Research ArticleExplicit Deconvolution of Well Test Data Dominated byWellbore Storage

K Razminia1 A Hashemi1 A Razminia2 and D Baleanu345

1 Department of Petroleum Engineering Petroleum University of Technology Ahwaz Iran2Dynamical Systems amp Control (DSC) Research Laboratory Department of Electrical Engineering School of EngineeringPersian Gulf University P O Box 75169 Bushehr Iran

3Department of Chemical and Materials Engineering Faculty of Engineering King Abdulaziz University Jeddah Saudi Arabia4Department of Mathematics and Computer Sciences Faculty of Arts and Sciences Cankaya University 06530 Ankara Turkey5 Institute of Space Sciences Magurele-Bucharest Romania

Correspondence should be addressed to A Razminia arazminiagmailcom

Received 27 December 2013 Accepted 28 February 2014 Published 3 April 2014

Academic Editor Abdon Atangana

Copyright copy 2014 K Razminia et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper addresses some methods for interpretation of oil and gas well test data distorted by wellbore storage effects Using thesetechniques we can deconvolve pressure and rate data from drawdown and buildup tests dominated by wellbore storage Some ofthese methods have the advantage of deconvolving the pressure data without rate measurement The two important methods thatare applied in this study are an explicit deconvolution method and a modification of material balance deconvolution method Incases with no rate measurements we use a blind deconvolution method to restore the pressure response free of wellbore storageeffects Our techniques detect the afterflowunloading rate function with explicit deconvolution of the observed pressure data Thepresented techniques can unveil the early time behavior of a reservoir system masked by wellbore storage effects and thus providepowerful tools to improve pressure transient test interpretation Eachmethod has been validated using both synthetic data and fieldcases and each method should be considered valid for practical applications

1 Introduction

In conventional well test analysis the pressure responseto constant rate production is essential information thatpresents the distinct characteristics for a specific type ofreservoir system However in many cases it is difficultto acquire sufficient constant rate pressure response dataThe recorded early time pressure data are often hidden bywellbore storage (variable sandface rates) In some casesouter boundary effects may appear before wellbore storageeffects disappear Therefore it is often imperative to restorethe early time pressure response in the absence of wellborestorage effects to provide a confident well test interpretation[1ndash4]

A considerable amount of work has been done onmultirate (variable) tests during the last 50 years Howeverthese are basically sequential constant rate drawdowns onlytransient pressure is measured and rate is assumed to be

constant during each drawdown test [5] All these worksdeal with the direct problem In other words the constantrate solution (the influence or the unit response function)is convolved (superimposed) with the time dependent innerboundary condition to obtain solutions to the diffusivityequation This process is called convolution The process ofdetermining the influence function is called deconvolution[6ndash8]

Based on [1] deconvolution is a technique used to convertmeasured pressure and sandface rate data into the constantrate pressure response of the reservoir In other words decon-volution provides the pressure response of a wellreservoirsystem free of wellbore storage effects as if the reservoir isproducing at a constant rate Once the deconvolved pressureis obtained conventional interpretationmethods can be usedfor reservoir system identification and parameter estimation

The purpose of the well test interpretation as stated byGringarten et al [9] is to identify the system and determine

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 912395 12 pageshttpdxdoiorg1011552014912395

2 Abstract and Applied Analysis

its governing parameters frommeasured data in the wellboreand at the wellhead This problem is known as the inverseproblem The solution of the inverse problem usually is notunique Gringarten et al [9] pointed out that if the numberand the rage measurements increased the nonuniqueness ofthe inverse problem will be reduced Thus combining sand-face flow rate with pressure measurements will enhance theconventional (including type curve) well test interpretationmethods On the other hand mathematically deconvolutionis also a highly unstable inverse problem because smallerrors in the data can result in large uncertainties in thedeconvolution solution [10 11]

Unfortunately standard deconvolution techniques re-quire accurate measurements of flow rate and pressure atdownhole (or sandface) conditions While accurate pressuremeasurements are commonplace the measurement of sand-face flow rates is rare essentially nonexistent in practiceAs such the deconvolution of wellbore storage distortedpressure test data is problematic In theory this process ispossible but in practice without accurate measurements offlow rates this process cannot be employed [12 13]

In the past 50 years a variety of deconvolution tech-niques have been proposed in petroleum engineering Russell[14] used the approximate direct method to correct thepressure transient data distorted by wellbore storage intothe equivalent pressure function for the constant rate caseGladfelter et al [15] and Fetkovich and Vienot [16] used therate normalization techniques to correct for thewellbore stor-age effectsThese rate normalizationmethods were successfulin some cases So Johnston [17] provided thematerial balancedeconvolution that is a practical approach for the analysis ofpressure transient data distorted by wellbore storage effects

Essentially rate normalization techniques and materialbalance deconvolution are restricted when the lack of ratemeasurement exists van Everdingen [18] and Hurst [19]demonstrated empirically that the sandface flow rate profilecan be modeled approximately using an exponential relationfor the duration of wellbore storage distortion during a pres-sure transient testThe vanEverdingen andHurst exponentialrate model is given in dimensionless form as

119902119863(119905119863) = 1 minus 119890

minus120573119905119863 (1)

Further they showed that the rate-time relationship duringafterflow (for a pressure buildup test) or unloading (in apressure drawdown test) is a function of the pressure dropchange with respect to time and a relatively constant wellborestorage coefficient

Based on a material balance in the wellbore the sandfaceflow rate can be calculated by the following relation given indimensionless form

119902119863(119905119863) = 1 minus 119862

119863

119889119901119908119863

119889119905119863

(2)

where it can be noted that in the development of wellborestorage models solutions (eg type curves) a constant well-bore storage coefficient is always assumed

Equations (1) and (2) laid the groundwork for 120573-deconvolution Joseph and Koederitz [20] and Kuchuk [21]

applied 120573-deconvolution for the analysis of wellbore stor-age distorted pressure transient data The 120573-deconvolutionformula which computes the undistorted pressure dropfunction directly from the wellbore storage affected data isgiven as

119901119904119863

(119905119863) =

1

120573

119889119901119908119863

(119905119863)

119889119905119863

+ 119901119908119863

(119905119863) (3)

It should be noted that (3) is only validwhen the sandface flowrate profile follows an exponential trend as prescribed by (1)[12] In thisworkwe develop an approach that overcomes thisrestriction by treating the 120573 quantity as a variable in generalrather than a constant

The obvious advantage of 120573-deconvolution is that thewellbore storage effects are eliminated and the restoredpressure is obtained using only the given pressure data

This paper reports a modification of Russell method(Appendix A) a modification of 120573-deconvolution and anexplicit deconvolution formula to restore the pressure andrate data and a review on material balance deconvolutionis also presented to deconvolve the pressure and rate dataafter the explicit deconvolution method Then a generalapproach to analyze the deconvolution problem is provided(Appendix B) In verification examples section we show thepower and practical applicability of our approaches withvarious synthetic and field case examples for oil and gas wells

2 Main Results

21 120573-Deconvolution and Its Modification van EverdingenandHurst [22] presented the dimensionless wellbore pressurefor a continuously varying flow rate as

119901119908119863

(119905119863) = 119902119863 (0) 119901119904119863 (119905119863) + int

119905119863

0

1199011015840

119863(120591) 119901119904119863 (119905119863 minus 120591) 119889120591

(4)

An alternative form to (4) can be obtained by integration byparts as

119901119908119863

(119905119863) = 119902119863(119905119863) 119901119904119863 (0) + int

119905119863

0

119902119863 (120591) 119901

1015840

119904119863(119905119863minus 120591) 119889120591

(5)

where 119901119908119863

(119905119863) = (119896ℎ1412119902119861120583)(119901

119894minus 119901119908119891(119905)) 119905

119863=

(000026371198961199051206011205831198881199051199032

119908) 119901119904119863

= 119901119863+ 119878 119901

119908119863(119905119863) is dimen-

sionless wellbore pressure with wellbore storage and skineffects 119901

119863(119905119863) is dimensionless sandface pressure for the

constant rate case without wellbore storage and skin effects 119878is steady state skin factor 1199011015840

119904119863(119905119863) = 119889119901

119904119863(119905119863)119889119905119863 119902119863(119905119863) =

119902119904119891(119905119863)119902119903 1199021015840119863(119905119863) = 119889119902

119863(119905119863)119889119905119863 and 119902

119903is reference

flow rate if the stabilized constant rate is available then 119902119903

should be replaced by 119902119877 119902119904119891(119905) is variable sandface flow rate

(flowmeter reading) and 119902119863(119905119863) is dimensionless sandface

rateIt should be emphasized that (4) and (5) can be applied

for many reservoir engineering problems The linearity ofdiffusivity equation allows us to use (4) and (5) for fractured

Abstract and Applied Analysis 3

layered anisotropic and heterogeneous systems as long as thefluid in the reservoir is single phaseThese equations can alsobe applied to both drawdown and buildup tests if the initialconditions are known For a reservoir with an initial constantand uniform pressure distribution (119901

119863(0) = 0) (4) and (5)

can be expressed as

119901119908119863

(119905119863) = int

119905119863

0

1199021015840

119863(120591) 119901119904119863 (119905119863 minus 120591) 119889120591 (6)

119901119908119863

(119905119863) = 119878119902

119863(119905119863) + int

119905119863

0

119902119863 (120591) 119901

1015840

119904119863(119905119863minus 120591) 119889120591 (7)

Furthermore (6) and (7) also can be expressed as

119901119908119863

(119905119863) = int

119905119863

0

1199021015840

119863(119905119863minus 120591) 119901

119904119863 (120591) 119889120591 (8)

119901119908119863

(119905119863) = 119878119902

119863(119905119863) + int

119905119863

0

119902119863(119905119863minus 120591) 119901

1015840

119904119863(120591) 119889120591 (9)

In (6) and (8) it is assumed that 1199021015840119863(119905119863) exists If 119902

119863(119905119863) is

constant then (7) and (9) must be used Equations (6) and(8) are known as a Volterra integral equation of first kind andthe convolution type

Taking the Laplace transform of (6) yields

119901119908119863

(119904) = 119904119902119863(119904) 119901119904119863

(119904) (10)

Rearranging (10) for the equivalent constant rate pressuredrop function 119901

119904119863(119904) we obtain

119901119904119863

(119904) =119901119908119863

(119904)

119904119902119863(119904)

(11)

By taking the Laplace transform of (1) one can easily obtain

119902119863(119904) =

1

119904minus

1

119904 + 120573 (12)

Substituting (12) into (11) and then taking the inverse Laplacetransform of (11) yield the beta deconvolution formula

119901119904119863

(119905119863) =

1

1205731199011015840

119908119863(119905119863) + 119901119908119863

(119905119863) (13)

where

1199011015840

119908119863(119905119863) =

119889119901119908119863

119889119905119863

(14)

where we note that (13) is specifically valid only for theexponential sandface flow rate profile given by (1) This maypresent a serious limitation in terms of practical applicationof the 120573-deconvolutionmethod To overcome this restrictionwe propose an idea It would be assumed that120573 is not constantin general If 119902

119863(119905119863)data are given the procedure is as follows

119902119863(119905119863) = 1 minus 119890

minus120573(119905119863)119905119863 (15)

Rearranging (15) yields

1 minus 119902119863(119905119863) = 119890minus120573(119905119863)119905119863 (16)

then

minus120573 (119905119863) 119905119863= ln (1 minus 119902

119863(119905119863)) (17)

so

120573 (119905119863) = minus

ln (1 minus 119902119863(119905119863))

119905119863

(18)

Substituting of (18) into (13) instead of 120573 term yields themodified 120573-deconvolution

119901119904119863

(119905119863) =

1

120573 (119905119863)1199011015840

119908119863(119905119863) + 119901119908119863

(119905119863) (19)

Plot of (19) versus (int119905119863

0119902119863(120591)119889120591119902

119863(119905119863)) and (int

119905119863

0(1 minus

119902119863(120591))1198891205911 minus 119902

119863(119905119863)) for drawdown and buildup tests

respectively should be considered more accurate to use as apractical tool for field applications As mentioned before thevariable 120573 quantity (rather than a constant) shows the generaland more accurate behavior of 119902

119863(119905119863) So accuracy of (19) is

also more than (13) It can be emphasized that (13) and (19)are applicable in the wellbore storage duration and for thelater times the given pressure data are correct However theresults of (19) show that this equation is not bad for the latertimes

22 Explicit Deconvolution Formula A few types of approx-imation functions can be used to approximate 119902

119863(119905119863) An

approximate proof is given here to show that the exponentialfunction gives a good representation of 119902

119863(119905119863) By setting

119902119863(119905119863) = 1 minus 119891 (119905

119863) (20)

119891 (0) = 1 (21)

Equation (21) is an initial condition for (20)Taking the time derivative of (20) gives

1199021015840

119863(119905119863) = minus119891

1015840(119905119863) (22)

Taking the time derivative of (2) gives

1199021015840

119863(119905119863) = minus119862

11986311990110158401015840

119908119863 (23)

From (2) and (20) we have

119891 (119905119863) = 1198621198631199011015840

119908119863 (24)

And from (22) and (23) we have

1198911015840(119905119863) = 11986211986311990110158401015840

119908119863 (25)

After dividing right and left hand sides of (25) by (24) theresult is

1198911015840(119905119863)

119891 (119905119863)=11990110158401015840

119908119863(119905119863)

1199011015840119908119863

(119905119863) (26)

Integrating both sides of (26) with respect to 119905119863from 0 to 119905

119863

yields

int

119905119863

0

1198911015840(120591)

119891 (120591)119889120591 = int

119905119863

0

11990110158401015840

119908119863(120591)

1199011015840119908119863

(120591)119889120591 (27)

4 Abstract and Applied Analysis

Using elementary calculus yields

ln (1003816100381610038161003816119891(120591)

1003816100381610038161003816)1003816100381610038161003816119905119863

0= ln (1003816100381610038161003816119891 (119905

119863)1003816100381610038161003816) minus ln (1) = int

119905119863

0

11990110158401015840

119908119863(120591)

1199011015840119908119863

(120591)119889120591

(28)

So

119891 (119905119863) = 119890120572(119905119863) (29)

where

120572 (119905119863) = int

119905119863

0

11990110158401015840

119908119863(120591)

1199011015840119908119863

(120591)119889120591 (30)

Now the explicit deconvolution of data based on (20) can bewritten as follows

119901119904119863

(119905119863) = minus

1

120572 (119905119863) 119905119863

1199011015840

119908119863+ 119901119908119863

(119905119863) (31)

or

119901119904119863

(119905119863) = minus

119905119863

120572 (119905119863)1199011015840

119908119863+ 119901119908119863

(119905119863) (32)

that is similar to (19) The other similarity between (19) and(32) is that the plot of (32) versus (int119905119863

0119902119863(120591)119889120591119902

119863(119905119863)) and

(int119905119863

0(1 minus 119902

119863(120591))119889120591(1 minus 119902

119863(119905119863))) for drawdown and buildup

tests respectively yields the more accurate results and isusable for practical applications The applicability of (32) isalso for duration of wellbore storage effects And for the othertimes the given pressure data are correct However using thismethod for the later times yields not bad results

3 Material Balance Deconvolution

Material balance deconvolution is an extension of the ratenormalization method Johnston [17] defines a new 119909-axisplotting function (material balance time) that provides anapproximate deconvolution of the variable rate pressuretransient problem There are numerous assumptions associ-ated with the material balance deconvolution methods oneof the most widely accepted assumptions is that the rateprofilemust change smoothly andmonotonically In practicalterms this condition should be met for the wellbore storageproblem

The general form of material balance deconvolution isprovided for the pressure drawdown case in terms of thematerial balance time function and the rate-normalizedpressure drop function The material balance time functionis given as

119905119898119887

=119873119901

119902119863 (119905)

(33)

where

119873119901= int

119905

0

119902119863 (120591) 119889120591 (34)

The rate-normalized pressure drop function is given by

Δ119901119908119891 (119905)

119902119863 (119905)

=119901119894minus 119901119908119891 (119905)

119902119863 (119905)

(35)

The material balance time function for the pressure buildupcase is given as

119905119898119887

=119873119901

1 minus 119902119863 (Δ119905)

(36)

where

119873119901= int

Δ119905

0

(1 minus 119902119863 (120591)) 119889120591 (37)

The rate-normalized pressure drop function for the pressurebuildup case is given as

Δ119901119908119904 (Δ119905)

1 minus 119902119863 (Δ119905)

=119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)

1 minus 119902119863 (Δ119905)

(38)

4 Methodology

In this work several methods are provided to restore thepressure datawithout any sandface flow rate informationThevalues of sandface flow rate data for all methods that requirethese data may be obtained from (20) We evaluated a veryold correctionmethod by Russell [14] However we modifiedthis method by least square method but it was found thatthis method is to be unacceptable for all applications Thederivation of this method and its modification is given inAppendix A

The 120573-deconvolution and its modification are anothertechnique that is mentioned in this study The formulationof this method is presented in Section 21 The other explicitdeconvolutionmethod is an approach similar to themodified120573-deconvolution that uses only the pressure data for decon-volution of pressure and rate data The formulation of thismethod was derived in Section 22 The 120573-deconvolution itsmodification and explicit deconvolution formula can be usedfor practical applications The applicability of these methodsis for duration of wellbore storage effects However usingthese approaches for the other times yields not bad results

The other technique is material balance deconvolutionmethod [17] that has sufficiently accurate resultsThismethodcan also be used as a good approach for practical applicationsThe formulas for this method were given in Section 3

In Appendix B deconvolution is expressed as an inverseproblem that is analyzed by a tricky idea Based on thisidea an interesting proof is given for rate normalizationmethod and material balance deconvolution that is a goodapproach to find the errors and limitations of these methodsand overcome them

Abstract and Applied Analysis 5

10minus3 10minus2 10minus1 100 101 102 1030

500

1000

1500

2000

2500

3000

3500

4000

Time (hr)

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolution methodd(Dps)d(ln(t)) material balance deconvolution methodDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 1 Deconvolution for a drawdown test in a homogeneousreservoir

From the above-mentioned methods we use the fol-lowing two techniques to show the verification of theseapproaches

