is also fixed at 280.081 psi. The conventional rate-normalized buildup derivatives based on the superposition time ofmeasured and estimated rates are also shown in the same figure forcomparison. The results of Fig. 26 indicate that all derivative re-sponses are reasonably consistent with each other, and that thechoice of rate history does not have a significant influence on thedeconvolved unit-rate and conventional rate-normalized deriva-tives from buildup pressures. Particularly, it is quite puzzling to uswhy the estimated derivative response obtained by processing thepressures for the entire test sequence (all four flow periods andbuildup period) by treating initial pressure and rates as unknown indeconvolution agrees well with the estimated derivative response

Fig. 24—Pressure matches obtained by deconvolution of pres-sures for the entire test sequence with unknown initial pressureand rates, and for the buildup pressures alone with fixed initialpressure of 280.081 psi and measured rates. Both deconvolu-tions were performed using the von Schroeter et al. methodwith for �=�def and �=101×�def; field example.

Fig. 25—Comparison of reconstructed unit-rate derivative re-sponses from deconvolutions based on pressures for second,third, and buildup periods with initial pressure fixed at 280.081psi and rates fixed at their estimated values shown in Fig. 23;deconvolutions were performed using the von Schroeter et al.method with for �=�def and �=101×�def; field example.

Fig. 26—Comparison of reconstructed unit-rate derivative re-sponses from various deconvolutions with rates fixed at theirestimated (in Fig. 23) and measured (Fig. 20) values with initialpressure fixed at 280.081 psi; deconvolutions were performedusing the von Schroeter et al. method with for �=�def and�=101×�def; field example.


obtained by processing only the pressures for the buildup period indeconvolution with initial pressure fixed at 280.081 psi and ratesfixed at their measured values.

In an attempt to clarify this point, we generated a pseudo-fieldtest sequence with the flow rate history shown in Fig. 20 andrate-dependent skin and wellbore-storage effects—wellbore-storage coefficient is constant and the same for each period. Forthe purpose of our discussion, we did not add errors to rate history.Our results not shown here indicate that processing the pressuresfor the entire test sequence at once (or also subsets including flowperiods) by jointly estimating initial pressure and rates in decon-volution produces correctly the constant unit-rate drawdown de-rivative response and the initial pressure. However, the deconvo-lution algorithm adjusts flow rate history (actually incorrectly) forthe sake of providing a perfect match of the measured pressurehistory, similar to the pressure match obtained in Fig. 24 for thefield example (solid curve in Fig. 24). Processing pressures for thebuildup period with rates and initial pressure fixed at their truevalues also produces the constant unit-rate drawdown derivativescorrectly, but the pressure match for the flow periods is not good,which should be expected because of rate-dependent skin, as simi-lar to the pressure match obtained in Fig. 24 for the field example(dashed curve).

In summary, we admit that the field example considered here iscomplex in the sense that pressures include rate-dependent skinand changing wellbore-storage effects; additionally, measuredrates contain errors that do not follow Gaussian statistics and ourlack of information on expected error margins on rate data. How-ever, on the basis of our deconvolution analyses of this data andthe pseudo-field example with rate-dependent skin, we believe thatestimated initial pressure of 280.081 is reasonably correct and thatthe unit-rate derivative response derived from buildup pressureswith this initial pressure represents the actual behavior of the geo-thermal reservoir system under consideration.

Next, we compare the performance of each deconvolutionmethod in generating the constant unit-rate drawdown responsefrom buildup pressures alone, using the initial pressure value of280.081 psi and the measured flow rate history as fixed in decon-volution. The deconvolved pressure drop and derivative responsesobtained from the von Schroeter et al., Levitan, and Ilk et al.methods are shown in Fig. 27, and the deconvolution results arecompared to the conventional pressure buildup derivative. Thealgorithmic parameters for deconvolution are again given in Table 5.In this case, for our implementation of the Levitan method, wefixed pu(t1) in Eq. 16 by using the value of the deconvolved unit-rate derivative response estimated at the first node from our imple-mentation of the von Schroeter et al. method. This action wasnecessary, because for this application, we estimated pu(t1) to be a

negative value* when pu(t1) is treated as an unknown model pa-rameter in our implementation of the Levitan method. As dis-cussed in detail previously, when pu(t1) is treated as an unknownmodel parameter in our implementation of his method, the resultingvalue may be negative because the minimization is unconstrained.