Explicit Deconvolution Formula The final result of explicitdeconvolution method is given as follows

119901119904119863

(119905119863) = minus

119905119863

120572 (119905119863)1199011015840

119908119863+ 119901119908119863

(119905119863) (39)

Plotting of this term versus (int119905119863

0119902119863(120591)119889120591119902

119863(119905119863)) and

(int119905119863

0(1 minus 119902

119863(120591))119889120591(1 minus 119902

119863(119905119863))) for drawdown and buildup

tests respectively yields more accurate resultsThis approachperforms well in field applications

Material Balance Deconvolution The material balance pres-sure drop function for the pressure drawdown test is givenas

Δ119901119908119891 (119905)

119902119863 (119905)

=119901119894minus 119901119908119891 (119905)

119902119863 (119905)

(40)

and the material balance time function for the pressuredrawdown test is given as

119905119898119887

=int119905119863

0119902119863 (120591) 119889120591

119902119863 (119905)

(41)

For the buildup test the material balance pressure dropfunction can be written as

Δ119901119908119904 (Δ119905)

1 minus 119902119863 (Δ119905)

=119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)

1 minus 119902119863 (Δ119905)

(42)

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

104

103

102

101Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolution methodd(Dps)d(ln(t)) material balance deconvolution methodDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Figure 2 Deconvolution for a drawdown test in a homogeneousreservoir

And the material balance time function can be written as

119905119898119887

=int119905119863

0(1 minus 119902

119863 (120591)) 119889120591

1 minus 119902119863 (Δ119905)

(43)

This technique is a good approach for practical applications

5 Verification Examples

We first use few synthetic cases for different reser-voirwellbore systems to validate the applicability of explicitdeconvolution formula and material balance deconvolutionmethod Then the field applications of these techniqueswill be shown by two real cases The unloading rates arecalculated with (20)

The first case is a synthetic test derived from a test designin a vertical oil wellThe reservoir is homogenous and infiniteacting with a constant wellbore storage The wellbore storagecoefficient is 119862

119904= 0015 bblpsi The pressure data are

obtained from the cylindrical source solution (test design)We used the explicit deconvolution formula and material

balance deconvolution method to deconvolve the pressureand unloading rate distorting by wellbore storage effectsTheresults are presented in Figures 1 and 2 for the semilog andlog-log plots in the case of drawdown test and Figures 3and 4 for the semilog and log-log plots in the case of builduptest Besides the deconvolved pressures and their logarithmicderivatives pressure responses with and without wellborestorage are also plotted on the figures for comparisonThrough the deconvolution process we have successfullyremoved the wellbore storage effects so that radial flow canbe identified at early test times The deconvolved pressuresand their logarithmic derivatives are almost identical to

6 Abstract and Applied Analysis

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dp wellbore storage

Dps explicit deconvolution method(ln

Dps material balance deconvolution

Dps no wellbore storage

ws

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpws

d(Dp )d(ln )) wellbore storagews (HTR

d(Dps)d )) explicit deconvolution method(HTR

d(Dps)d(ln )) no wellbore storage(HTR)

d(Dps)d(ln )) material balance deconvolution (HTR

Figure 3 Deconvolution for a buildup test in a homogeneousreservoir

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

104

103

102

101

ws

ws

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpws

Dp wellbore storage

Dps explicit deconvolution method

Dps material balance deconvolution

Dps no wellbore storage

d(Dp )d(ln )) wellbore storagews (HTR

d(Dps)d )) explicit deconvolution method(HTR

d(Dps)d(ln )) no wellbore storage(HTR)

d(Dps)d(ln )) material balance deconvolution (HTR

(ln

Figure 4 Deconvolution for a buildup test in a homogeneousreservoir

the analytic solutions (without wellbore storage) whichenables us to estimate reservoir parameters accurately

The second case is an oil well drawdown test in adual porosity reservoir The storativity ratio (120596) and inter-porosity flow coefficient (120582) are assumed to be 01 and10minus6 respectively The constant wellbore storage coefficient

is 119862119904= 015 bblpsi The dimensionless wellbore pressure

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

600

500

400

300

200

100

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 5 Deconvolution for a drawdown test in a dual porosityreservoir

data were obtained from the cylindrical source solution withpseudosteady-state interporosity flow (test design) It is notpossible to identify the dual porosity behavior characteristicof the reservoir from wellbore pressure data because thisbehavior has been masked completely by wellbore storage asindicated in Figures 5 and 6

We deconvolved the pressure and unloading rate andpresent the results in Figures 5 and 6 The deconvolvedresults are reasonably consistent with the analytical solutionAfter deconvolution the dual porosity behavior can beidentified easily from the valley in the logarithmic derivativeof pressure If wellbore storage effects were not removed atearly times it easily could have incorrectly identified thereservoir system as infinite actingThe estimation of reservoirparameters can be obtained now from deconvolved pressure

The third case is an oil well drawdown test in a reservoirwith a sealing fault 250 ft from the well Synthetic pressureresponses were generated by simulation (test design) with aconstant wellbore storage coefficient of 01 bblpsi As we cansee in Figures 7 and 8 the early time infinite acting reservoirbehavior has been masked completely by wellbore storage

To recover the hidden reservoir behavior we performeddeconvolution on the pressure change to remove the wellborestorage effect As in Figures 7 and 8 the deconvolved pressurechange and logarithmic derivative data are almost identicalto the responses without wellbore storageThe deconvolutionresults clearly indicate parallel horizontal lines for the loga-rithmic derivative However without wellbore storage effectselimination this maymislead to identify the reservoir systemas infinite acting So this allows us to correctly identify thereservoir system having a sealing fault

The fourth case is a field example that is taken from theliterature [23] The basic reservoir and fluid properties are

Abstract and Applied Analysis 7

Table 1 Reservoir rock and fluid properties for the first field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119902

119900(STBD) 119861

119900(RBSTB) 119888

119905(psiminus1) 120583

119900(cp) 120601

1533 107 029 174 106 42 times 10minus6 25 025

Table 2 Reservoir rock and fluid properties for the second field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119879 (∘F) 119901 (psi) 119902

119900(STBD) 119902

119892(MscfD) 119888

119891(psiminus1) 119879

119889119901(∘F) 119901

119889119901(psi) ∘API 120574

119892

143111 100 0354 2795 7322 104 13925 3 times 10minus6 212 5445 45375 068

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

10minus1

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 6 Deconvolution for a drawdown test in a dual porosityreservoir

shown in Table 1 In this case we provide the deconvolvedpressure data using the explicit deconvolution formula andthe material balance deconvolution method that is presentedin this work The data are taken from a pressure buildup testand can be reasonably used as field data The deconvolutionresults are shown in Figure 9 in semilog plot As it isillustrated from Figure 9 the corrected pressure provides aclear straight line starting as early as 15 hours in contrast tothe vague line from the uncorrected pressure response

And the last (field) example is a gas condensatewell buildup test following a variable rate productionhistory The duration of last pressure buildup test is361556 hours (from 1889 to 2250556 hours) The basicreservoir and fluid properties for the last drawdown testare shown in Table 2 However the properties of otherproduction tests are not shown here To use our newtechniques effectively for gas wells we use normalizedpseudopressure and pseudotime functions as defined byMeunier et al [24]The final results of explicit deconvolutionformula and material balance deconvolution methodshow two different slopes (Figure 10) where the earlytimes slope is greater than the last one This identifiesthe radial composite behavior of the reservoir With

the corrected pressures the reservoir parameters can also beestimated

6 Conclusions

In this work we have provided some explicit deconvolutionmethods for restoration of constant rate pressure responsesfrom measured pressure data dominated by wellbore storageeffects without sandface rate measurements Based on ourinvestigations we can state the following specific conclusionsrelated to these techniques

(1) The results obtained from the synthetic and fieldexamples show the practical and computational effi-ciency of our deconvolution methods

(2) Modification of 120573-deconvolution is presented wherethe beta quantity is variable rather than a constant

(3) For situations in which there are no downhole ratemeasurements the explicit deconvolution methods(blind deconvolution approaches) can be used toderive the downhole rate function from the pressuremeasurements and restore the reservoir pressureresponse

(4) We can unveil the early time behavior of a reservoirsystemmasked by wellbore storage effects using thesenew approaches Wellbore geometries and reservoirtypes can be identified from this restored pressuredata Furthermore the conventional methods canbe used to analyze this restored formation pressureto estimate formation parameters So the presentedtechniques can be used as the practical tools toimprove pressure transient test interpretation

(5) Deconvolution in well test analysis is often expressedas an inverse problem This approach can be used asan idea to justify the rate normalization method andthe material balance deconvolution method

Appendices

A Russel Method and Its Modification

A method for correcting pressure buildup pressure data freeof wellbore storage effects at early times has been developedby Russell [14] In this method it is not necessary to measurethe rate of influx after closing in at the surface Instead oneuses a theoretical equation which gives the form that thebottom hole pressure should have fluid accumulates in the

8 Abstract and Applied Analysis

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

250

200

150

100

50

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 7 Deconvolution for a drawdown test in a reservoir with asealing fault

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 8 Deconvolution for a drawdown test in a reservoir with asealing fault

wellbore during buildup This leads to the result that oneshould plot

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

(A1)

versus log (Δ119905) in analyzing pressure buildup data during theearly fill-up period The denominator makes a correction for

104103102101100

Horner time ratio

800

700

600

500

400

300

200

100

0

Dpws

(psi)

Dpws wellbore storageDps explicit deconvolution methodDps material balance deconvolution

Figure 9 Restored pressures for a buildup test in a homogeneousreservoir

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dpa

(psi)

Dpaw wellbore storage explicit deconvolution method material balance deconvolution

DpasDpas

Figure 10 Restored pressures for a buildup test in a radial compositereservoir

the gradually decreasing flow into the wellbore The quantity1198622is obtained by trial and error as the value which makes the

curve straight at early times After obtaining the straight linesection the rest of the analysis is the same as for any otherpressure buildup This method has the advantage or requiresno additional data over that taken routinely during a shut intest

Due to the time consuming and unconfident of trial anderror to determine 119862

2in Russell method we modify this

method Russellrsquos wellbore storage correction is given as

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

= 119891 (Δ119905 = 1 hr) + 119898119904119897log (Δ119905)

(A2)

Abstract and Applied Analysis 9

Defining the following function and using least squarestechnique obtain the final result

119864 (1198622 119891119898)

=

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891(1198622) minus 119898(119862

2) log(Δ119905

119894))

2

(A3)

Tominimize the function119864 it can differentiate119864with respectto 1198622 119891 and 119898 and setting the results equal to zero After

differentiation it reduces to

120597119864

1205971198622

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894) ) = 0

(A4)

120597119864

120597119891

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622) minus 119898 (119862

2) log (Δ119905

119894))

= 0

(A5)120597119864

120597119898

= minus2

119873

sum

119894=1

log (Δ119905119894) (

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus 119898 (1198622) log (Δ119905

119894)) = 0

(A6)

From (A4) we define the following function

119883(1198622)

=

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times(Δ119901119908119904119894

(1minus (11198622Δ119905119894))minus119891 (119862

2) minus119898 (119862

2) log (Δ119905

119894))

= 0

(A7)

The roots of (A7) can be found via the following algorithm

1198622(119894+1)

= 1198622(119894)

minus119883 (1198622)

1198831015840 (1198622) (A8)

where

119883(1198622) =

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894))

1198831015840(1198622) =

119873

sum

119894=1

(minus2Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3)

times (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2

times [Δ119901119908119904119894

(1 minus (11198622Δ119905))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894)])

+ (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (minusΔ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3minus 1198911015840(1198622)

minus1198981015840(1198622) log (Δ119905

119894))

(A9)

where the unknown functions are given by the followingrelations

119891 (1198622) =

120573119906 (1198622) minus 120572V (119862

2)

1205732 minus 119873120572(A10)

whose derivative is

1198911015840(1198622) =

1205731199061015840(1198622) minus 120572V1015840 (119862

2)

1205732 minus 119873120572

119898 (1198622) =

119873119906 (1198622) minus 120573V (119862

2)

119873120572 minus 1205732

(A11)

where its derivative is

1198981015840(1198622) =

1198731199061015840(1198622) minus 120573V1015840 (119862

2)

119873120572 minus 1205732

119906 (1198622) =

119873

sum

119894=1

log (Δ119905119894)

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A12)

10 Abstract and Applied Analysis

and after differentiation it yields

1199061015840(1198622) = minus

119873

sum

119894=1

log (Δ119905

119894)

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2

V (1198622) =

119873

sum

119894=1

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A13)

which has the following derivative

V1015840 (1198622) = minus

119873

sum

119894=1

1

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2 (A14)

finally the values of 120572 and 120573 parameters are given by thefollowing formulas

120572 =

119873

sum

119894=1

(log(Δ119905119894))2

120573 =

119873

sum

119894=1

log (Δ119905119894)

(A15)

B General Formula for Inverse Problem

The convolution of (6) yields

119901119904119863

(119905119863) = int

119905119863

0

119870 (120591) 119901119908119863 (119905119863 minus 120591) 119889120591

= int

119905119863

0

119870(119905119863minus 120591) 119901

119908119863 (120591) 119889120591

(B1)

where

119870(119905119863) = 119871minus1(

1

119904119902 (119904)) (B2)

where 119871minus1 is the inverse Lapalce transform operator 119870(119905119863)

can be computed either from the Laplace transforms of119902119863(119905119863) data a curve-fitted equation of 119902

119863(119905119863) data or using

another technique with a relation between119870(119905119863) and 119902

119863(119905119863)

However it would be time consuming to invert all the 119902119863(119905119863)

data in Laplace space and transform it back to real space inaccordance with (B2) Once an approximation is obtained itwill be easy to compute119870(119905

119863) and integrate (B1) to determine

119901119904119863(119905119863) However finding some approximation functions for

119902119863(119905119863) data may be difficult We provide an approach to

overcome this problemThis idea is based on finding anotherrelation between 119870(119905

119863) and 119902

119863(119905119863) that is

ℓ (1) =1

119904=

1

119904119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B3)or

1 = int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591 = int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591

(B4)

The other relations that can be used are

ℓ (119905119863) =

1

1199042=

1

1199042119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B5)

or

119905119863= int

119905119863

0

119871 (120591) 119902119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

119863(120591) 119889120591

(B6)

ℓ (1) =1

119904=

1

1199042119902119863(119904)

119904119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591)

(B7)

or

1 = int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591

(B8)

where

119871 (119905119863) = ℓminus1(

1

1199042119902 (119904)) (B9)

Equation (B4) can be used simultaneously with (B1) toanalyze the deconvolution problem

Equation (11) can be written as

119901119904119863

(119904) = 119904119901119908119863

(119904)1

1199042119902119863(119904)

(B10)

or

119901119904119863

(119905119863) = int

119905119863

0

119871 (120591) 1199011015840

119908119863(119905119863minus 120591) 119889120591

= int

119905119863

0

119871 (119905119863minus 120591) 119901

1015840

119908119863(120591) 119889120591

(B11)

The applications of (B6) and (B8) are with (B11)For justifying the rate normalizationmethod andmaterial

balance deconvolution it can use (B1) (B4) ((B11) can alsobe used with (B6) or (B8) for this goal) and generalizedmean value theorem for integrals This theorem is expressedas follows

Theorem 1 Let 119891 and 119892 be continuous on [119886 119887] and supposethat 119892 does not change sign on [119886 119887] Then there is a point 119888 in[119886 119887] such that

int

119887

119886

119891 (119905) 119892 (119905) 119889119905 = 119891 (119888) int

119887

119886

119892 (119905) 119889119905 (B12)

Abstract and Applied Analysis 11

By Theorem 1 there are some values 1198881and 1198882between 0

and 119905119863such that

119901119904119863

(119905119863) = 119901119908119863

(1198881) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B13)

1 = 119902119863(1198882) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B14)

By setting 1198881

= 1198882

= 119905119863and dividing the right and left

side of (B13) respectively the rate normalization techniqueis developed However the values 119888

1and 1198882are not equal

to each other and are not equal to 119905119863

in general Thisis the reason of error in rate normalization and materialbalance deconvolution methods 119888

1and 1198882are less than 119905

119863in

these methods for the duration of wellbore storage distortionduring a pressure transient test Since these conditions areapproximately satisfied in material balance deconvolutionthe results of material balance are more accurate and moreconfident than rate normalization

Nomenclature

API API gravity ∘API119861119900 Oil formation volume factor RBSTB

119888119905 Total isothermal compressibility factor

psiminus1

1198622 Arbitrary constant hr

minus1

119862119863 Wellbore storage constant dimensionless

119862119904 Wellbore storage coefficient bblpsi

ℎ Formation thickness ftHTR Horner time ratio dimensionless119896 Formation permeability md119870 Kernel of the convolution integral119871 Kernel of the convolution integral119873119901 Cumulative oil production bbl

119901 Pressure psi119901119886 Normalized pseudopressure psi

119901 Average reservoir pressure psi119901119863 Formation pressure dimensionless

119901119889119901 Dewpoint pressure psi

119901119894 Initial reservoir pressure psi

119901119904119863 Formation pressure including skin

dimensionless119901119904 Formation pressure including skin psi

119901119908119863

Pressure draw down dimensionless119901119908119891 Flowing bottom hole pressure psi

119901119908119904 Shut-in bottom hole pressure psi

119902119863 Sandface flow rate dimensionless

119902119892 Gas flow rate MscfD

119902119900 Oil flow rate STBD

119903119908 Wellbore radius ft

119904 Laplace variable119878 Skin factor119905 Time hr119905119863 Time dimensionless

119905119901 Production time hr

119879 Reservoir temperature ∘F

119879119889119901 Dewpoint temperature ∘F

120572 Explicit deconvolution variabledimensionless

120573 Beta deconvolution variabledimensionless

120574119892 Gas gravity (air = 10)

120582 Interporosity flow coefficient of dualporosity reservoir

120601 Porosity of reservoir rock dimensionless120583119892 Gas viscosity cp

120583119900 Oil viscosity cp

120596 Storativity ratio of dual porosity reservoirminus Laplace transform of a function1015840 Derivative of a function

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Cheng W J Lee and D A McVay ldquoFast-Fourier-transform-based deconvolution for interpretation of pressure-transient-test data dominated by Wellbore storagerdquo SPE Reservoir Eval-uation and Engineering vol 8 no 3 pp 224ndash239 2005