The prior geologic model (not shown here) indicates that thewell is nearly centered between two parallel faults, but no infor-mation exists whether the faults are no-flow or conductive. Thedeconvolved unit-rate response indicates that these faults may beno-flow because of a well-defined 1/2 slope line after 1 hour (Fig.27). The deconvolved derivative response at early times from0.003 to 0.2 hr gives an indication of a partially or fully penetratingwell with changing wellbore-storage effect. In addition, the –1slope line in the time interval from 0.3 to 1 hr exhibited by thedeconvolved derivative response may indicate that the well is neara highly conductive fault or permeability-thickness product in-creasing away from the well. Unfortunately, we do not have theanalytical models in our catalogue to account for all these flowregimes together. Therefore, we consider rather a simpler modelbased on a partially-penetrating well producing [with changingwellbore-storage (Hegeman et al. 1993) and rate-dependent skineffects] between two no-flow parallel faults at least to honor the1/2 slope line observed after 1 hour on the deconvolved derivativeresponse (Fig. 27). Fig. 28 presents the match of the unit-ratederivative response derived from deconvolution of the builduppressures, while Fig. 29 presents the match of the buildup pressurechange and its derivative (based on superposition time). Fig. 30presents a comparison of the pressure data generated from thismodel with the test pressure data during the whole test sequence.The results shown in Figs. 28 through 30 are based on measuredrates and the initial pressure of 280.081 psi. The model cannotreproduce the measured pressures for the flowing periods, eventhough we included rate-dependent skin effects (Fig. 30). As dis-cussed previously in detail, this discrepancy may be caused byerrors in measured flow rates; though we do not know how tocorrect them or if the estimated (or adjusted) rates by deconvolu-tion are correct. However, the model reproduces the constant unit-rate drawdown buildup responses derived from deconvolution ofbuildup pressures (Fig. 28) and rate-normalized buildup responsesbased on superposition time (Fig. 29) fairly well at late times. Itdoes not, however, reproduce the estimated derivative responseswell at early times in the time interval from 0.03 to 0.5 hr. We

* Based on a personal communication between Levitan and the first author of this paperthrough an e-mail on 16 October 2006, by sending his results in a figure (digital data werenot provided by Levitan) generated from his implementation of the algorithm for the this fieldexample, Levitan showed that his implementation of the algorithm did not produce theproblem [i.e., a negative value of pu(t1)] that our implementation did. Based on this com-munication, it was also brought to our attention that some details of Levitan’s algorithm werenot revealed in his paper (2005) for purposes of protection of intellectual property rights.Without complete details of the algorithm, we could not fully evaluate it and reproduce hisresults for the field example.

Fig. 27—Comparison of deconvolved responses derived frombuildup pressure data using the von Schroeter et al., Levitan,and Ilk et al. algorithms with the conventional pressure builduppressure change and its logarithmic derivative (normalized bythe last rate prior to buildup); field example.


believe that this is because the model we used cannot incorporatea finite-conductivity fault nearby the well. We note that we obtaina much better match for the early-time portions of the derivativeresponses by using the model based on a finite-conductivity fault(for a fully penetrating well with changing wellbore-storage ef-fects) given by Abbaszadeh and Cinco-Ley (1995), as shown inFig. 31. The model fails, however, to reproduce the late-time por-tions of the buildup after 1 hr, as expected, because it assumes thatthe zones on both sides of the fault plane are of infinite extent. Tomatch both the early- and late-time portions of the buildup re-sponses we need a finite-conductivity fault model that considersone of the zones as bounded by two parallel no-flow boundaries.The permeability-thickness product (∼295**−957 Darcy-m), me-chanical skin (∼−1.0**−–2.5), and distances to the finite conduc-tivity fault (∼30** m) and no-flow faults (∼340 m) estimated fromboth models were reasonable and consistent with the availablegeologic model and estimated parameters from the tests conductedin other nearby wells. Finally, we should note that we also per-formed history matches of the test data by using the same models,