[2] M M Levitan and M R Wilson ldquoDeconvolution of pressureand rate data from gas reservoirs with significant pressuredepletionrdquo SPE Journal vol 17 no 3 pp 727ndash741 2012

[3] Y Liu and R N Horne ldquoInterpreting pressure and flow-ratedata from permanent downhole gauges by use of data-miningapproachesrdquo SPE Journal vol 18 no 1 pp 69ndash82 2013

[4] O Ogunrewo and A C Gringarten ldquoDeconvolution of welltest data in lean and rich gas condensate and volatile oil wellsbelow saturation pressurerdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Document ID 166340-MS New Orleans La USA September-October 2013

[5] R C Earlougher Jr Advances in Well Test Analysis vol 5 ofMonogragh Series SPE Richardson Tex USA 1977

[6] F Kucuk and L Ayestaran ldquoAnalysis of simultaneously mea-sured pressure and sandface flow rate in transient well testingrdquoJournal of PetroleumTechnology vol 37 no 2 pp 323ndash334 1985

[7] M Onur M Cinar D IIk P P Valko T A Blasingame and PHegeman ldquoAn investigation of recent deconvolution methodsfor well-test data analysisrdquo SPE Journal vol 13 no 2 pp 226ndash247 2008

[8] F J KuchukMOnur andFHollaender ldquoPressure transient testdesign and interpretationrdquo Developments in Petroleum Sciencevol 57 pp 303ndash359 2010

[9] A C Gringarten D P Bourdet P A Landel and V J KniazeffldquoA comparison between different skin and wellbore storagetype-curves for early-time transient analysisrdquo in Proceedings ofthe SPE Annual Technical Conference and Exhibition Paper SPE8205 Las Vegas Nev USA 1979

[10] A C Gringarten ldquoFrom straight lines to deconvolution theevolution of the state of the art in well test analysisrdquo SPEReservoir Evaluation and Engineering vol 11 no 1 pp 41ndash622008

[11] J A Cumming D A Wooff T Whittle and A C GringartenldquoMultiple well deconvolutionrdquo in Proceedings of the SPE Annual

12 Abstract and Applied Analysis

Technical Conference and Exhibition Document ID 166458-MS New Orleans La USA September-October 2013

[12] O Bahabanian D Ilk N Hosseinpour-Zonoozi and T ABlasingame ldquoExplicit deconvolution of well test data distortedby wellbore storagerdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Paper SPE 103216 pp 4444ndash4453San Antonio Tex USA September 2006

[13] J Jones and A Chen ldquoSuccessful applications of pressure-ratedeconvolution in theCad-Nik tight gas formations of the BritishColumbia foothillsrdquo Journal of Canadian Petroleum Technologyvol 51 no 3 pp 176ndash192 2012

[14] D G Russell ldquoExtension of pressure buildup analysis methodsrdquoJournal of Petroleum Technology vol 18 no 12 pp 1624ndash16361966

[15] R E Gladfelter G W Tracy and L E Wilsey ldquoSelecting wellswhich will respond to production-stimulation treatmentrdquo inDrilling and Production Practice API Dallas Tex USA 1955

[16] M J Fetkovich and M E Vienot ldquoRate normalization ofbuildup pressure by using afterflow datardquo Journal of PetroleumTechnology vol 36 no 13 pp 2211ndash2224 1984

[17] J L Johnston Variable rate analysis of transient well testdata using semi-analytical methods [MS thesis] Texas AampMUniversity College Station Tex USA 1992

[18] A F van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 6 pp 171ndash176 1953

[19] W Hurst ldquoEstablishment of the skin effect and its impedimentto fluid flow into a well borerdquo The Petroleum Engineer pp B6ndashB16 1953

[20] J A Joseph and L F Koederitz ldquoA simple nonlinear model forrepresentation of field transient responsesrdquo Paper SPE 11435SPE Richardson Tex USA 1982

[21] F J Kuchuk ldquoGladfelter deconvolutionrdquo SPE Formation Evalu-ation vol 5 no 3 pp 285ndash292 1990

[22] A F van Everdingen and W Hurst ldquoThe application of thelaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[23] D Bourdet T M Whittle A A Douglas and Y M Pirard ldquoAnew set of type curves simplifies well test analysisrdquo World Oilpp 95ndash106 1983

[24] D F Meunier C S Kabir and M J Wittmann ldquoGas well testanalysis use of normalized pseudovariablesrdquo SPE FormationEvaluation vol 2 no 4 pp 629ndash636 1987

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

its governing parameters frommeasured data in the wellboreand at the wellhead This problem is known as the inverseproblem The solution of the inverse problem usually is notunique Gringarten et al [9] pointed out that if the numberand the rage measurements increased the nonuniqueness ofthe inverse problem will be reduced Thus combining sand-face flow rate with pressure measurements will enhance theconventional (including type curve) well test interpretationmethods On the other hand mathematically deconvolutionis also a highly unstable inverse problem because smallerrors in the data can result in large uncertainties in thedeconvolution solution [10 11]

Unfortunately standard deconvolution techniques re-quire accurate measurements of flow rate and pressure atdownhole (or sandface) conditions While accurate pressuremeasurements are commonplace the measurement of sand-face flow rates is rare essentially nonexistent in practiceAs such the deconvolution of wellbore storage distortedpressure test data is problematic In theory this process ispossible but in practice without accurate measurements offlow rates this process cannot be employed [12 13]

In the past 50 years a variety of deconvolution tech-niques have been proposed in petroleum engineering Russell[14] used the approximate direct method to correct thepressure transient data distorted by wellbore storage intothe equivalent pressure function for the constant rate caseGladfelter et al [15] and Fetkovich and Vienot [16] used therate normalization techniques to correct for thewellbore stor-age effectsThese rate normalizationmethods were successfulin some cases So Johnston [17] provided thematerial balancedeconvolution that is a practical approach for the analysis ofpressure transient data distorted by wellbore storage effects

Essentially rate normalization techniques and materialbalance deconvolution are restricted when the lack of ratemeasurement exists van Everdingen [18] and Hurst [19]demonstrated empirically that the sandface flow rate profilecan be modeled approximately using an exponential relationfor the duration of wellbore storage distortion during a pres-sure transient testThe vanEverdingen andHurst exponentialrate model is given in dimensionless form as

119902119863(119905119863) = 1 minus 119890

minus120573119905119863 (1)

Further they showed that the rate-time relationship duringafterflow (for a pressure buildup test) or unloading (in apressure drawdown test) is a function of the pressure dropchange with respect to time and a relatively constant wellborestorage coefficient

Based on a material balance in the wellbore the sandfaceflow rate can be calculated by the following relation given indimensionless form

119902119863(119905119863) = 1 minus 119862

119863

119889119901119908119863

119889119905119863

(2)

where it can be noted that in the development of wellborestorage models solutions (eg type curves) a constant well-bore storage coefficient is always assumed

Equations (1) and (2) laid the groundwork for 120573-deconvolution Joseph and Koederitz [20] and Kuchuk [21]

applied 120573-deconvolution for the analysis of wellbore stor-age distorted pressure transient data The 120573-deconvolutionformula which computes the undistorted pressure dropfunction directly from the wellbore storage affected data isgiven as

119901119904119863

(119905119863) =

1

120573

119889119901119908119863

(119905119863)

119889119905119863

+ 119901119908119863

(119905119863) (3)

It should be noted that (3) is only validwhen the sandface flowrate profile follows an exponential trend as prescribed by (1)[12] In thisworkwe develop an approach that overcomes thisrestriction by treating the 120573 quantity as a variable in generalrather than a constant

The obvious advantage of 120573-deconvolution is that thewellbore storage effects are eliminated and the restoredpressure is obtained using only the given pressure data

This paper reports a modification of Russell method(Appendix A) a modification of 120573-deconvolution and anexplicit deconvolution formula to restore the pressure andrate data and a review on material balance deconvolutionis also presented to deconvolve the pressure and rate dataafter the explicit deconvolution method Then a generalapproach to analyze the deconvolution problem is provided(Appendix B) In verification examples section we show thepower and practical applicability of our approaches withvarious synthetic and field case examples for oil and gas wells

2 Main Results

21 120573-Deconvolution and Its Modification van EverdingenandHurst [22] presented the dimensionless wellbore pressurefor a continuously varying flow rate as

119901119908119863

(119905119863) = 119902119863 (0) 119901119904119863 (119905119863) + int

119905119863

0

1199011015840

119863(120591) 119901119904119863 (119905119863 minus 120591) 119889120591

(4)

An alternative form to (4) can be obtained by integration byparts as

119901119908119863

(119905119863) = 119902119863(119905119863) 119901119904119863 (0) + int

119905119863

0

119902119863 (120591) 119901

1015840

119904119863(119905119863minus 120591) 119889120591

(5)

where 119901119908119863

(119905119863) = (119896ℎ1412119902119861120583)(119901

119894minus 119901119908119891(119905)) 119905

119863=

(000026371198961199051206011205831198881199051199032

119908) 119901119904119863

= 119901119863+ 119878 119901

119908119863(119905119863) is dimen-

sionless wellbore pressure with wellbore storage and skineffects 119901

119863(119905119863) is dimensionless sandface pressure for the

constant rate case without wellbore storage and skin effects 119878is steady state skin factor 1199011015840

119904119863(119905119863) = 119889119901

119904119863(119905119863)119889119905119863 119902119863(119905119863) =

119902119904119891(119905119863)119902119903 1199021015840119863(119905119863) = 119889119902

119863(119905119863)119889119905119863 and 119902

119903is reference

flow rate if the stabilized constant rate is available then 119902119903

should be replaced by 119902119877 119902119904119891(119905) is variable sandface flow rate

(flowmeter reading) and 119902119863(119905119863) is dimensionless sandface

rateIt should be emphasized that (4) and (5) can be applied

for many reservoir engineering problems The linearity ofdiffusivity equation allows us to use (4) and (5) for fractured

Abstract and Applied Analysis 3

layered anisotropic and heterogeneous systems as long as thefluid in the reservoir is single phaseThese equations can alsobe applied to both drawdown and buildup tests if the initialconditions are known For a reservoir with an initial constantand uniform pressure distribution (119901

119863(0) = 0) (4) and (5)

can be expressed as

119901119908119863

(119905119863) = int

119905119863

0

1199021015840

119863(120591) 119901119904119863 (119905119863 minus 120591) 119889120591 (6)

119901119908119863

(119905119863) = 119878119902

119863(119905119863) + int

119905119863

0

119902119863 (120591) 119901

1015840

119904119863(119905119863minus 120591) 119889120591 (7)

Furthermore (6) and (7) also can be expressed as

119901119908119863

(119905119863) = int

119905119863

0

1199021015840

119863(119905119863minus 120591) 119901

119904119863 (120591) 119889120591 (8)

119901119908119863

(119905119863) = 119878119902

119863(119905119863) + int

119905119863

0

119902119863(119905119863minus 120591) 119901

1015840

119904119863(120591) 119889120591 (9)

In (6) and (8) it is assumed that 1199021015840119863(119905119863) exists If 119902

119863(119905119863) is

constant then (7) and (9) must be used Equations (6) and(8) are known as a Volterra integral equation of first kind andthe convolution type

Taking the Laplace transform of (6) yields

119901119908119863

(119904) = 119904119902119863(119904) 119901119904119863

(119904) (10)

Rearranging (10) for the equivalent constant rate pressuredrop function 119901

119904119863(119904) we obtain

119901119904119863

(119904) =119901119908119863

(119904)

119904119902119863(119904)

(11)

By taking the Laplace transform of (1) one can easily obtain

119902119863(119904) =

1

119904minus

1

119904 + 120573 (12)

Substituting (12) into (11) and then taking the inverse Laplacetransform of (11) yield the beta deconvolution formula

119901119904119863

(119905119863) =

1

1205731199011015840

119908119863(119905119863) + 119901119908119863

(119905119863) (13)

where

1199011015840

119908119863(119905119863) =

119889119901119908119863

119889119905119863

(14)

where we note that (13) is specifically valid only for theexponential sandface flow rate profile given by (1) This maypresent a serious limitation in terms of practical applicationof the 120573-deconvolutionmethod To overcome this restrictionwe propose an idea It would be assumed that120573 is not constantin general If 119902

119863(119905119863)data are given the procedure is as follows

119902119863(119905119863) = 1 minus 119890

minus120573(119905119863)119905119863 (15)

Rearranging (15) yields

1 minus 119902119863(119905119863) = 119890minus120573(119905119863)119905119863 (16)

then

minus120573 (119905119863) 119905119863= ln (1 minus 119902

119863(119905119863)) (17)

so

120573 (119905119863) = minus

ln (1 minus 119902119863(119905119863))

119905119863

(18)

Substituting of (18) into (13) instead of 120573 term yields themodified 120573-deconvolution

119901119904119863

(119905119863) =

1

120573 (119905119863)1199011015840

119908119863(119905119863) + 119901119908119863

(119905119863) (19)

Plot of (19) versus (int119905119863

0119902119863(120591)119889120591119902

119863(119905119863)) and (int

119905119863

0(1 minus

119902119863(120591))1198891205911 minus 119902

119863(119905119863)) for drawdown and buildup tests

respectively should be considered more accurate to use as apractical tool for field applications As mentioned before thevariable 120573 quantity (rather than a constant) shows the generaland more accurate behavior of 119902

119863(119905119863) So accuracy of (19) is

also more than (13) It can be emphasized that (13) and (19)are applicable in the wellbore storage duration and for thelater times the given pressure data are correct However theresults of (19) show that this equation is not bad for the latertimes

22 Explicit Deconvolution Formula A few types of approx-imation functions can be used to approximate 119902

119863(119905119863) An

approximate proof is given here to show that the exponentialfunction gives a good representation of 119902

119863(119905119863) By setting

119902119863(119905119863) = 1 minus 119891 (119905

119863) (20)

119891 (0) = 1 (21)

Equation (21) is an initial condition for (20)Taking the time derivative of (20) gives

1199021015840

119863(119905119863) = minus119891

1015840(119905119863) (22)

Taking the time derivative of (2) gives

1199021015840

119863(119905119863) = minus119862

11986311990110158401015840

119908119863 (23)

From (2) and (20) we have

119891 (119905119863) = 1198621198631199011015840

119908119863 (24)

And from (22) and (23) we have

1198911015840(119905119863) = 11986211986311990110158401015840

119908119863 (25)

After dividing right and left hand sides of (25) by (24) theresult is

1198911015840(119905119863)

119891 (119905119863)=11990110158401015840

119908119863(119905119863)

1199011015840119908119863

(119905119863) (26)

Integrating both sides of (26) with respect to 119905119863from 0 to 119905

119863

yields

int

119905119863

0

1198911015840(120591)

119891 (120591)119889120591 = int

119905119863

0

11990110158401015840

119908119863(120591)

1199011015840119908119863

(120591)119889120591 (27)

4 Abstract and Applied Analysis

Using elementary calculus yields

ln (1003816100381610038161003816119891(120591)

1003816100381610038161003816)1003816100381610038161003816119905119863

0= ln (1003816100381610038161003816119891 (119905

119863)1003816100381610038161003816) minus ln (1) = int

119905119863

0

11990110158401015840

119908119863(120591)

1199011015840119908119863

(120591)119889120591

(28)

So

119891 (119905119863) = 119890120572(119905119863) (29)

where

120572 (119905119863) = int

119905119863

0

11990110158401015840

119908119863(120591)

1199011015840119908119863

(120591)119889120591 (30)

Now the explicit deconvolution of data based on (20) can bewritten as follows

119901119904119863

(119905119863) = minus

1

120572 (119905119863) 119905119863

1199011015840

119908119863+ 119901119908119863

(119905119863) (31)

or

119901119904119863

(119905119863) = minus

119905119863

120572 (119905119863)1199011015840

119908119863+ 119901119908119863

(119905119863) (32)

that is similar to (19) The other similarity between (19) and(32) is that the plot of (32) versus (int119905119863

0119902119863(120591)119889120591119902

119863(119905119863)) and

(int119905119863

0(1 minus 119902

119863(120591))119889120591(1 minus 119902

119863(119905119863))) for drawdown and buildup

tests respectively yields the more accurate results and isusable for practical applications The applicability of (32) isalso for duration of wellbore storage effects And for the othertimes the given pressure data are correct However using thismethod for the later times yields not bad results

3 Material Balance Deconvolution

Material balance deconvolution is an extension of the ratenormalization method Johnston [17] defines a new 119909-axisplotting function (material balance time) that provides anapproximate deconvolution of the variable rate pressuretransient problem There are numerous assumptions associ-ated with the material balance deconvolution methods oneof the most widely accepted assumptions is that the rateprofilemust change smoothly andmonotonically In practicalterms this condition should be met for the wellbore storageproblem

The general form of material balance deconvolution isprovided for the pressure drawdown case in terms of thematerial balance time function and the rate-normalizedpressure drop function The material balance time functionis given as

119905119898119887

=119873119901

119902119863 (119905)

(33)

where

119873119901= int

119905

0

119902119863 (120591) 119889120591 (34)

The rate-normalized pressure drop function is given by

Δ119901119908119891 (119905)

119902119863 (119905)

=119901119894minus 119901119908119891 (119905)

119902119863 (119905)

(35)

The material balance time function for the pressure buildupcase is given as

119905119898119887

=119873119901

1 minus 119902119863 (Δ119905)

(36)

where

119873119901= int

Δ119905

0

(1 minus 119902119863 (120591)) 119889120591 (37)

The rate-normalized pressure drop function for the pressurebuildup case is given as

Δ119901119908119904 (Δ119905)

1 minus 119902119863 (Δ119905)

=119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)

1 minus 119902119863 (Δ119905)

(38)

4 Methodology

In this work several methods are provided to restore thepressure datawithout any sandface flow rate informationThevalues of sandface flow rate data for all methods that requirethese data may be obtained from (20) We evaluated a veryold correctionmethod by Russell [14] However we modifiedthis method by least square method but it was found thatthis method is to be unacceptable for all applications Thederivation of this method and its modification is given inAppendix A

The 120573-deconvolution and its modification are anothertechnique that is mentioned in this study The formulationof this method is presented in Section 21 The other explicitdeconvolutionmethod is an approach similar to themodified120573-deconvolution that uses only the pressure data for decon-volution of pressure and rate data The formulation of thismethod was derived in Section 22 The 120573-deconvolution itsmodification and explicit deconvolution formula can be usedfor practical applications The applicability of these methodsis for duration of wellbore storage effects However usingthese approaches for the other times yields not bad results

The other technique is material balance deconvolutionmethod [17] that has sufficiently accurate resultsThismethodcan also be used as a good approach for practical applicationsThe formulas for this method were given in Section 3

In Appendix B deconvolution is expressed as an inverseproblem that is analyzed by a tricky idea Based on thisidea an interesting proof is given for rate normalizationmethod and material balance deconvolution that is a goodapproach to find the errors and limitations of these methodsand overcome them

Abstract and Applied Analysis 5

10minus3 10minus2 10minus1 100 101 102 1030

500

1000

1500

2000

2500

3000

3500

4000

Time (hr)

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolution methodd(Dps)d(ln(t)) material balance deconvolution methodDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 1 Deconvolution for a drawdown test in a homogeneousreservoir

From the above-mentioned methods we use the fol-lowing two techniques to show the verification of theseapproaches

Explicit Deconvolution Formula The final result of explicitdeconvolution method is given as follows

119901119904119863

(119905119863) = minus

119905119863

120572 (119905119863)1199011015840

119908119863+ 119901119908119863

(119905119863) (39)

Plotting of this term versus (int119905119863

0119902119863(120591)119889120591119902

119863(119905119863)) and

(int119905119863

0(1 minus 119902

119863(120591))119889120591(1 minus 119902

119863(119905119863))) for drawdown and buildup

tests respectively yields more accurate resultsThis approachperforms well in field applications

Material Balance Deconvolution The material balance pres-sure drop function for the pressure drawdown test is givenas

Δ119901119908119891 (119905)

119902119863 (119905)

=119901119894minus 119901119908119891 (119905)

119902119863 (119905)

(40)

and the material balance time function for the pressuredrawdown test is given as

119905119898119887

=int119905119863

0119902119863 (120591) 119889120591

119902119863 (119905)

(41)

For the buildup test the material balance pressure dropfunction can be written as

Δ119901119908119904 (Δ119905)

1 minus 119902119863 (Δ119905)

=119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)

1 minus 119902119863 (Δ119905)

(42)

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

104

103

102

101Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolution methodd(Dps)d(ln(t)) material balance deconvolution methodDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Figure 2 Deconvolution for a drawdown test in a homogeneousreservoir

And the material balance time function can be written as

119905119898119887

=int119905119863

0(1 minus 119902

119863 (120591)) 119889120591

1 minus 119902119863 (Δ119905)

(43)

This technique is a good approach for practical applications

5 Verification Examples

We first use few synthetic cases for different reser-voirwellbore systems to validate the applicability of explicitdeconvolution formula and material balance deconvolutionmethod Then the field applications of these techniqueswill be shown by two real cases The unloading rates arecalculated with (20)

The first case is a synthetic test derived from a test designin a vertical oil wellThe reservoir is homogenous and infiniteacting with a constant wellbore storage The wellbore storagecoefficient is 119862

119904= 0015 bblpsi The pressure data are

obtained from the cylindrical source solution (test design)We used the explicit deconvolution formula and material

balance deconvolution method to deconvolve the pressureand unloading rate distorting by wellbore storage effectsTheresults are presented in Figures 1 and 2 for the semilog andlog-log plots in the case of drawdown test and Figures 3and 4 for the semilog and log-log plots in the case of builduptest Besides the deconvolved pressures and their logarithmicderivatives pressure responses with and without wellborestorage are also plotted on the figures for comparisonThrough the deconvolution process we have successfullyremoved the wellbore storage effects so that radial flow canbe identified at early test times The deconvolved pressuresand their logarithmic derivatives are almost identical to

6 Abstract and Applied Analysis

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dp wellbore storage

Dps explicit deconvolution method(ln

Dps material balance deconvolution

Dps no wellbore storage

ws

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpws

d(Dp )d(ln )) wellbore storagews (HTR

d(Dps)d )) explicit deconvolution method(HTR

d(Dps)d(ln )) no wellbore storage(HTR)

d(Dps)d(ln )) material balance deconvolution (HTR

Figure 3 Deconvolution for a buildup test in a homogeneousreservoir

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

104

103

102

101

ws

ws

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpws

Dp wellbore storage

Dps explicit deconvolution method

Dps material balance deconvolution

Dps no wellbore storage

d(Dp )d(ln )) wellbore storagews (HTR

d(Dps)d )) explicit deconvolution method(HTR

d(Dps)d(ln )) no wellbore storage(HTR)

d(Dps)d(ln )) material balance deconvolution (HTR

(ln

Figure 4 Deconvolution for a buildup test in a homogeneousreservoir

the analytic solutions (without wellbore storage) whichenables us to estimate reservoir parameters accurately

The second case is an oil well drawdown test in adual porosity reservoir The storativity ratio (120596) and inter-porosity flow coefficient (120582) are assumed to be 01 and10minus6 respectively The constant wellbore storage coefficient

is 119862119904= 015 bblpsi The dimensionless wellbore pressure

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

600

500

400

300

200

100

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 5 Deconvolution for a drawdown test in a dual porosityreservoir

data were obtained from the cylindrical source solution withpseudosteady-state interporosity flow (test design) It is notpossible to identify the dual porosity behavior characteristicof the reservoir from wellbore pressure data because thisbehavior has been masked completely by wellbore storage asindicated in Figures 5 and 6

We deconvolved the pressure and unloading rate andpresent the results in Figures 5 and 6 The deconvolvedresults are reasonably consistent with the analytical solutionAfter deconvolution the dual porosity behavior can beidentified easily from the valley in the logarithmic derivativeof pressure If wellbore storage effects were not removed atearly times it easily could have incorrectly identified thereservoir system as infinite actingThe estimation of reservoirparameters can be obtained now from deconvolved pressure

The third case is an oil well drawdown test in a reservoirwith a sealing fault 250 ft from the well Synthetic pressureresponses were generated by simulation (test design) with aconstant wellbore storage coefficient of 01 bblpsi As we cansee in Figures 7 and 8 the early time infinite acting reservoirbehavior has been masked completely by wellbore storage

To recover the hidden reservoir behavior we performeddeconvolution on the pressure change to remove the wellborestorage effect As in Figures 7 and 8 the deconvolved pressurechange and logarithmic derivative data are almost identicalto the responses without wellbore storageThe deconvolutionresults clearly indicate parallel horizontal lines for the loga-rithmic derivative However without wellbore storage effectselimination this maymislead to identify the reservoir systemas infinite acting So this allows us to correctly identify thereservoir system having a sealing fault

The fourth case is a field example that is taken from theliterature [23] The basic reservoir and fluid properties are

Abstract and Applied Analysis 7

Table 1 Reservoir rock and fluid properties for the first field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119902

119900(STBD) 119861

119900(RBSTB) 119888

119905(psiminus1) 120583

119900(cp) 120601

1533 107 029 174 106 42 times 10minus6 25 025

Table 2 Reservoir rock and fluid properties for the second field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119879 (∘F) 119901 (psi) 119902

119900(STBD) 119902

119892(MscfD) 119888

119891(psiminus1) 119879

119889119901(∘F) 119901

119889119901(psi) ∘API 120574

119892

143111 100 0354 2795 7322 104 13925 3 times 10minus6 212 5445 45375 068

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

10minus1

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 6 Deconvolution for a drawdown test in a dual porosityreservoir

shown in Table 1 In this case we provide the deconvolvedpressure data using the explicit deconvolution formula andthe material balance deconvolution method that is presentedin this work The data are taken from a pressure buildup testand can be reasonably used as field data The deconvolutionresults are shown in Figure 9 in semilog plot As it isillustrated from Figure 9 the corrected pressure provides aclear straight line starting as early as 15 hours in contrast tothe vague line from the uncorrected pressure response

And the last (field) example is a gas condensatewell buildup test following a variable rate productionhistory The duration of last pressure buildup test is361556 hours (from 1889 to 2250556 hours) The basicreservoir and fluid properties for the last drawdown testare shown in Table 2 However the properties of otherproduction tests are not shown here To use our newtechniques effectively for gas wells we use normalizedpseudopressure and pseudotime functions as defined byMeunier et al [24]The final results of explicit deconvolutionformula and material balance deconvolution methodshow two different slopes (Figure 10) where the earlytimes slope is greater than the last one This identifiesthe radial composite behavior of the reservoir With

the corrected pressures the reservoir parameters can also beestimated

6 Conclusions

In this work we have provided some explicit deconvolutionmethods for restoration of constant rate pressure responsesfrom measured pressure data dominated by wellbore storageeffects without sandface rate measurements Based on ourinvestigations we can state the following specific conclusionsrelated to these techniques

(1) The results obtained from the synthetic and fieldexamples show the practical and computational effi-ciency of our deconvolution methods

(2) Modification of 120573-deconvolution is presented wherethe beta quantity is variable rather than a constant

(3) For situations in which there are no downhole ratemeasurements the explicit deconvolution methods(blind deconvolution approaches) can be used toderive the downhole rate function from the pressuremeasurements and restore the reservoir pressureresponse

(4) We can unveil the early time behavior of a reservoirsystemmasked by wellbore storage effects using thesenew approaches Wellbore geometries and reservoirtypes can be identified from this restored pressuredata Furthermore the conventional methods canbe used to analyze this restored formation pressureto estimate formation parameters So the presentedtechniques can be used as the practical tools toimprove pressure transient test interpretation

(5) Deconvolution in well test analysis is often expressedas an inverse problem This approach can be used asan idea to justify the rate normalization method andthe material balance deconvolution method

Appendices

A Russel Method and Its Modification

A method for correcting pressure buildup pressure data freeof wellbore storage effects at early times has been developedby Russell [14] In this method it is not necessary to measurethe rate of influx after closing in at the surface Instead oneuses a theoretical equation which gives the form that thebottom hole pressure should have fluid accumulates in the

8 Abstract and Applied Analysis

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

250

200

150

100

50

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 7 Deconvolution for a drawdown test in a reservoir with asealing fault

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 8 Deconvolution for a drawdown test in a reservoir with asealing fault

wellbore during buildup This leads to the result that oneshould plot

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

(A1)

versus log (Δ119905) in analyzing pressure buildup data during theearly fill-up period The denominator makes a correction for

104103102101100

Horner time ratio

800

700

600

500

400

300

200

100

0

Dpws

(psi)

Dpws wellbore storageDps explicit deconvolution methodDps material balance deconvolution

Figure 9 Restored pressures for a buildup test in a homogeneousreservoir

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dpa

(psi)

Dpaw wellbore storage explicit deconvolution method material balance deconvolution

DpasDpas

Figure 10 Restored pressures for a buildup test in a radial compositereservoir

the gradually decreasing flow into the wellbore The quantity1198622is obtained by trial and error as the value which makes the

curve straight at early times After obtaining the straight linesection the rest of the analysis is the same as for any otherpressure buildup This method has the advantage or requiresno additional data over that taken routinely during a shut intest

Due to the time consuming and unconfident of trial anderror to determine 119862

2in Russell method we modify this

method Russellrsquos wellbore storage correction is given as

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

= 119891 (Δ119905 = 1 hr) + 119898119904119897log (Δ119905)

(A2)

Abstract and Applied Analysis 9

Defining the following function and using least squarestechnique obtain the final result

119864 (1198622 119891119898)

=

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891(1198622) minus 119898(119862

2) log(Δ119905

119894))

2

(A3)

Tominimize the function119864 it can differentiate119864with respectto 1198622 119891 and 119898 and setting the results equal to zero After

differentiation it reduces to

120597119864

1205971198622

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894) ) = 0

(A4)

120597119864

120597119891

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622) minus 119898 (119862

2) log (Δ119905

119894))

= 0

(A5)120597119864

120597119898

= minus2

119873

sum

119894=1

log (Δ119905119894) (

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus 119898 (1198622) log (Δ119905

119894)) = 0

(A6)

From (A4) we define the following function

119883(1198622)

=

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times(Δ119901119908119904119894

(1minus (11198622Δ119905119894))minus119891 (119862

2) minus119898 (119862

2) log (Δ119905

119894))

= 0

(A7)

The roots of (A7) can be found via the following algorithm

1198622(119894+1)

= 1198622(119894)

minus119883 (1198622)

1198831015840 (1198622) (A8)

where

119883(1198622) =

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894))

1198831015840(1198622) =

119873

sum

119894=1

(minus2Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3)

times (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2

times [Δ119901119908119904119894

(1 minus (11198622Δ119905))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894)])

+ (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (minusΔ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3minus 1198911015840(1198622)

minus1198981015840(1198622) log (Δ119905

119894))

(A9)

where the unknown functions are given by the followingrelations

119891 (1198622) =

120573119906 (1198622) minus 120572V (119862

2)

1205732 minus 119873120572(A10)

whose derivative is

1198911015840(1198622) =

1205731199061015840(1198622) minus 120572V1015840 (119862

2)

1205732 minus 119873120572

119898 (1198622) =

119873119906 (1198622) minus 120573V (119862

2)

119873120572 minus 1205732

(A11)

where its derivative is

1198981015840(1198622) =

1198731199061015840(1198622) minus 120573V1015840 (119862

2)

119873120572 minus 1205732

119906 (1198622) =

119873

sum

119894=1

log (Δ119905119894)

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A12)

10 Abstract and Applied Analysis

and after differentiation it yields

1199061015840(1198622) = minus

119873

sum

119894=1

log (Δ119905

119894)

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2

V (1198622) =

119873

sum

119894=1

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A13)

which has the following derivative

V1015840 (1198622) = minus

119873

sum

119894=1

1

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2 (A14)

finally the values of 120572 and 120573 parameters are given by thefollowing formulas

120572 =

119873

sum

119894=1

(log(Δ119905119894))2

120573 =

119873

sum

119894=1

log (Δ119905119894)

(A15)

B General Formula for Inverse Problem

The convolution of (6) yields

119901119904119863

(119905119863) = int

119905119863

0

119870 (120591) 119901119908119863 (119905119863 minus 120591) 119889120591

= int

119905119863

0

119870(119905119863minus 120591) 119901

119908119863 (120591) 119889120591

(B1)

where

119870(119905119863) = 119871minus1(

1

119904119902 (119904)) (B2)

where 119871minus1 is the inverse Lapalce transform operator 119870(119905119863)

can be computed either from the Laplace transforms of119902119863(119905119863) data a curve-fitted equation of 119902

119863(119905119863) data or using

another technique with a relation between119870(119905119863) and 119902

119863(119905119863)

However it would be time consuming to invert all the 119902119863(119905119863)

data in Laplace space and transform it back to real space inaccordance with (B2) Once an approximation is obtained itwill be easy to compute119870(119905

119863) and integrate (B1) to determine

119901119904119863(119905119863) However finding some approximation functions for

119902119863(119905119863) data may be difficult We provide an approach to

overcome this problemThis idea is based on finding anotherrelation between 119870(119905

119863) and 119902

119863(119905119863) that is

ℓ (1) =1

119904=

1

119904119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B3)or

1 = int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591 = int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591

(B4)

The other relations that can be used are

ℓ (119905119863) =

1

1199042=

1

1199042119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B5)

or

119905119863= int

119905119863

0

119871 (120591) 119902119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

119863(120591) 119889120591

(B6)

ℓ (1) =1

119904=

1

1199042119902119863(119904)

119904119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591)

(B7)

or

1 = int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591

(B8)

where

119871 (119905119863) = ℓminus1(

1

1199042119902 (119904)) (B9)

Equation (B4) can be used simultaneously with (B1) toanalyze the deconvolution problem

Equation (11) can be written as

119901119904119863

(119904) = 119904119901119908119863

(119904)1

1199042119902119863(119904)

(B10)

or

119901119904119863

(119905119863) = int

119905119863

0

119871 (120591) 1199011015840

119908119863(119905119863minus 120591) 119889120591

= int

119905119863

0

119871 (119905119863minus 120591) 119901

1015840

119908119863(120591) 119889120591

(B11)

The applications of (B6) and (B8) are with (B11)For justifying the rate normalizationmethod andmaterial

balance deconvolution it can use (B1) (B4) ((B11) can alsobe used with (B6) or (B8) for this goal) and generalizedmean value theorem for integrals This theorem is expressedas follows

Theorem 1 Let 119891 and 119892 be continuous on [119886 119887] and supposethat 119892 does not change sign on [119886 119887] Then there is a point 119888 in[119886 119887] such that

int

119887

119886

119891 (119905) 119892 (119905) 119889119905 = 119891 (119888) int

119887

119886

119892 (119905) 119889119905 (B12)

Abstract and Applied Analysis 11

By Theorem 1 there are some values 1198881and 1198882between 0

and 119905119863such that

119901119904119863

(119905119863) = 119901119908119863

(1198881) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B13)

1 = 119902119863(1198882) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B14)

By setting 1198881

= 1198882

= 119905119863and dividing the right and left

side of (B13) respectively the rate normalization techniqueis developed However the values 119888

1and 1198882are not equal

to each other and are not equal to 119905119863

in general Thisis the reason of error in rate normalization and materialbalance deconvolution methods 119888

1and 1198882are less than 119905

119863in

these methods for the duration of wellbore storage distortionduring a pressure transient test Since these conditions areapproximately satisfied in material balance deconvolutionthe results of material balance are more accurate and moreconfident than rate normalization

Nomenclature

API API gravity ∘API119861119900 Oil formation volume factor RBSTB

119888119905 Total isothermal compressibility factor

psiminus1

1198622 Arbitrary constant hr

minus1

119862119863 Wellbore storage constant dimensionless

119862119904 Wellbore storage coefficient bblpsi

ℎ Formation thickness ftHTR Horner time ratio dimensionless119896 Formation permeability md119870 Kernel of the convolution integral119871 Kernel of the convolution integral119873119901 Cumulative oil production bbl