but with the estimated rates from deconvolution (Fig. 23). Theresults were similar to those obtained by using measured rates. Forexample, with the model used to generate the results given in Figs.28 through 30, we obtained the permeability-thickness product as1175 Darcy-m, mechanical skin as –2.3, and the distances to theno-flow faults as 330 m. The model matches for deconvolved andbuildup responses were similar to those shown in Figs. 28 and 29,but the model match for the pressures during the flow periods wasmuch better than that shown in Fig. 30 based on measured rates.

DiscussionBased on the results of our three synthetic and one field examples,we propose, as a general statement, that the von Schroeter et al.(2002, 2004), Levitan (2005), and Ilk et al. (2006a) methods makedeconvolution a viable tool, and are very useful for the analysis ofvariable rate/pressure well test and production data. We believethat these deconvolution methods open a new era in the interpre-tation of well-test and production data.

However, the proper use of these methods requires an under-standing of their assumptions, and also an understanding of thedeconvolution problem itself, particularly its ill-posed nature. Inpractice, we never know a priori to what extent a given data set isconsistent with the deconvolution model (Eq. 1), and hence, weconcur with both the von Schroeter et al. (2002, 2004) and Levitan(2005, 2006) perspectives to avoid problems (or failure) and to iden-tify any inconsistencies in the data, when applying deconvolution.

** Estimated values based on the finite-conductivity fault model (Abbaszadeh and Cinco-Ley 1995).

Fig. 29—Model match of buildup pressure change and its de-rivative using measured rates and initial pressure of 280.081psi; field example.

Fig. 30—Model match of measured pressure data during thewhole test sequence, using measured rates and initial pressureof 280.081 psi. The model is a partially penetrating well in areservoir with two parallel no-flow faults; field example.

Fig. 31—Model match of the unit-rate responses derived fromdeconvolution of buildup pressures using measured rates andinitial pressure of 280.081 psi. The model is a finite conductivityfault (Abbaszadeh and Cinco-Ley 1995); field example.

Fig. 28—Model match of the unit-rate responses derived fromdeconvolution of buildup pressures using measured ratesand initial pressure of 280.081 psi. The model is a partially pene-trating well in a reservoir with two parallel no-flow faults,field example.


The von Schroeter et al. perspective is to jointly estimate ratesand/or initial pressure together with the unit-rate drawdown de-rivative response, and then check whether the results honor thedata to within expected error margins, coupled with the statisticalanalysis (e.g., rms errors in pressure and rate as well as 95%confidence intervals) based on Gaussian statistics. As shown pre-viously, their perspective works very well if we have consistentdata sets and most importantly a priori knowledge of error levelsin rate and pressure, but more importantly in rate. In cases ofinconsistent data sets as in the field example case considered, theinconsistency of the data with the deconvolution model, as well asthe validity of such prior knowledge, may also be detected bychecking pressure and rate matches coupled with the statisticalanalysis. However, as we faced in the case of our field exampleapplication, for which the data subsets are not consistent becauseof changing wellbore-storage and rate-dependent skin effect, wedo advise caution that adjusted rates obtained by processing pres-sures based on the von Schroeter et al. strategy may not necessarilybe correct because deconvolution adjusts the rates for the sake of“honoring” measured pressures. Therefore, we do not view thisdeconvolution strategy or deconvolution in general as a tool tocorrect the rates from pressure data alone for possible measure-ment errors. In fact, we could easily show that there is no decon-volution method or algorithm that can estimate the rate historyfrom the pressure data, or “correct” the measured rate, even in thecase of consistent data sets unless we know the spread of the error(or expected error margins) in rate data. Here, we will give twosimple examples to show this point.

Suppose we have a constant rate drawdown followed by abuildup. Furthermore, let us assume that pressure/rate data areconsistent with the deconvolution model of Eq. 1. Further, supposethat we process the entire drawdown and buildup pressure data bydeconvolution with the drawdown rate and unit-rate (or z–) re-sponse as unknown. Can the deconvolution methods (von Schroe-ter 2002, 2004; Levitan 2005, 2006) estimate the true (unknown)drawdown rate if an initial guess used for the rate is different fromthe true value? The answer is “no,” because the algorithms will(and should) give back the same initial input rate without provid-ing any correction, but the unit-rate responses (pressure changeand derivative) will be vertically displaced depending on the inputvalue of drawdown flow rate, though pressure match for both thedrawdown and buildup periods will be perfect. The reason is thatwe do not have a unique solution for the z-response and for thedrawdown rate, and hence any value of drawdown flow rate withthe estimated z-response (based on this flow rate) will reproducethe same pressure history. We can give another example: Supposethat we have a test with rate 1, buildup, rate 2, buildup. Again, webelieve that there is no deconvolution method that could estimatethe rate history, or correct the measured rate. At best, it might beable to estimate the ratio of the two rates, or it might identify aninconsistency in the two measured rates, but it would never knowwhich is correct and which would be wrong. A prior knowledge oferror level in rate data will be a prerequisite for one to be able todetermine the appropriate values of rates estimated from decon-volution. Of course, if we have further inconsistency of pressuredata (e.g., in the form of changing skin from flow period to flowperiod) with the deconvolution model of Eq. 1, then matchingpressure data pertaining to only the buildups or drawdowns or bothdrawdown and buildups while letting flow rate and initial pressurego unknown in deconvolution could easily lead to nonunique re-sults. In such cases, inspecting statistics of the deconvolution re-sults may prove useful, as shown with the synthetic and field dataexample applications.

The Levitan perspective is to apply deconvolution for eachflow (preferably buildup) period separately with rates and initialpressure fixed in each run in order to check the consistency of datawith the deconvolution model of Eq. 1. As shown with the syn-thetic and field data applications, this strategy also works very wellif the test sequence contains a few buildups in the sense that wecan identify the correct reservoir response at late times, and detectany inconsistencies of the pressure/rate data sets with the decon-volution model of Eq. 1, identify any inconsistency of rate data

with pressure data, and inaccuracy in initial pressure. However, forthe cases where a well-test sequence or production data do notcontain buildup periods or contain only a single buildup period, theuse of this strategy is limited because a single buildup or flow (ordrawdown) period do not contain enough information to identifythe correct value of initial pressure and rate history required forreconstruction of correct constant unit-rate drawdown pressure.However, variable-rate drawdown data, if of “good quality,” canbear significant information, and does not rule out successful useof deconvolution.

Here, we provide some further general remarks on deconvolu-tion. Regardless of the deconvolution method used, the quality ofa deconvolved response depends on (i) the validity of the pressure-rate data pair with the deconvolution model of Eq. 1, (ii) datasubsets used for deconvolution, (iii) regularization parameters(“weights”) used, and (iv) possible use of constraints [i.e., con-straining initial pressure or rates (as fixed) in deconvolution]. Inaddition, deconvolution could be very sensitive to small errors inthe data. For example, small time lags or time synchronizationerrors between rates and pressure could easily lead to large aber-rations or oscillations in the deconvolved response, as pointed outby von Schroeter et al. (2004) and demonstrated by Levitan et al.(2006). Some data may not be representative of reservoir flowbecause of wellbore/surface effects, and even some rate data maybe missing. Hence, due care must be taken in preparing and se-lecting data to be used in deconvolution. When working with verylarge datasets, data filtering and flow rate decimation and simpli-fication become critical in practical use of deconvolution.

In summary, deconvolution is an interpretation tool for the dataitself, and it depends on the user running it, and therefore requiresdata preparation as well as the user’s judgment. We believe that thedeconvolved response must always be compared with the conven-tional derivative response, and more importantly must be consis-tent with geological prior information, and be validated throughpressure history matching based on the possible well/reservoirmodel(s). Furthermore, we think that it is too much to expect adeconvolution algorithm to correct both rate history and initialpressure. It is also unfortunate that the late-time portion of thedeconvolved unit-rate response is so sensitive to the value of initialpressure. Irrespective of all these issues, we believe that it shouldalso be the task of technology to provide more accurate measure-ments of rates and initial pressure to be used in the deconvolutionmethods considered in this work.