119901 Pressure psi119901119886 Normalized pseudopressure psi

119901 Average reservoir pressure psi119901119863 Formation pressure dimensionless

119901119889119901 Dewpoint pressure psi

119901119894 Initial reservoir pressure psi

119901119904119863 Formation pressure including skin

dimensionless119901119904 Formation pressure including skin psi

119901119908119863

Pressure draw down dimensionless119901119908119891 Flowing bottom hole pressure psi

119901119908119904 Shut-in bottom hole pressure psi

119902119863 Sandface flow rate dimensionless

119902119892 Gas flow rate MscfD

119902119900 Oil flow rate STBD

119903119908 Wellbore radius ft

119904 Laplace variable119878 Skin factor119905 Time hr119905119863 Time dimensionless

119905119901 Production time hr

119879 Reservoir temperature ∘F

119879119889119901 Dewpoint temperature ∘F

120572 Explicit deconvolution variabledimensionless

120573 Beta deconvolution variabledimensionless

120574119892 Gas gravity (air = 10)

120582 Interporosity flow coefficient of dualporosity reservoir

120601 Porosity of reservoir rock dimensionless120583119892 Gas viscosity cp

120583119900 Oil viscosity cp

120596 Storativity ratio of dual porosity reservoirminus Laplace transform of a function1015840 Derivative of a function

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Cheng W J Lee and D A McVay ldquoFast-Fourier-transform-based deconvolution for interpretation of pressure-transient-test data dominated by Wellbore storagerdquo SPE Reservoir Eval-uation and Engineering vol 8 no 3 pp 224ndash239 2005

[2] M M Levitan and M R Wilson ldquoDeconvolution of pressureand rate data from gas reservoirs with significant pressuredepletionrdquo SPE Journal vol 17 no 3 pp 727ndash741 2012

[3] Y Liu and R N Horne ldquoInterpreting pressure and flow-ratedata from permanent downhole gauges by use of data-miningapproachesrdquo SPE Journal vol 18 no 1 pp 69ndash82 2013

[4] O Ogunrewo and A C Gringarten ldquoDeconvolution of welltest data in lean and rich gas condensate and volatile oil wellsbelow saturation pressurerdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Document ID 166340-MS New Orleans La USA September-October 2013

[5] R C Earlougher Jr Advances in Well Test Analysis vol 5 ofMonogragh Series SPE Richardson Tex USA 1977

[6] F Kucuk and L Ayestaran ldquoAnalysis of simultaneously mea-sured pressure and sandface flow rate in transient well testingrdquoJournal of PetroleumTechnology vol 37 no 2 pp 323ndash334 1985

[7] M Onur M Cinar D IIk P P Valko T A Blasingame and PHegeman ldquoAn investigation of recent deconvolution methodsfor well-test data analysisrdquo SPE Journal vol 13 no 2 pp 226ndash247 2008

[8] F J KuchukMOnur andFHollaender ldquoPressure transient testdesign and interpretationrdquo Developments in Petroleum Sciencevol 57 pp 303ndash359 2010

[9] A C Gringarten D P Bourdet P A Landel and V J KniazeffldquoA comparison between different skin and wellbore storagetype-curves for early-time transient analysisrdquo in Proceedings ofthe SPE Annual Technical Conference and Exhibition Paper SPE8205 Las Vegas Nev USA 1979

[10] A C Gringarten ldquoFrom straight lines to deconvolution theevolution of the state of the art in well test analysisrdquo SPEReservoir Evaluation and Engineering vol 11 no 1 pp 41ndash622008

[11] J A Cumming D A Wooff T Whittle and A C GringartenldquoMultiple well deconvolutionrdquo in Proceedings of the SPE Annual

12 Abstract and Applied Analysis

Technical Conference and Exhibition Document ID 166458-MS New Orleans La USA September-October 2013

[12] O Bahabanian D Ilk N Hosseinpour-Zonoozi and T ABlasingame ldquoExplicit deconvolution of well test data distortedby wellbore storagerdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Paper SPE 103216 pp 4444ndash4453San Antonio Tex USA September 2006

[13] J Jones and A Chen ldquoSuccessful applications of pressure-ratedeconvolution in theCad-Nik tight gas formations of the BritishColumbia foothillsrdquo Journal of Canadian Petroleum Technologyvol 51 no 3 pp 176ndash192 2012

[14] D G Russell ldquoExtension of pressure buildup analysis methodsrdquoJournal of Petroleum Technology vol 18 no 12 pp 1624ndash16361966

[15] R E Gladfelter G W Tracy and L E Wilsey ldquoSelecting wellswhich will respond to production-stimulation treatmentrdquo inDrilling and Production Practice API Dallas Tex USA 1955

[16] M J Fetkovich and M E Vienot ldquoRate normalization ofbuildup pressure by using afterflow datardquo Journal of PetroleumTechnology vol 36 no 13 pp 2211ndash2224 1984

[17] J L Johnston Variable rate analysis of transient well testdata using semi-analytical methods [MS thesis] Texas AampMUniversity College Station Tex USA 1992

[18] A F van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 6 pp 171ndash176 1953

[19] W Hurst ldquoEstablishment of the skin effect and its impedimentto fluid flow into a well borerdquo The Petroleum Engineer pp B6ndashB16 1953

[20] J A Joseph and L F Koederitz ldquoA simple nonlinear model forrepresentation of field transient responsesrdquo Paper SPE 11435SPE Richardson Tex USA 1982

[21] F J Kuchuk ldquoGladfelter deconvolutionrdquo SPE Formation Evalu-ation vol 5 no 3 pp 285ndash292 1990

[22] A F van Everdingen and W Hurst ldquoThe application of thelaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[23] D Bourdet T M Whittle A A Douglas and Y M Pirard ldquoAnew set of type curves simplifies well test analysisrdquo World Oilpp 95ndash106 1983

[24] D F Meunier C S Kabir and M J Wittmann ldquoGas well testanalysis use of normalized pseudovariablesrdquo SPE FormationEvaluation vol 2 no 4 pp 629ndash636 1987

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

layered anisotropic and heterogeneous systems as long as thefluid in the reservoir is single phaseThese equations can alsobe applied to both drawdown and buildup tests if the initialconditions are known For a reservoir with an initial constantand uniform pressure distribution (119901

119863(0) = 0) (4) and (5)

can be expressed as

119901119908119863

(119905119863) = int

119905119863

0

1199021015840

119863(120591) 119901119904119863 (119905119863 minus 120591) 119889120591 (6)

119901119908119863

(119905119863) = 119878119902

119863(119905119863) + int

119905119863

0

119902119863 (120591) 119901

1015840

119904119863(119905119863minus 120591) 119889120591 (7)

Furthermore (6) and (7) also can be expressed as

119901119908119863

(119905119863) = int

119905119863

0

1199021015840

119863(119905119863minus 120591) 119901

119904119863 (120591) 119889120591 (8)

119901119908119863

(119905119863) = 119878119902

119863(119905119863) + int

119905119863

0

119902119863(119905119863minus 120591) 119901

1015840

119904119863(120591) 119889120591 (9)

In (6) and (8) it is assumed that 1199021015840119863(119905119863) exists If 119902

119863(119905119863) is

constant then (7) and (9) must be used Equations (6) and(8) are known as a Volterra integral equation of first kind andthe convolution type

Taking the Laplace transform of (6) yields

119901119908119863

(119904) = 119904119902119863(119904) 119901119904119863

(119904) (10)

Rearranging (10) for the equivalent constant rate pressuredrop function 119901

119904119863(119904) we obtain

119901119904119863

(119904) =119901119908119863

(119904)

119904119902119863(119904)

(11)

By taking the Laplace transform of (1) one can easily obtain

119902119863(119904) =

1

119904minus

1

119904 + 120573 (12)

Substituting (12) into (11) and then taking the inverse Laplacetransform of (11) yield the beta deconvolution formula

119901119904119863

(119905119863) =

1

1205731199011015840

119908119863(119905119863) + 119901119908119863

(119905119863) (13)

where

1199011015840

119908119863(119905119863) =

119889119901119908119863

119889119905119863

(14)

where we note that (13) is specifically valid only for theexponential sandface flow rate profile given by (1) This maypresent a serious limitation in terms of practical applicationof the 120573-deconvolutionmethod To overcome this restrictionwe propose an idea It would be assumed that120573 is not constantin general If 119902

119863(119905119863)data are given the procedure is as follows

119902119863(119905119863) = 1 minus 119890

minus120573(119905119863)119905119863 (15)

Rearranging (15) yields

1 minus 119902119863(119905119863) = 119890minus120573(119905119863)119905119863 (16)

then

minus120573 (119905119863) 119905119863= ln (1 minus 119902

119863(119905119863)) (17)

so

120573 (119905119863) = minus

ln (1 minus 119902119863(119905119863))

119905119863

(18)

Substituting of (18) into (13) instead of 120573 term yields themodified 120573-deconvolution

119901119904119863

(119905119863) =

1

120573 (119905119863)1199011015840

119908119863(119905119863) + 119901119908119863

(119905119863) (19)

Plot of (19) versus (int119905119863

0119902119863(120591)119889120591119902

119863(119905119863)) and (int

119905119863

0(1 minus

119902119863(120591))1198891205911 minus 119902

119863(119905119863)) for drawdown and buildup tests

respectively should be considered more accurate to use as apractical tool for field applications As mentioned before thevariable 120573 quantity (rather than a constant) shows the generaland more accurate behavior of 119902

119863(119905119863) So accuracy of (19) is

also more than (13) It can be emphasized that (13) and (19)are applicable in the wellbore storage duration and for thelater times the given pressure data are correct However theresults of (19) show that this equation is not bad for the latertimes

22 Explicit Deconvolution Formula A few types of approx-imation functions can be used to approximate 119902

119863(119905119863) An

approximate proof is given here to show that the exponentialfunction gives a good representation of 119902

119863(119905119863) By setting

119902119863(119905119863) = 1 minus 119891 (119905

119863) (20)

119891 (0) = 1 (21)

Equation (21) is an initial condition for (20)Taking the time derivative of (20) gives

1199021015840

119863(119905119863) = minus119891

1015840(119905119863) (22)

Taking the time derivative of (2) gives

1199021015840

119863(119905119863) = minus119862

11986311990110158401015840

119908119863 (23)

From (2) and (20) we have

119891 (119905119863) = 1198621198631199011015840

119908119863 (24)

And from (22) and (23) we have

1198911015840(119905119863) = 11986211986311990110158401015840

119908119863 (25)

After dividing right and left hand sides of (25) by (24) theresult is

1198911015840(119905119863)

119891 (119905119863)=11990110158401015840

119908119863(119905119863)

1199011015840119908119863

(119905119863) (26)

Integrating both sides of (26) with respect to 119905119863from 0 to 119905

119863

yields

int

119905119863

0

1198911015840(120591)

119891 (120591)119889120591 = int

119905119863

0

11990110158401015840

119908119863(120591)

1199011015840119908119863

(120591)119889120591 (27)

4 Abstract and Applied Analysis

Using elementary calculus yields

ln (1003816100381610038161003816119891(120591)

1003816100381610038161003816)1003816100381610038161003816119905119863

0= ln (1003816100381610038161003816119891 (119905

119863)1003816100381610038161003816) minus ln (1) = int

119905119863

0

11990110158401015840

119908119863(120591)

1199011015840119908119863

(120591)119889120591

(28)

So

119891 (119905119863) = 119890120572(119905119863) (29)

where

120572 (119905119863) = int

119905119863

0

11990110158401015840

119908119863(120591)

1199011015840119908119863

(120591)119889120591 (30)

Now the explicit deconvolution of data based on (20) can bewritten as follows

119901119904119863

(119905119863) = minus

1

120572 (119905119863) 119905119863

1199011015840

119908119863+ 119901119908119863

(119905119863) (31)

or

119901119904119863

(119905119863) = minus

119905119863

120572 (119905119863)1199011015840

119908119863+ 119901119908119863

(119905119863) (32)

that is similar to (19) The other similarity between (19) and(32) is that the plot of (32) versus (int119905119863

0119902119863(120591)119889120591119902

119863(119905119863)) and

(int119905119863

0(1 minus 119902

119863(120591))119889120591(1 minus 119902

119863(119905119863))) for drawdown and buildup

tests respectively yields the more accurate results and isusable for practical applications The applicability of (32) isalso for duration of wellbore storage effects And for the othertimes the given pressure data are correct However using thismethod for the later times yields not bad results

3 Material Balance Deconvolution

Material balance deconvolution is an extension of the ratenormalization method Johnston [17] defines a new 119909-axisplotting function (material balance time) that provides anapproximate deconvolution of the variable rate pressuretransient problem There are numerous assumptions associ-ated with the material balance deconvolution methods oneof the most widely accepted assumptions is that the rateprofilemust change smoothly andmonotonically In practicalterms this condition should be met for the wellbore storageproblem

The general form of material balance deconvolution isprovided for the pressure drawdown case in terms of thematerial balance time function and the rate-normalizedpressure drop function The material balance time functionis given as

119905119898119887

=119873119901

119902119863 (119905)

(33)

where

119873119901= int

119905

0

119902119863 (120591) 119889120591 (34)

The rate-normalized pressure drop function is given by

Δ119901119908119891 (119905)

119902119863 (119905)

=119901119894minus 119901119908119891 (119905)

119902119863 (119905)

(35)

The material balance time function for the pressure buildupcase is given as

119905119898119887

=119873119901

1 minus 119902119863 (Δ119905)

(36)

where

119873119901= int

Δ119905

0

(1 minus 119902119863 (120591)) 119889120591 (37)

The rate-normalized pressure drop function for the pressurebuildup case is given as

Δ119901119908119904 (Δ119905)

1 minus 119902119863 (Δ119905)

=119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)

1 minus 119902119863 (Δ119905)

(38)

4 Methodology

In this work several methods are provided to restore thepressure datawithout any sandface flow rate informationThevalues of sandface flow rate data for all methods that requirethese data may be obtained from (20) We evaluated a veryold correctionmethod by Russell [14] However we modifiedthis method by least square method but it was found thatthis method is to be unacceptable for all applications Thederivation of this method and its modification is given inAppendix A

The 120573-deconvolution and its modification are anothertechnique that is mentioned in this study The formulationof this method is presented in Section 21 The other explicitdeconvolutionmethod is an approach similar to themodified120573-deconvolution that uses only the pressure data for decon-volution of pressure and rate data The formulation of thismethod was derived in Section 22 The 120573-deconvolution itsmodification and explicit deconvolution formula can be usedfor practical applications The applicability of these methodsis for duration of wellbore storage effects However usingthese approaches for the other times yields not bad results

The other technique is material balance deconvolutionmethod [17] that has sufficiently accurate resultsThismethodcan also be used as a good approach for practical applicationsThe formulas for this method were given in Section 3

In Appendix B deconvolution is expressed as an inverseproblem that is analyzed by a tricky idea Based on thisidea an interesting proof is given for rate normalizationmethod and material balance deconvolution that is a goodapproach to find the errors and limitations of these methodsand overcome them

Abstract and Applied Analysis 5

10minus3 10minus2 10minus1 100 101 102 1030

500

1000

1500

2000

2500

3000

3500

4000

Time (hr)

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolution methodd(Dps)d(ln(t)) material balance deconvolution methodDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 1 Deconvolution for a drawdown test in a homogeneousreservoir

From the above-mentioned methods we use the fol-lowing two techniques to show the verification of theseapproaches

Explicit Deconvolution Formula The final result of explicitdeconvolution method is given as follows

119901119904119863

(119905119863) = minus

119905119863

120572 (119905119863)1199011015840

119908119863+ 119901119908119863

(119905119863) (39)

Plotting of this term versus (int119905119863

0119902119863(120591)119889120591119902

119863(119905119863)) and

(int119905119863

0(1 minus 119902

119863(120591))119889120591(1 minus 119902

119863(119905119863))) for drawdown and buildup

tests respectively yields more accurate resultsThis approachperforms well in field applications

Material Balance Deconvolution The material balance pres-sure drop function for the pressure drawdown test is givenas

Δ119901119908119891 (119905)

119902119863 (119905)

=119901119894minus 119901119908119891 (119905)

119902119863 (119905)

(40)

and the material balance time function for the pressuredrawdown test is given as

119905119898119887

=int119905119863

0119902119863 (120591) 119889120591

119902119863 (119905)

(41)

For the buildup test the material balance pressure dropfunction can be written as

Δ119901119908119904 (Δ119905)

1 minus 119902119863 (Δ119905)

=119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)

1 minus 119902119863 (Δ119905)

(42)

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

104

103

102

101Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolution methodd(Dps)d(ln(t)) material balance deconvolution methodDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Figure 2 Deconvolution for a drawdown test in a homogeneousreservoir

And the material balance time function can be written as

119905119898119887

=int119905119863

0(1 minus 119902

119863 (120591)) 119889120591

1 minus 119902119863 (Δ119905)

(43)

This technique is a good approach for practical applications

5 Verification Examples

We first use few synthetic cases for different reser-voirwellbore systems to validate the applicability of explicitdeconvolution formula and material balance deconvolutionmethod Then the field applications of these techniqueswill be shown by two real cases The unloading rates arecalculated with (20)

The first case is a synthetic test derived from a test designin a vertical oil wellThe reservoir is homogenous and infiniteacting with a constant wellbore storage The wellbore storagecoefficient is 119862

119904= 0015 bblpsi The pressure data are

obtained from the cylindrical source solution (test design)We used the explicit deconvolution formula and material

balance deconvolution method to deconvolve the pressureand unloading rate distorting by wellbore storage effectsTheresults are presented in Figures 1 and 2 for the semilog andlog-log plots in the case of drawdown test and Figures 3and 4 for the semilog and log-log plots in the case of builduptest Besides the deconvolved pressures and their logarithmicderivatives pressure responses with and without wellborestorage are also plotted on the figures for comparisonThrough the deconvolution process we have successfullyremoved the wellbore storage effects so that radial flow canbe identified at early test times The deconvolved pressuresand their logarithmic derivatives are almost identical to

6 Abstract and Applied Analysis

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dp wellbore storage

Dps explicit deconvolution method(ln

Dps material balance deconvolution

Dps no wellbore storage

ws

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpws

d(Dp )d(ln )) wellbore storagews (HTR

d(Dps)d )) explicit deconvolution method(HTR

d(Dps)d(ln )) no wellbore storage(HTR)

d(Dps)d(ln )) material balance deconvolution (HTR

Figure 3 Deconvolution for a buildup test in a homogeneousreservoir

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

104

103

102

101

ws

ws

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpws

Dp wellbore storage

Dps explicit deconvolution method

Dps material balance deconvolution

Dps no wellbore storage

d(Dp )d(ln )) wellbore storagews (HTR

d(Dps)d )) explicit deconvolution method(HTR

d(Dps)d(ln )) no wellbore storage(HTR)

d(Dps)d(ln )) material balance deconvolution (HTR

(ln

Figure 4 Deconvolution for a buildup test in a homogeneousreservoir

the analytic solutions (without wellbore storage) whichenables us to estimate reservoir parameters accurately

The second case is an oil well drawdown test in adual porosity reservoir The storativity ratio (120596) and inter-porosity flow coefficient (120582) are assumed to be 01 and10minus6 respectively The constant wellbore storage coefficient

is 119862119904= 015 bblpsi The dimensionless wellbore pressure

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

600

500

400

300

200

100

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 5 Deconvolution for a drawdown test in a dual porosityreservoir

data were obtained from the cylindrical source solution withpseudosteady-state interporosity flow (test design) It is notpossible to identify the dual porosity behavior characteristicof the reservoir from wellbore pressure data because thisbehavior has been masked completely by wellbore storage asindicated in Figures 5 and 6

We deconvolved the pressure and unloading rate andpresent the results in Figures 5 and 6 The deconvolvedresults are reasonably consistent with the analytical solutionAfter deconvolution the dual porosity behavior can beidentified easily from the valley in the logarithmic derivativeof pressure If wellbore storage effects were not removed atearly times it easily could have incorrectly identified thereservoir system as infinite actingThe estimation of reservoirparameters can be obtained now from deconvolved pressure

The third case is an oil well drawdown test in a reservoirwith a sealing fault 250 ft from the well Synthetic pressureresponses were generated by simulation (test design) with aconstant wellbore storage coefficient of 01 bblpsi As we cansee in Figures 7 and 8 the early time infinite acting reservoirbehavior has been masked completely by wellbore storage

To recover the hidden reservoir behavior we performeddeconvolution on the pressure change to remove the wellborestorage effect As in Figures 7 and 8 the deconvolved pressurechange and logarithmic derivative data are almost identicalto the responses without wellbore storageThe deconvolutionresults clearly indicate parallel horizontal lines for the loga-rithmic derivative However without wellbore storage effectselimination this maymislead to identify the reservoir systemas infinite acting So this allows us to correctly identify thereservoir system having a sealing fault

The fourth case is a field example that is taken from theliterature [23] The basic reservoir and fluid properties are

Abstract and Applied Analysis 7

Table 1 Reservoir rock and fluid properties for the first field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119902

119900(STBD) 119861

119900(RBSTB) 119888

119905(psiminus1) 120583

119900(cp) 120601

1533 107 029 174 106 42 times 10minus6 25 025

Table 2 Reservoir rock and fluid properties for the second field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119879 (∘F) 119901 (psi) 119902

119900(STBD) 119902

119892(MscfD) 119888

119891(psiminus1) 119879

119889119901(∘F) 119901

119889119901(psi) ∘API 120574

119892

143111 100 0354 2795 7322 104 13925 3 times 10minus6 212 5445 45375 068

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

10minus1

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 6 Deconvolution for a drawdown test in a dual porosityreservoir

shown in Table 1 In this case we provide the deconvolvedpressure data using the explicit deconvolution formula andthe material balance deconvolution method that is presentedin this work The data are taken from a pressure buildup testand can be reasonably used as field data The deconvolutionresults are shown in Figure 9 in semilog plot As it isillustrated from Figure 9 the corrected pressure provides aclear straight line starting as early as 15 hours in contrast tothe vague line from the uncorrected pressure response

And the last (field) example is a gas condensatewell buildup test following a variable rate productionhistory The duration of last pressure buildup test is361556 hours (from 1889 to 2250556 hours) The basicreservoir and fluid properties for the last drawdown testare shown in Table 2 However the properties of otherproduction tests are not shown here To use our newtechniques effectively for gas wells we use normalizedpseudopressure and pseudotime functions as defined byMeunier et al [24]The final results of explicit deconvolutionformula and material balance deconvolution methodshow two different slopes (Figure 10) where the earlytimes slope is greater than the last one This identifiesthe radial composite behavior of the reservoir With

the corrected pressures the reservoir parameters can also beestimated

6 Conclusions

In this work we have provided some explicit deconvolutionmethods for restoration of constant rate pressure responsesfrom measured pressure data dominated by wellbore storageeffects without sandface rate measurements Based on ourinvestigations we can state the following specific conclusionsrelated to these techniques

(1) The results obtained from the synthetic and fieldexamples show the practical and computational effi-ciency of our deconvolution methods

(2) Modification of 120573-deconvolution is presented wherethe beta quantity is variable rather than a constant

(3) For situations in which there are no downhole ratemeasurements the explicit deconvolution methods(blind deconvolution approaches) can be used toderive the downhole rate function from the pressuremeasurements and restore the reservoir pressureresponse

(4) We can unveil the early time behavior of a reservoirsystemmasked by wellbore storage effects using thesenew approaches Wellbore geometries and reservoirtypes can be identified from this restored pressuredata Furthermore the conventional methods canbe used to analyze this restored formation pressureto estimate formation parameters So the presentedtechniques can be used as the practical tools toimprove pressure transient test interpretation

(5) Deconvolution in well test analysis is often expressedas an inverse problem This approach can be used asan idea to justify the rate normalization method andthe material balance deconvolution method

Appendices

A Russel Method and Its Modification

A method for correcting pressure buildup pressure data freeof wellbore storage effects at early times has been developedby Russell [14] In this method it is not necessary to measurethe rate of influx after closing in at the surface Instead oneuses a theoretical equation which gives the form that thebottom hole pressure should have fluid accumulates in the

8 Abstract and Applied Analysis

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

250

200

150

100

50

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 7 Deconvolution for a drawdown test in a reservoir with asealing fault

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 8 Deconvolution for a drawdown test in a reservoir with asealing fault

wellbore during buildup This leads to the result that oneshould plot

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

(A1)

versus log (Δ119905) in analyzing pressure buildup data during theearly fill-up period The denominator makes a correction for

104103102101100

Horner time ratio

800

700

600

500

400

300

200

100

0

Dpws

(psi)

Dpws wellbore storageDps explicit deconvolution methodDps material balance deconvolution

Figure 9 Restored pressures for a buildup test in a homogeneousreservoir

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dpa

(psi)

Dpaw wellbore storage explicit deconvolution method material balance deconvolution

DpasDpas

Figure 10 Restored pressures for a buildup test in a radial compositereservoir

the gradually decreasing flow into the wellbore The quantity1198622is obtained by trial and error as the value which makes the

curve straight at early times After obtaining the straight linesection the rest of the analysis is the same as for any otherpressure buildup This method has the advantage or requiresno additional data over that taken routinely during a shut intest

Due to the time consuming and unconfident of trial anderror to determine 119862

2in Russell method we modify this

method Russellrsquos wellbore storage correction is given as

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

= 119891 (Δ119905 = 1 hr) + 119898119904119897log (Δ119905)

(A2)

Abstract and Applied Analysis 9

Defining the following function and using least squarestechnique obtain the final result

119864 (1198622 119891119898)

=

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891(1198622) minus 119898(119862

2) log(Δ119905

119894))

2

(A3)

Tominimize the function119864 it can differentiate119864with respectto 1198622 119891 and 119898 and setting the results equal to zero After

differentiation it reduces to

120597119864

1205971198622

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894) ) = 0

(A4)

120597119864

120597119891

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622) minus 119898 (119862

2) log (Δ119905

119894))

= 0

(A5)120597119864

120597119898

= minus2

119873

sum

119894=1

log (Δ119905119894) (

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus 119898 (1198622) log (Δ119905

119894)) = 0

(A6)

From (A4) we define the following function

119883(1198622)

=

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times(Δ119901119908119904119894

(1minus (11198622Δ119905119894))minus119891 (119862

2) minus119898 (119862

2) log (Δ119905

119894))

= 0

(A7)

The roots of (A7) can be found via the following algorithm

1198622(119894+1)

= 1198622(119894)

minus119883 (1198622)

1198831015840 (1198622) (A8)

where

119883(1198622) =

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894))

1198831015840(1198622) =

119873

sum

119894=1

(minus2Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3)

times (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2

times [Δ119901119908119904119894

(1 minus (11198622Δ119905))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894)])

+ (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (minusΔ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3minus 1198911015840(1198622)

minus1198981015840(1198622) log (Δ119905

119894))

(A9)

where the unknown functions are given by the followingrelations

119891 (1198622) =

120573119906 (1198622) minus 120572V (119862

2)

1205732 minus 119873120572(A10)

whose derivative is

1198911015840(1198622) =

1205731199061015840(1198622) minus 120572V1015840 (119862

2)

1205732 minus 119873120572

119898 (1198622) =

119873119906 (1198622) minus 120573V (119862

2)

119873120572 minus 1205732

(A11)

where its derivative is

1198981015840(1198622) =

1198731199061015840(1198622) minus 120573V1015840 (119862

2)

119873120572 minus 1205732

119906 (1198622) =

119873

sum

119894=1

log (Δ119905119894)

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A12)

10 Abstract and Applied Analysis

and after differentiation it yields

1199061015840(1198622) = minus

119873

sum

119894=1

log (Δ119905

119894)

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2

V (1198622) =

119873

sum

119894=1

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A13)

which has the following derivative

V1015840 (1198622) = minus

119873

sum

119894=1

1

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2 (A14)

finally the values of 120572 and 120573 parameters are given by thefollowing formulas

120572 =

119873

sum

119894=1

(log(Δ119905119894))2

120573 =

119873

sum

119894=1

log (Δ119905119894)

(A15)

B General Formula for Inverse Problem

The convolution of (6) yields

119901119904119863

(119905119863) = int

119905119863

0

119870 (120591) 119901119908119863 (119905119863 minus 120591) 119889120591

= int

119905119863

0

119870(119905119863minus 120591) 119901

119908119863 (120591) 119889120591

(B1)

where

119870(119905119863) = 119871minus1(

1

119904119902 (119904)) (B2)

where 119871minus1 is the inverse Lapalce transform operator 119870(119905119863)

can be computed either from the Laplace transforms of119902119863(119905119863) data a curve-fitted equation of 119902

119863(119905119863) data or using

another technique with a relation between119870(119905119863) and 119902

119863(119905119863)

However it would be time consuming to invert all the 119902119863(119905119863)

data in Laplace space and transform it back to real space inaccordance with (B2) Once an approximation is obtained itwill be easy to compute119870(119905

119863) and integrate (B1) to determine

119901119904119863(119905119863) However finding some approximation functions for

119902119863(119905119863) data may be difficult We provide an approach to

overcome this problemThis idea is based on finding anotherrelation between 119870(119905

119863) and 119902

119863(119905119863) that is

ℓ (1) =1

119904=

1

119904119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B3)or

1 = int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591 = int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591

(B4)

The other relations that can be used are

ℓ (119905119863) =

1

1199042=

1

1199042119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B5)

or

119905119863= int

119905119863

0

119871 (120591) 119902119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

119863(120591) 119889120591

(B6)

ℓ (1) =1

119904=

1

1199042119902119863(119904)

119904119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591)

(B7)

or

1 = int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591

(B8)

where

119871 (119905119863) = ℓminus1(

1

1199042119902 (119904)) (B9)

Equation (B4) can be used simultaneously with (B1) toanalyze the deconvolution problem

Equation (11) can be written as

119901119904119863

(119904) = 119904119901119908119863

(119904)1

1199042119902119863(119904)

(B10)

or

119901119904119863

(119905119863) = int

119905119863

0

119871 (120591) 1199011015840

119908119863(119905119863minus 120591) 119889120591

= int

119905119863

0

119871 (119905119863minus 120591) 119901

1015840

119908119863(120591) 119889120591

(B11)

The applications of (B6) and (B8) are with (B11)For justifying the rate normalizationmethod andmaterial

balance deconvolution it can use (B1) (B4) ((B11) can alsobe used with (B6) or (B8) for this goal) and generalizedmean value theorem for integrals This theorem is expressedas follows

Theorem 1 Let 119891 and 119892 be continuous on [119886 119887] and supposethat 119892 does not change sign on [119886 119887] Then there is a point 119888 in[119886 119887] such that

int

119887

119886

119891 (119905) 119892 (119905) 119889119905 = 119891 (119888) int

119887

119886

119892 (119905) 119889119905 (B12)

Abstract and Applied Analysis 11

By Theorem 1 there are some values 1198881and 1198882between 0

and 119905119863such that

119901119904119863

(119905119863) = 119901119908119863

(1198881) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B13)

1 = 119902119863(1198882) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B14)

By setting 1198881

= 1198882

= 119905119863and dividing the right and left

side of (B13) respectively the rate normalization techniqueis developed However the values 119888

1and 1198882are not equal

to each other and are not equal to 119905119863

in general Thisis the reason of error in rate normalization and materialbalance deconvolution methods 119888

1and 1198882are less than 119905

119863in

these methods for the duration of wellbore storage distortionduring a pressure transient test Since these conditions areapproximately satisfied in material balance deconvolutionthe results of material balance are more accurate and moreconfident than rate normalization

Nomenclature

API API gravity ∘API119861119900 Oil formation volume factor RBSTB

119888119905 Total isothermal compressibility factor

psiminus1

1198622 Arbitrary constant hr

minus1

119862119863 Wellbore storage constant dimensionless

119862119904 Wellbore storage coefficient bblpsi

ℎ Formation thickness ftHTR Horner time ratio dimensionless119896 Formation permeability md119870 Kernel of the convolution integral119871 Kernel of the convolution integral119873119901 Cumulative oil production bbl

119901 Pressure psi119901119886 Normalized pseudopressure psi

119901 Average reservoir pressure psi119901119863 Formation pressure dimensionless

119901119889119901 Dewpoint pressure psi

119901119894 Initial reservoir pressure psi

119901119904119863 Formation pressure including skin

dimensionless119901119904 Formation pressure including skin psi

119901119908119863

Pressure draw down dimensionless119901119908119891 Flowing bottom hole pressure psi

119901119908119904 Shut-in bottom hole pressure psi

119902119863 Sandface flow rate dimensionless

119902119892 Gas flow rate MscfD

119902119900 Oil flow rate STBD

119903119908 Wellbore radius ft

119904 Laplace variable119878 Skin factor119905 Time hr119905119863 Time dimensionless

119905119901 Production time hr

119879 Reservoir temperature ∘F

119879119889119901 Dewpoint temperature ∘F

120572 Explicit deconvolution variabledimensionless

120573 Beta deconvolution variabledimensionless

120574119892 Gas gravity (air = 10)

120582 Interporosity flow coefficient of dualporosity reservoir

120601 Porosity of reservoir rock dimensionless120583119892 Gas viscosity cp

120583119900 Oil viscosity cp

120596 Storativity ratio of dual porosity reservoirminus Laplace transform of a function1015840 Derivative of a function

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Cheng W J Lee and D A McVay ldquoFast-Fourier-transform-based deconvolution for interpretation of pressure-transient-test data dominated by Wellbore storagerdquo SPE Reservoir Eval-uation and Engineering vol 8 no 3 pp 224ndash239 2005

[2] M M Levitan and M R Wilson ldquoDeconvolution of pressureand rate data from gas reservoirs with significant pressuredepletionrdquo SPE Journal vol 17 no 3 pp 727ndash741 2012

[3] Y Liu and R N Horne ldquoInterpreting pressure and flow-ratedata from permanent downhole gauges by use of data-miningapproachesrdquo SPE Journal vol 18 no 1 pp 69ndash82 2013

[4] O Ogunrewo and A C Gringarten ldquoDeconvolution of welltest data in lean and rich gas condensate and volatile oil wellsbelow saturation pressurerdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Document ID 166340-MS New Orleans La USA September-October 2013

[5] R C Earlougher Jr Advances in Well Test Analysis vol 5 ofMonogragh Series SPE Richardson Tex USA 1977

[6] F Kucuk and L Ayestaran ldquoAnalysis of simultaneously mea-sured pressure and sandface flow rate in transient well testingrdquoJournal of PetroleumTechnology vol 37 no 2 pp 323ndash334 1985

[7] M Onur M Cinar D IIk P P Valko T A Blasingame and PHegeman ldquoAn investigation of recent deconvolution methodsfor well-test data analysisrdquo SPE Journal vol 13 no 2 pp 226ndash247 2008

[8] F J KuchukMOnur andFHollaender ldquoPressure transient testdesign and interpretationrdquo Developments in Petroleum Sciencevol 57 pp 303ndash359 2010

[9] A C Gringarten D P Bourdet P A Landel and V J KniazeffldquoA comparison between different skin and wellbore storagetype-curves for early-time transient analysisrdquo in Proceedings ofthe SPE Annual Technical Conference and Exhibition Paper SPE8205 Las Vegas Nev USA 1979

[10] A C Gringarten ldquoFrom straight lines to deconvolution theevolution of the state of the art in well test analysisrdquo SPEReservoir Evaluation and Engineering vol 11 no 1 pp 41ndash622008

[11] J A Cumming D A Wooff T Whittle and A C GringartenldquoMultiple well deconvolutionrdquo in Proceedings of the SPE Annual

12 Abstract and Applied Analysis

Technical Conference and Exhibition Document ID 166458-MS New Orleans La USA September-October 2013