ConclusionsAs mentioned previously, the algorithms used in this study forevaluating the von Schroeter et al. (2002,2004) and Levitan (2005)methods represent our independent implementations based on thematerial presented in their papers. These implementations form thebasis of our conclusions.

Based on our investigation, we can state the following specificconclusions related to each algorithm:1. Our algorithm based on the von Schroeter et al. method:

• Algorithm ensures positivity in deconvolved pressure and de-rivative responses.

• Algorithm addresses pressure and rate uncertainties.• Algorithm allows the treatment of initial reservoir pressure and

rates as unknown in deconvolution.• Algorithm requires only two input parameters, namely � and

�—the relative error weight and the regularization parameter.The prescribed (default) values are typically sufficient. Theparameter � can be estimated directly from measured pressureand rate data available as the ratio of the “pressure error” to“rate error” and can be fixed at its default value, while theregularization parameter � is adjusted manually by powers often of �def (Eq. 11) computed from pressure data in deconvo-lution until a smooth enough response, interpretable withoutlosing its dominant features, is obtained.

• Assumption of a wellbore-storage unit-slope trend at/beforethe first node causes artifacts on the unit-rate derivative re-sponse near the first node for data sets with no storage orminimal storage effects, but these artifacts disappear after the


first log-cycle. For data with no storage or minimal storageeffects, the value of ez1 estimated at the first time node repre-sents the constant unit-rate drawdown pressure drop includingthe skin effect, not the value of constant unit-rate drawdownderivative response.

2. Our algorithm based on Levitan method:• Algorithm ensures positivity in deconvolved deriva-

tive responses.• Algorithm addresses pressure and rate uncertainties.• Algorithm allows the treatment of initial reservoir pressure and

rates as unknown in deconvolution.• Algorithm requires three input (control) parameters: p, q, and

c, where p and p represent the standard deviation of errors(“error band” or noise levels) for pressure and rate data and c

represents the “weight” for the curvature constraint. The val-ues of p and q to be used in the algorithm are determinedfrom our prior knowledge of noise levels in pressure and ratedata, and the curvature constraint weight is adjusted manually.As shown, if each measured pressure point in the pressure dataset has the same standard deviation of error p, and each mea-sured rate point in the rate data set has the same standarddeviation of error q, and if we ignore the differences in en-coding between the von Schroeter et al. and Levitan methods,the objective functions considered by these authors are iden-tical provided that we define ��(p/q)2 and ��(p/c)

2. Hence,only the ratios p/q and p/c are required, and as a result thenumber of control parameters is the same in both methods.

• Algorithm removes the assumption of unit-slope trend at/before the first node, and hence can reconstruct the unit-rateresponses correctly at the very early times even if the data donot exhibit wellbore-storage unit-slope line, provided that timevalue t1 for the first node is chosen less than or equal to theminimum elapsed time in the pressure signal. Use of L−2curvature constraint equations in the objective function (Eq.16) provides slightly better representation of the reconstructedunit-rate response than that of L−1 curvature constraint equations.

• There is no positivity constraint on the constant unit-rate draw-down pressure response at the first grid node [i.e., pu(t1)],which is treated as an unknown model parameter in the algo-rithm. Therefore, the resulting value may be negative. As seenin our Synthetic Example 2 and field data application, this cancause an artifact at the reconstructed unit-rate responses at thevery early times.

• When the algorithm is applied to individual buildup periods,the sensitivity of the model pressure with respect to pu(t1)depends only on the flow rate at the instant of shut-in.

• An improper selection of t1 value and of curvature constraintc, and/or measurement errors in pressure/rate and time syn-chronization errors in rate and pressure data, are possiblecauses of driving the unconstrained nonlinear minimizationalgorithm to a negative value for pu(t1).

3. Ilk et al. algorithm:• Algorithm offers the flexibility of modeling rates as continu-

ous measurements.• Algorithm handles the earliest part of the unit-rate derivative

response for the data with a skin effect (but minimal orno wellbore-storage effects) by adding additional “anchor”B-splines. Four additional B-splines added to the left of thefirst observed time point in the signal are sufficient for virtu-ally any scenario.