[12] O Bahabanian D Ilk N Hosseinpour-Zonoozi and T ABlasingame ldquoExplicit deconvolution of well test data distortedby wellbore storagerdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Paper SPE 103216 pp 4444ndash4453San Antonio Tex USA September 2006

[13] J Jones and A Chen ldquoSuccessful applications of pressure-ratedeconvolution in theCad-Nik tight gas formations of the BritishColumbia foothillsrdquo Journal of Canadian Petroleum Technologyvol 51 no 3 pp 176ndash192 2012

[14] D G Russell ldquoExtension of pressure buildup analysis methodsrdquoJournal of Petroleum Technology vol 18 no 12 pp 1624ndash16361966

[15] R E Gladfelter G W Tracy and L E Wilsey ldquoSelecting wellswhich will respond to production-stimulation treatmentrdquo inDrilling and Production Practice API Dallas Tex USA 1955

[16] M J Fetkovich and M E Vienot ldquoRate normalization ofbuildup pressure by using afterflow datardquo Journal of PetroleumTechnology vol 36 no 13 pp 2211ndash2224 1984

[17] J L Johnston Variable rate analysis of transient well testdata using semi-analytical methods [MS thesis] Texas AampMUniversity College Station Tex USA 1992

[18] A F van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 6 pp 171ndash176 1953

[19] W Hurst ldquoEstablishment of the skin effect and its impedimentto fluid flow into a well borerdquo The Petroleum Engineer pp B6ndashB16 1953

[20] J A Joseph and L F Koederitz ldquoA simple nonlinear model forrepresentation of field transient responsesrdquo Paper SPE 11435SPE Richardson Tex USA 1982

[21] F J Kuchuk ldquoGladfelter deconvolutionrdquo SPE Formation Evalu-ation vol 5 no 3 pp 285ndash292 1990

[22] A F van Everdingen and W Hurst ldquoThe application of thelaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[23] D Bourdet T M Whittle A A Douglas and Y M Pirard ldquoAnew set of type curves simplifies well test analysisrdquo World Oilpp 95ndash106 1983

[24] D F Meunier C S Kabir and M J Wittmann ldquoGas well testanalysis use of normalized pseudovariablesrdquo SPE FormationEvaluation vol 2 no 4 pp 629ndash636 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Using elementary calculus yields

ln (1003816100381610038161003816119891(120591)

1003816100381610038161003816)1003816100381610038161003816119905119863

0= ln (1003816100381610038161003816119891 (119905

119863)1003816100381610038161003816) minus ln (1) = int

119905119863

0

11990110158401015840

119908119863(120591)

1199011015840119908119863

(120591)119889120591

(28)

So

119891 (119905119863) = 119890120572(119905119863) (29)

where

120572 (119905119863) = int

119905119863

0

11990110158401015840

119908119863(120591)

1199011015840119908119863

(120591)119889120591 (30)

Now the explicit deconvolution of data based on (20) can bewritten as follows

119901119904119863

(119905119863) = minus

1

120572 (119905119863) 119905119863

1199011015840

119908119863+ 119901119908119863

(119905119863) (31)

or

119901119904119863

(119905119863) = minus

119905119863

120572 (119905119863)1199011015840

119908119863+ 119901119908119863

(119905119863) (32)

that is similar to (19) The other similarity between (19) and(32) is that the plot of (32) versus (int119905119863

0119902119863(120591)119889120591119902

119863(119905119863)) and

(int119905119863

0(1 minus 119902

119863(120591))119889120591(1 minus 119902

119863(119905119863))) for drawdown and buildup

tests respectively yields the more accurate results and isusable for practical applications The applicability of (32) isalso for duration of wellbore storage effects And for the othertimes the given pressure data are correct However using thismethod for the later times yields not bad results

3 Material Balance Deconvolution

Material balance deconvolution is an extension of the ratenormalization method Johnston [17] defines a new 119909-axisplotting function (material balance time) that provides anapproximate deconvolution of the variable rate pressuretransient problem There are numerous assumptions associ-ated with the material balance deconvolution methods oneof the most widely accepted assumptions is that the rateprofilemust change smoothly andmonotonically In practicalterms this condition should be met for the wellbore storageproblem

The general form of material balance deconvolution isprovided for the pressure drawdown case in terms of thematerial balance time function and the rate-normalizedpressure drop function The material balance time functionis given as

119905119898119887

=119873119901

119902119863 (119905)

(33)

where

119873119901= int

119905

0

119902119863 (120591) 119889120591 (34)

The rate-normalized pressure drop function is given by

Δ119901119908119891 (119905)

119902119863 (119905)

=119901119894minus 119901119908119891 (119905)

119902119863 (119905)

(35)

The material balance time function for the pressure buildupcase is given as

119905119898119887

=119873119901

1 minus 119902119863 (Δ119905)

(36)

where

119873119901= int

Δ119905

0

(1 minus 119902119863 (120591)) 119889120591 (37)

The rate-normalized pressure drop function for the pressurebuildup case is given as

Δ119901119908119904 (Δ119905)

1 minus 119902119863 (Δ119905)

=119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)

1 minus 119902119863 (Δ119905)

(38)

4 Methodology

In this work several methods are provided to restore thepressure datawithout any sandface flow rate informationThevalues of sandface flow rate data for all methods that requirethese data may be obtained from (20) We evaluated a veryold correctionmethod by Russell [14] However we modifiedthis method by least square method but it was found thatthis method is to be unacceptable for all applications Thederivation of this method and its modification is given inAppendix A

The 120573-deconvolution and its modification are anothertechnique that is mentioned in this study The formulationof this method is presented in Section 21 The other explicitdeconvolutionmethod is an approach similar to themodified120573-deconvolution that uses only the pressure data for decon-volution of pressure and rate data The formulation of thismethod was derived in Section 22 The 120573-deconvolution itsmodification and explicit deconvolution formula can be usedfor practical applications The applicability of these methodsis for duration of wellbore storage effects However usingthese approaches for the other times yields not bad results

The other technique is material balance deconvolutionmethod [17] that has sufficiently accurate resultsThismethodcan also be used as a good approach for practical applicationsThe formulas for this method were given in Section 3

In Appendix B deconvolution is expressed as an inverseproblem that is analyzed by a tricky idea Based on thisidea an interesting proof is given for rate normalizationmethod and material balance deconvolution that is a goodapproach to find the errors and limitations of these methodsand overcome them

Abstract and Applied Analysis 5

10minus3 10minus2 10minus1 100 101 102 1030

500

1000

1500

2000

2500

3000

3500

4000

Time (hr)

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolution methodd(Dps)d(ln(t)) material balance deconvolution methodDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 1 Deconvolution for a drawdown test in a homogeneousreservoir

From the above-mentioned methods we use the fol-lowing two techniques to show the verification of theseapproaches

Explicit Deconvolution Formula The final result of explicitdeconvolution method is given as follows

119901119904119863

(119905119863) = minus

119905119863

120572 (119905119863)1199011015840

119908119863+ 119901119908119863

(119905119863) (39)

Plotting of this term versus (int119905119863

0119902119863(120591)119889120591119902

119863(119905119863)) and

(int119905119863

0(1 minus 119902

119863(120591))119889120591(1 minus 119902

119863(119905119863))) for drawdown and buildup

tests respectively yields more accurate resultsThis approachperforms well in field applications

Material Balance Deconvolution The material balance pres-sure drop function for the pressure drawdown test is givenas

Δ119901119908119891 (119905)

119902119863 (119905)

=119901119894minus 119901119908119891 (119905)

119902119863 (119905)

(40)

and the material balance time function for the pressuredrawdown test is given as

119905119898119887

=int119905119863

0119902119863 (120591) 119889120591

119902119863 (119905)

(41)

For the buildup test the material balance pressure dropfunction can be written as

Δ119901119908119904 (Δ119905)

1 minus 119902119863 (Δ119905)

=119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)

1 minus 119902119863 (Δ119905)

(42)

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

104

103

102

101Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolution methodd(Dps)d(ln(t)) material balance deconvolution methodDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Figure 2 Deconvolution for a drawdown test in a homogeneousreservoir

And the material balance time function can be written as

119905119898119887

=int119905119863

0(1 minus 119902

119863 (120591)) 119889120591

1 minus 119902119863 (Δ119905)

(43)

This technique is a good approach for practical applications

5 Verification Examples

We first use few synthetic cases for different reser-voirwellbore systems to validate the applicability of explicitdeconvolution formula and material balance deconvolutionmethod Then the field applications of these techniqueswill be shown by two real cases The unloading rates arecalculated with (20)

The first case is a synthetic test derived from a test designin a vertical oil wellThe reservoir is homogenous and infiniteacting with a constant wellbore storage The wellbore storagecoefficient is 119862

119904= 0015 bblpsi The pressure data are

obtained from the cylindrical source solution (test design)We used the explicit deconvolution formula and material

balance deconvolution method to deconvolve the pressureand unloading rate distorting by wellbore storage effectsTheresults are presented in Figures 1 and 2 for the semilog andlog-log plots in the case of drawdown test and Figures 3and 4 for the semilog and log-log plots in the case of builduptest Besides the deconvolved pressures and their logarithmicderivatives pressure responses with and without wellborestorage are also plotted on the figures for comparisonThrough the deconvolution process we have successfullyremoved the wellbore storage effects so that radial flow canbe identified at early test times The deconvolved pressuresand their logarithmic derivatives are almost identical to

6 Abstract and Applied Analysis

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dp wellbore storage

Dps explicit deconvolution method(ln

Dps material balance deconvolution

Dps no wellbore storage

ws

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpws

d(Dp )d(ln )) wellbore storagews (HTR

d(Dps)d )) explicit deconvolution method(HTR

d(Dps)d(ln )) no wellbore storage(HTR)

d(Dps)d(ln )) material balance deconvolution (HTR

Figure 3 Deconvolution for a buildup test in a homogeneousreservoir

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

104

103

102

101

ws

ws

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpws

Dp wellbore storage

Dps explicit deconvolution method

Dps material balance deconvolution

Dps no wellbore storage

d(Dp )d(ln )) wellbore storagews (HTR

d(Dps)d )) explicit deconvolution method(HTR

d(Dps)d(ln )) no wellbore storage(HTR)

d(Dps)d(ln )) material balance deconvolution (HTR

(ln

Figure 4 Deconvolution for a buildup test in a homogeneousreservoir

the analytic solutions (without wellbore storage) whichenables us to estimate reservoir parameters accurately

The second case is an oil well drawdown test in adual porosity reservoir The storativity ratio (120596) and inter-porosity flow coefficient (120582) are assumed to be 01 and10minus6 respectively The constant wellbore storage coefficient

is 119862119904= 015 bblpsi The dimensionless wellbore pressure

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

600

500

400

300

200

100

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 5 Deconvolution for a drawdown test in a dual porosityreservoir

data were obtained from the cylindrical source solution withpseudosteady-state interporosity flow (test design) It is notpossible to identify the dual porosity behavior characteristicof the reservoir from wellbore pressure data because thisbehavior has been masked completely by wellbore storage asindicated in Figures 5 and 6

We deconvolved the pressure and unloading rate andpresent the results in Figures 5 and 6 The deconvolvedresults are reasonably consistent with the analytical solutionAfter deconvolution the dual porosity behavior can beidentified easily from the valley in the logarithmic derivativeof pressure If wellbore storage effects were not removed atearly times it easily could have incorrectly identified thereservoir system as infinite actingThe estimation of reservoirparameters can be obtained now from deconvolved pressure

The third case is an oil well drawdown test in a reservoirwith a sealing fault 250 ft from the well Synthetic pressureresponses were generated by simulation (test design) with aconstant wellbore storage coefficient of 01 bblpsi As we cansee in Figures 7 and 8 the early time infinite acting reservoirbehavior has been masked completely by wellbore storage

To recover the hidden reservoir behavior we performeddeconvolution on the pressure change to remove the wellborestorage effect As in Figures 7 and 8 the deconvolved pressurechange and logarithmic derivative data are almost identicalto the responses without wellbore storageThe deconvolutionresults clearly indicate parallel horizontal lines for the loga-rithmic derivative However without wellbore storage effectselimination this maymislead to identify the reservoir systemas infinite acting So this allows us to correctly identify thereservoir system having a sealing fault

The fourth case is a field example that is taken from theliterature [23] The basic reservoir and fluid properties are

Abstract and Applied Analysis 7

Table 1 Reservoir rock and fluid properties for the first field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119902

119900(STBD) 119861

119900(RBSTB) 119888

119905(psiminus1) 120583

119900(cp) 120601

1533 107 029 174 106 42 times 10minus6 25 025

Table 2 Reservoir rock and fluid properties for the second field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119879 (∘F) 119901 (psi) 119902

119900(STBD) 119902

119892(MscfD) 119888

119891(psiminus1) 119879

119889119901(∘F) 119901

119889119901(psi) ∘API 120574

119892

143111 100 0354 2795 7322 104 13925 3 times 10minus6 212 5445 45375 068

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

10minus1

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 6 Deconvolution for a drawdown test in a dual porosityreservoir

shown in Table 1 In this case we provide the deconvolvedpressure data using the explicit deconvolution formula andthe material balance deconvolution method that is presentedin this work The data are taken from a pressure buildup testand can be reasonably used as field data The deconvolutionresults are shown in Figure 9 in semilog plot As it isillustrated from Figure 9 the corrected pressure provides aclear straight line starting as early as 15 hours in contrast tothe vague line from the uncorrected pressure response

And the last (field) example is a gas condensatewell buildup test following a variable rate productionhistory The duration of last pressure buildup test is361556 hours (from 1889 to 2250556 hours) The basicreservoir and fluid properties for the last drawdown testare shown in Table 2 However the properties of otherproduction tests are not shown here To use our newtechniques effectively for gas wells we use normalizedpseudopressure and pseudotime functions as defined byMeunier et al [24]The final results of explicit deconvolutionformula and material balance deconvolution methodshow two different slopes (Figure 10) where the earlytimes slope is greater than the last one This identifiesthe radial composite behavior of the reservoir With

the corrected pressures the reservoir parameters can also beestimated

6 Conclusions

In this work we have provided some explicit deconvolutionmethods for restoration of constant rate pressure responsesfrom measured pressure data dominated by wellbore storageeffects without sandface rate measurements Based on ourinvestigations we can state the following specific conclusionsrelated to these techniques

(1) The results obtained from the synthetic and fieldexamples show the practical and computational effi-ciency of our deconvolution methods

(2) Modification of 120573-deconvolution is presented wherethe beta quantity is variable rather than a constant

(3) For situations in which there are no downhole ratemeasurements the explicit deconvolution methods(blind deconvolution approaches) can be used toderive the downhole rate function from the pressuremeasurements and restore the reservoir pressureresponse

(4) We can unveil the early time behavior of a reservoirsystemmasked by wellbore storage effects using thesenew approaches Wellbore geometries and reservoirtypes can be identified from this restored pressuredata Furthermore the conventional methods canbe used to analyze this restored formation pressureto estimate formation parameters So the presentedtechniques can be used as the practical tools toimprove pressure transient test interpretation

(5) Deconvolution in well test analysis is often expressedas an inverse problem This approach can be used asan idea to justify the rate normalization method andthe material balance deconvolution method

Appendices

A Russel Method and Its Modification

A method for correcting pressure buildup pressure data freeof wellbore storage effects at early times has been developedby Russell [14] In this method it is not necessary to measurethe rate of influx after closing in at the surface Instead oneuses a theoretical equation which gives the form that thebottom hole pressure should have fluid accumulates in the

8 Abstract and Applied Analysis

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

250

200

150

100

50

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 7 Deconvolution for a drawdown test in a reservoir with asealing fault

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 8 Deconvolution for a drawdown test in a reservoir with asealing fault

wellbore during buildup This leads to the result that oneshould plot

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

(A1)

versus log (Δ119905) in analyzing pressure buildup data during theearly fill-up period The denominator makes a correction for

104103102101100

Horner time ratio

800

700

600

500

400

300

200

100

0

Dpws

(psi)

Dpws wellbore storageDps explicit deconvolution methodDps material balance deconvolution

Figure 9 Restored pressures for a buildup test in a homogeneousreservoir

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dpa

(psi)

Dpaw wellbore storage explicit deconvolution method material balance deconvolution

DpasDpas

Figure 10 Restored pressures for a buildup test in a radial compositereservoir

the gradually decreasing flow into the wellbore The quantity1198622is obtained by trial and error as the value which makes the

curve straight at early times After obtaining the straight linesection the rest of the analysis is the same as for any otherpressure buildup This method has the advantage or requiresno additional data over that taken routinely during a shut intest

Due to the time consuming and unconfident of trial anderror to determine 119862

2in Russell method we modify this

method Russellrsquos wellbore storage correction is given as

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

= 119891 (Δ119905 = 1 hr) + 119898119904119897log (Δ119905)

(A2)

Abstract and Applied Analysis 9

Defining the following function and using least squarestechnique obtain the final result

119864 (1198622 119891119898)

=

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891(1198622) minus 119898(119862

2) log(Δ119905

119894))

2

(A3)

Tominimize the function119864 it can differentiate119864with respectto 1198622 119891 and 119898 and setting the results equal to zero After

differentiation it reduces to

120597119864

1205971198622

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894) ) = 0

(A4)

120597119864

120597119891

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622) minus 119898 (119862

2) log (Δ119905

119894))

= 0

(A5)120597119864

120597119898

= minus2

119873

sum

119894=1

log (Δ119905119894) (

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus 119898 (1198622) log (Δ119905

119894)) = 0

(A6)

From (A4) we define the following function

119883(1198622)

=

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times(Δ119901119908119904119894

(1minus (11198622Δ119905119894))minus119891 (119862

2) minus119898 (119862

2) log (Δ119905

119894))

= 0

(A7)

The roots of (A7) can be found via the following algorithm

1198622(119894+1)

= 1198622(119894)

minus119883 (1198622)

1198831015840 (1198622) (A8)

where

119883(1198622) =

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894))

1198831015840(1198622) =

119873

sum

119894=1

(minus2Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3)

times (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2

times [Δ119901119908119904119894

(1 minus (11198622Δ119905))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894)])