• Algorithm considers a linear weighted least-squares objectivefunction based on a diagonal weighting matrix to account forerrors only in pressure data. The weights in diagonal elementsof the matrix are defined as the reciprocal of the observedvalue of measured pressure drop (i.e., the measured pressuresare corrupted by a constant level of relative error). However, inview of the concept of maximum likelihood estimation, theweights should actually represent the inverse of standard de-viation of errors in data to have the correct assessment ofstatistics of the match, and hence using a weighting matrix asproposed by Ilk et al. may yield incorrect statistics (e.g., in-

correct confidence intervals), though the resulting estimates ofthe unit-rate response may be reasonable.

• Algorithm does not ensure positivity in deconvoled responses.• The regularization parameter � in the algorithm is adjusted

manually, and the regularization approach used by Ilk et al.tend to make the algorithm less tolerant to the errors in pres-sure and rate data, compared to the von Schroeter et al. andLevitan algorithms.

Nomenclature

a� coefficients in rate function, Eq. 43, L3/t, bbl/day [cm3/s]

b � coefficients in rate function, Eq. 43, or base oflogarithmically evenly distributed knots

Bi2(t) � second-degree B-spline starting at bi

Bi,int2 (t) � integral of 2nd degree B-spline starting at bi

ct � total compressibility, Lt2/m, 1/psi, [1/atm]c � vector of unknown B-spline coefficients, Eq. 27

Cw � wellbore-storage coefficient, L4t2/m, stb/psi [cm3/atm]C � design matrix for pressure match in Levitan’s

objective function, Eq. 16.C � design matrix for pressure match in von Schroeter

et al.’s objective function, Eq. 8D � matrix in curvature measuree � vector of size m with each component equal to 1

ez1 � exponential of response function, z1 at the first nodeE � objective function, Eqs. 8, 16, and 31E � objective function, Eq. 37h � thickness, L, ft, [cm]k � permeability, L2, md [Darcy ]k � vector in curvature measureK � index of the last knot for B-splinel � index of the first knot for B-spline

L � number of nodes for the deconvolved responseLxf � fracture half length, L, ft [cm]L1 � perpendicular distance to the first no-flow boundary,

L, ft [cm]L2 � perpendicular distance to the second no-flow

boundary, L, ft [cm]L−1 � Laplace inverse transform operator

M � number of pressure data pointsn � number of unknown coefficients cis, Eq. 34 or

number of B-splinesN � number of flow periods or segmentsp � pressure, m/Lt2, psi [atm]

p0 � initial pressure, m/Lt2, psi [atm]pu �unit-rate drawdown-pressure response, m/L3t3, psi

/(stb/d) [atm/(cm3/s)]p � vector of pressures, m/Lt2, psi [atm]

�p � pressure change, m/Lt2, psi [atm]�p � (vector of) pressure change, m/Lt2, psi [atm]

�p* � (vector of) augmented pressure changes, Eq. 36,m/Lt2, psi [atm]

q � flow rate, L3/t, stb/day [cm3/s]re � the distance from the well to the circular boundary,

L, ft [cm]rw � wellbore radius, L, ft [cm]

s � Laplace transform variable, 1/t, hour-1 [s-1]S � mechanical skin, dimensionlesst � time, t, hour [s]

t� � time, t, hour [s] (integration variable)t1 � time for the first node, t, hour [s]W � weighting matrixX � sensitivity matrix, Eq. 27


X* � augmented matrix, Eq. 36Xa

+ � pseudoinverse matrix of the augmented matrix, Eq. 39y � (vector of) estimated rates, Eq. 8, L3/t, stb/day [cm3/s]y � (vector of) estimated rates (Levitan algorithm), L3/t,

stb/day [cm3/s]z � response function defined by Eq. 6

z1 � value of response function at the first nodez � (vector of) response coefficients� � regularization parameter (Ilk et al. algorithm),

unit-less� � (vector of) absolute rate errors, L3/t, bbl/day [cm3/s]� � (vector of) absolute pressure data errors, m/Lt2,

psi [atm] � error bounds used in Eq. 16� � error levels, Eqs. 2 and 3, fraction � Heaviside function� � regularization parameter in Eq. 8 or interporosity

flow coefficient, m2/L2t4, psi2 [atm2]�def � default value for regularization parameter defined by

Eq. 10, m2/L2t4, psi2 [atm2]�def � default value for regularization parameter defined by

Eq. 11, m2/L2t4, psi2 [atm2]� � viscosity, m/Lt, cp [cp]� � relative error weight used in Eq. 8, m2/L8t2,

(psi.d/stb)2 [(atm.s/cm3)2]�def � default value for relative error weight defined by Eq.

9, m2/L8t2, (psi.d/stb)2 [(atm.s/cm3)2]� � standard deviation of errors� � variable of integration or logarithm of time� � porosity, fraction� � storativity ratio for the double porosity model,

dimensionless � data model predictions for curvature constraints, Eqs.

17 and 18� �2 � l2-norm of a vector

Subscripts

c � curvaturedef � default

m � measuredF � Frobenius normp � pressureq � rateu � unit-rate

Superscripts

b � buildupT � transpose of a matrix^ � estimated (from least squares)

— � Laplace transform–1 � inverse of a matrix

Acknowledgments

The first author thanks Drs. Thomas von Schroeter of ImperialCollege and Michael Levitan of BP for providing data to verify ourversions of their algorithms developed during the course of thiswork, and for providing their comments to our questions. We alsoare thankful to the anonymous referees for their careful reviewingof the manuscript and useful suggestions. The first author alsothanks Kappa for providing Saphir free for educational and re-search purposes for the Department of Petroleum and Natural GasEngineering at ITU.

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Mustafa Onur is a professor and head of the Department ofPetroleum and Natural Gas Engineering at Istanbul TechnicalUniversity, Turkey. His interests include well testing, pressure-transient analysis, formation testing, and geothermal reservoirengineering. He holds a BS degree from the Middle East Tech-

nical University in Turkey and MS and PhD degrees from theUniversity of Tulsa, all in petroleum engineering. He has servedon the editorial committees of SPEREE and SPE Journal. MuratÇınar is a PhD student in the Department of Energy ResourcesEngineering at Stanford University. His research interest in-cludes well-test analysis, geothermal engineering and in-situcombustion. He holds a BS degree and MS degrees from Istan-bul Technical University, Turkey, both in petroleum engineering.Dilhan Ilk is a PhD student in the Department of PetroleumEngineering at Texas A&M University. His research interest in-cludes well-test/production-data analysis, reservoir engineer-ing, and numerical analysis. He holds a BS degree from IstanbulTechnical University, Turkey, and an MS degree from TexasA&M University, both in petroleum engineering. Peter P. Valkóis a professor of petroleum engineering at Texas A&M Univer-sity. Previously, he taught at academic institutions in Austriaand Hungary and worked for the Hungarian Oil Co. (MOL). Hisresearch interests include design and evaluation of hydraulicfracture stimulation treatments, rheology of fracturing fluids,and performance of advanced and stimulated wells. Valkóhas coauthored three books and published several dozen re-search papers. He holds BS and MS degrees from VeszprémUniversity, Hungary, and a PhD degree from the Institute ofCatalysis, Novosibirsk, Russia. Tom Blasingame is a professor ofpetroleum engineering at Texas A&M University, where heworks in the areas of reservoir engineering and analysis of res-ervoir performance. He holds BS, MS, and PhD degrees fromTexas A&M University, all in petroleum engineering. Blasingameis a Distinguished Member of SPE, and received both the 2005SPE Distinguished Service Award and 2006 Lester C. UrenAward. He has chaired Forums, Applied Technology Work-shops (ATWs), and committees. Peter Hegeman is a projectmanager and engineering advisor in the Reservoir Samplingand Pressure Discipline of Schlumberger Sugar Land ProductCenter. His interests include well testing, pressure-transientanalysis, formation testing, and production systems analysis. Heholds BS and MS degrees in petroleum engineering from thePennsylvania State University.


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