+ (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (minusΔ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3minus 1198911015840(1198622)

minus1198981015840(1198622) log (Δ119905

119894))

(A9)

where the unknown functions are given by the followingrelations

119891 (1198622) =

120573119906 (1198622) minus 120572V (119862

2)

1205732 minus 119873120572(A10)

whose derivative is

1198911015840(1198622) =

1205731199061015840(1198622) minus 120572V1015840 (119862

2)

1205732 minus 119873120572

119898 (1198622) =

119873119906 (1198622) minus 120573V (119862

2)

119873120572 minus 1205732

(A11)

where its derivative is

1198981015840(1198622) =

1198731199061015840(1198622) minus 120573V1015840 (119862

2)

119873120572 minus 1205732

119906 (1198622) =

119873

sum

119894=1

log (Δ119905119894)

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A12)

10 Abstract and Applied Analysis

and after differentiation it yields

1199061015840(1198622) = minus

119873

sum

119894=1

log (Δ119905

119894)

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2

V (1198622) =

119873

sum

119894=1

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A13)

which has the following derivative

V1015840 (1198622) = minus

119873

sum

119894=1

1

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2 (A14)

finally the values of 120572 and 120573 parameters are given by thefollowing formulas

120572 =

119873

sum

119894=1

(log(Δ119905119894))2

120573 =

119873

sum

119894=1

log (Δ119905119894)

(A15)

B General Formula for Inverse Problem

The convolution of (6) yields

119901119904119863

(119905119863) = int

119905119863

0

119870 (120591) 119901119908119863 (119905119863 minus 120591) 119889120591

= int

119905119863

0

119870(119905119863minus 120591) 119901

119908119863 (120591) 119889120591

(B1)

where

119870(119905119863) = 119871minus1(

1

119904119902 (119904)) (B2)

where 119871minus1 is the inverse Lapalce transform operator 119870(119905119863)

can be computed either from the Laplace transforms of119902119863(119905119863) data a curve-fitted equation of 119902

119863(119905119863) data or using

another technique with a relation between119870(119905119863) and 119902

119863(119905119863)

However it would be time consuming to invert all the 119902119863(119905119863)

data in Laplace space and transform it back to real space inaccordance with (B2) Once an approximation is obtained itwill be easy to compute119870(119905

119863) and integrate (B1) to determine

119901119904119863(119905119863) However finding some approximation functions for

119902119863(119905119863) data may be difficult We provide an approach to

overcome this problemThis idea is based on finding anotherrelation between 119870(119905

119863) and 119902

119863(119905119863) that is

ℓ (1) =1

119904=

1

119904119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B3)or

1 = int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591 = int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591

(B4)

The other relations that can be used are

ℓ (119905119863) =

1

1199042=

1

1199042119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B5)

or

119905119863= int

119905119863

0

119871 (120591) 119902119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

119863(120591) 119889120591

(B6)

ℓ (1) =1

119904=

1

1199042119902119863(119904)

119904119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591)

(B7)

or

1 = int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591

(B8)

where

119871 (119905119863) = ℓminus1(

1

1199042119902 (119904)) (B9)

Equation (B4) can be used simultaneously with (B1) toanalyze the deconvolution problem

Equation (11) can be written as

119901119904119863

(119904) = 119904119901119908119863

(119904)1

1199042119902119863(119904)

(B10)

or

119901119904119863

(119905119863) = int

119905119863

0

119871 (120591) 1199011015840

119908119863(119905119863minus 120591) 119889120591

= int

119905119863

0

119871 (119905119863minus 120591) 119901

1015840

119908119863(120591) 119889120591

(B11)

The applications of (B6) and (B8) are with (B11)For justifying the rate normalizationmethod andmaterial

balance deconvolution it can use (B1) (B4) ((B11) can alsobe used with (B6) or (B8) for this goal) and generalizedmean value theorem for integrals This theorem is expressedas follows

Theorem 1 Let 119891 and 119892 be continuous on [119886 119887] and supposethat 119892 does not change sign on [119886 119887] Then there is a point 119888 in[119886 119887] such that

int

119887

119886

119891 (119905) 119892 (119905) 119889119905 = 119891 (119888) int

119887

119886

119892 (119905) 119889119905 (B12)

Abstract and Applied Analysis 11

By Theorem 1 there are some values 1198881and 1198882between 0

and 119905119863such that

119901119904119863

(119905119863) = 119901119908119863

(1198881) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B13)

1 = 119902119863(1198882) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B14)

By setting 1198881

= 1198882

= 119905119863and dividing the right and left

side of (B13) respectively the rate normalization techniqueis developed However the values 119888

1and 1198882are not equal

to each other and are not equal to 119905119863

in general Thisis the reason of error in rate normalization and materialbalance deconvolution methods 119888

1and 1198882are less than 119905

119863in

these methods for the duration of wellbore storage distortionduring a pressure transient test Since these conditions areapproximately satisfied in material balance deconvolutionthe results of material balance are more accurate and moreconfident than rate normalization

Nomenclature

API API gravity ∘API119861119900 Oil formation volume factor RBSTB

119888119905 Total isothermal compressibility factor

psiminus1

1198622 Arbitrary constant hr

minus1

119862119863 Wellbore storage constant dimensionless

119862119904 Wellbore storage coefficient bblpsi

ℎ Formation thickness ftHTR Horner time ratio dimensionless119896 Formation permeability md119870 Kernel of the convolution integral119871 Kernel of the convolution integral119873119901 Cumulative oil production bbl

119901 Pressure psi119901119886 Normalized pseudopressure psi

119901 Average reservoir pressure psi119901119863 Formation pressure dimensionless

119901119889119901 Dewpoint pressure psi

119901119894 Initial reservoir pressure psi

119901119904119863 Formation pressure including skin

dimensionless119901119904 Formation pressure including skin psi

119901119908119863

Pressure draw down dimensionless119901119908119891 Flowing bottom hole pressure psi

119901119908119904 Shut-in bottom hole pressure psi

119902119863 Sandface flow rate dimensionless

119902119892 Gas flow rate MscfD

119902119900 Oil flow rate STBD

119903119908 Wellbore radius ft

119904 Laplace variable119878 Skin factor119905 Time hr119905119863 Time dimensionless

119905119901 Production time hr

119879 Reservoir temperature ∘F

119879119889119901 Dewpoint temperature ∘F

120572 Explicit deconvolution variabledimensionless

120573 Beta deconvolution variabledimensionless

120574119892 Gas gravity (air = 10)

120582 Interporosity flow coefficient of dualporosity reservoir

120601 Porosity of reservoir rock dimensionless120583119892 Gas viscosity cp

120583119900 Oil viscosity cp

120596 Storativity ratio of dual porosity reservoirminus Laplace transform of a function1015840 Derivative of a function

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Cheng W J Lee and D A McVay ldquoFast-Fourier-transform-based deconvolution for interpretation of pressure-transient-test data dominated by Wellbore storagerdquo SPE Reservoir Eval-uation and Engineering vol 8 no 3 pp 224ndash239 2005

[2] M M Levitan and M R Wilson ldquoDeconvolution of pressureand rate data from gas reservoirs with significant pressuredepletionrdquo SPE Journal vol 17 no 3 pp 727ndash741 2012

[3] Y Liu and R N Horne ldquoInterpreting pressure and flow-ratedata from permanent downhole gauges by use of data-miningapproachesrdquo SPE Journal vol 18 no 1 pp 69ndash82 2013

[4] O Ogunrewo and A C Gringarten ldquoDeconvolution of welltest data in lean and rich gas condensate and volatile oil wellsbelow saturation pressurerdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Document ID 166340-MS New Orleans La USA September-October 2013

[5] R C Earlougher Jr Advances in Well Test Analysis vol 5 ofMonogragh Series SPE Richardson Tex USA 1977

[6] F Kucuk and L Ayestaran ldquoAnalysis of simultaneously mea-sured pressure and sandface flow rate in transient well testingrdquoJournal of PetroleumTechnology vol 37 no 2 pp 323ndash334 1985

[7] M Onur M Cinar D IIk P P Valko T A Blasingame and PHegeman ldquoAn investigation of recent deconvolution methodsfor well-test data analysisrdquo SPE Journal vol 13 no 2 pp 226ndash247 2008

[8] F J KuchukMOnur andFHollaender ldquoPressure transient testdesign and interpretationrdquo Developments in Petroleum Sciencevol 57 pp 303ndash359 2010

[9] A C Gringarten D P Bourdet P A Landel and V J KniazeffldquoA comparison between different skin and wellbore storagetype-curves for early-time transient analysisrdquo in Proceedings ofthe SPE Annual Technical Conference and Exhibition Paper SPE8205 Las Vegas Nev USA 1979

[10] A C Gringarten ldquoFrom straight lines to deconvolution theevolution of the state of the art in well test analysisrdquo SPEReservoir Evaluation and Engineering vol 11 no 1 pp 41ndash622008

[11] J A Cumming D A Wooff T Whittle and A C GringartenldquoMultiple well deconvolutionrdquo in Proceedings of the SPE Annual

12 Abstract and Applied Analysis

Technical Conference and Exhibition Document ID 166458-MS New Orleans La USA September-October 2013

[12] O Bahabanian D Ilk N Hosseinpour-Zonoozi and T ABlasingame ldquoExplicit deconvolution of well test data distortedby wellbore storagerdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Paper SPE 103216 pp 4444ndash4453San Antonio Tex USA September 2006

[13] J Jones and A Chen ldquoSuccessful applications of pressure-ratedeconvolution in theCad-Nik tight gas formations of the BritishColumbia foothillsrdquo Journal of Canadian Petroleum Technologyvol 51 no 3 pp 176ndash192 2012

[14] D G Russell ldquoExtension of pressure buildup analysis methodsrdquoJournal of Petroleum Technology vol 18 no 12 pp 1624ndash16361966

[15] R E Gladfelter G W Tracy and L E Wilsey ldquoSelecting wellswhich will respond to production-stimulation treatmentrdquo inDrilling and Production Practice API Dallas Tex USA 1955

[16] M J Fetkovich and M E Vienot ldquoRate normalization ofbuildup pressure by using afterflow datardquo Journal of PetroleumTechnology vol 36 no 13 pp 2211ndash2224 1984

[17] J L Johnston Variable rate analysis of transient well testdata using semi-analytical methods [MS thesis] Texas AampMUniversity College Station Tex USA 1992

[18] A F van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 6 pp 171ndash176 1953

[19] W Hurst ldquoEstablishment of the skin effect and its impedimentto fluid flow into a well borerdquo The Petroleum Engineer pp B6ndashB16 1953

[20] J A Joseph and L F Koederitz ldquoA simple nonlinear model forrepresentation of field transient responsesrdquo Paper SPE 11435SPE Richardson Tex USA 1982

[21] F J Kuchuk ldquoGladfelter deconvolutionrdquo SPE Formation Evalu-ation vol 5 no 3 pp 285ndash292 1990

[22] A F van Everdingen and W Hurst ldquoThe application of thelaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[23] D Bourdet T M Whittle A A Douglas and Y M Pirard ldquoAnew set of type curves simplifies well test analysisrdquo World Oilpp 95ndash106 1983

[24] D F Meunier C S Kabir and M J Wittmann ldquoGas well testanalysis use of normalized pseudovariablesrdquo SPE FormationEvaluation vol 2 no 4 pp 629ndash636 1987

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

10minus3 10minus2 10minus1 100 101 102 1030

500

1000

1500

2000

2500

3000

3500

4000

Time (hr)

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolution methodd(Dps)d(ln(t)) material balance deconvolution methodDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 1 Deconvolution for a drawdown test in a homogeneousreservoir

From the above-mentioned methods we use the fol-lowing two techniques to show the verification of theseapproaches

Explicit Deconvolution Formula The final result of explicitdeconvolution method is given as follows

119901119904119863

(119905119863) = minus

119905119863

120572 (119905119863)1199011015840

119908119863+ 119901119908119863

(119905119863) (39)

Plotting of this term versus (int119905119863

0119902119863(120591)119889120591119902

119863(119905119863)) and

(int119905119863

0(1 minus 119902

119863(120591))119889120591(1 minus 119902

119863(119905119863))) for drawdown and buildup

tests respectively yields more accurate resultsThis approachperforms well in field applications

Material Balance Deconvolution The material balance pres-sure drop function for the pressure drawdown test is givenas

Δ119901119908119891 (119905)

119902119863 (119905)

=119901119894minus 119901119908119891 (119905)

119902119863 (119905)

(40)

and the material balance time function for the pressuredrawdown test is given as

119905119898119887

=int119905119863

0119902119863 (120591) 119889120591

119902119863 (119905)

(41)

For the buildup test the material balance pressure dropfunction can be written as

Δ119901119908119904 (Δ119905)

1 minus 119902119863 (Δ119905)

=119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)

1 minus 119902119863 (Δ119905)

(42)

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

104

103

102

101Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolution methodd(Dps)d(ln(t)) material balance deconvolution methodDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Figure 2 Deconvolution for a drawdown test in a homogeneousreservoir

And the material balance time function can be written as

119905119898119887

=int119905119863

0(1 minus 119902

119863 (120591)) 119889120591

1 minus 119902119863 (Δ119905)

(43)

This technique is a good approach for practical applications

5 Verification Examples

We first use few synthetic cases for different reser-voirwellbore systems to validate the applicability of explicitdeconvolution formula and material balance deconvolutionmethod Then the field applications of these techniqueswill be shown by two real cases The unloading rates arecalculated with (20)

The first case is a synthetic test derived from a test designin a vertical oil wellThe reservoir is homogenous and infiniteacting with a constant wellbore storage The wellbore storagecoefficient is 119862

119904= 0015 bblpsi The pressure data are

obtained from the cylindrical source solution (test design)We used the explicit deconvolution formula and material

balance deconvolution method to deconvolve the pressureand unloading rate distorting by wellbore storage effectsTheresults are presented in Figures 1 and 2 for the semilog andlog-log plots in the case of drawdown test and Figures 3and 4 for the semilog and log-log plots in the case of builduptest Besides the deconvolved pressures and their logarithmicderivatives pressure responses with and without wellborestorage are also plotted on the figures for comparisonThrough the deconvolution process we have successfullyremoved the wellbore storage effects so that radial flow canbe identified at early test times The deconvolved pressuresand their logarithmic derivatives are almost identical to

6 Abstract and Applied Analysis

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dp wellbore storage

Dps explicit deconvolution method(ln

Dps material balance deconvolution

Dps no wellbore storage

ws

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpws

d(Dp )d(ln )) wellbore storagews (HTR

d(Dps)d )) explicit deconvolution method(HTR

d(Dps)d(ln )) no wellbore storage(HTR)

d(Dps)d(ln )) material balance deconvolution (HTR

Figure 3 Deconvolution for a buildup test in a homogeneousreservoir

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

104

103

102

101

ws

ws

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpws

Dp wellbore storage

Dps explicit deconvolution method

Dps material balance deconvolution

Dps no wellbore storage

d(Dp )d(ln )) wellbore storagews (HTR

d(Dps)d )) explicit deconvolution method(HTR

d(Dps)d(ln )) no wellbore storage(HTR)

d(Dps)d(ln )) material balance deconvolution (HTR

(ln

Figure 4 Deconvolution for a buildup test in a homogeneousreservoir

the analytic solutions (without wellbore storage) whichenables us to estimate reservoir parameters accurately

The second case is an oil well drawdown test in adual porosity reservoir The storativity ratio (120596) and inter-porosity flow coefficient (120582) are assumed to be 01 and10minus6 respectively The constant wellbore storage coefficient

is 119862119904= 015 bblpsi The dimensionless wellbore pressure

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

600

500

400

300

200

100

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 5 Deconvolution for a drawdown test in a dual porosityreservoir

data were obtained from the cylindrical source solution withpseudosteady-state interporosity flow (test design) It is notpossible to identify the dual porosity behavior characteristicof the reservoir from wellbore pressure data because thisbehavior has been masked completely by wellbore storage asindicated in Figures 5 and 6

We deconvolved the pressure and unloading rate andpresent the results in Figures 5 and 6 The deconvolvedresults are reasonably consistent with the analytical solutionAfter deconvolution the dual porosity behavior can beidentified easily from the valley in the logarithmic derivativeof pressure If wellbore storage effects were not removed atearly times it easily could have incorrectly identified thereservoir system as infinite actingThe estimation of reservoirparameters can be obtained now from deconvolved pressure

The third case is an oil well drawdown test in a reservoirwith a sealing fault 250 ft from the well Synthetic pressureresponses were generated by simulation (test design) with aconstant wellbore storage coefficient of 01 bblpsi As we cansee in Figures 7 and 8 the early time infinite acting reservoirbehavior has been masked completely by wellbore storage

To recover the hidden reservoir behavior we performeddeconvolution on the pressure change to remove the wellborestorage effect As in Figures 7 and 8 the deconvolved pressurechange and logarithmic derivative data are almost identicalto the responses without wellbore storageThe deconvolutionresults clearly indicate parallel horizontal lines for the loga-rithmic derivative However without wellbore storage effectselimination this maymislead to identify the reservoir systemas infinite acting So this allows us to correctly identify thereservoir system having a sealing fault

The fourth case is a field example that is taken from theliterature [23] The basic reservoir and fluid properties are

Abstract and Applied Analysis 7

Table 1 Reservoir rock and fluid properties for the first field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119902

119900(STBD) 119861

119900(RBSTB) 119888

119905(psiminus1) 120583

119900(cp) 120601

1533 107 029 174 106 42 times 10minus6 25 025

Table 2 Reservoir rock and fluid properties for the second field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119879 (∘F) 119901 (psi) 119902

119900(STBD) 119902

119892(MscfD) 119888

119891(psiminus1) 119879

119889119901(∘F) 119901

119889119901(psi) ∘API 120574

119892

143111 100 0354 2795 7322 104 13925 3 times 10minus6 212 5445 45375 068

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

10minus1

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 6 Deconvolution for a drawdown test in a dual porosityreservoir

shown in Table 1 In this case we provide the deconvolvedpressure data using the explicit deconvolution formula andthe material balance deconvolution method that is presentedin this work The data are taken from a pressure buildup testand can be reasonably used as field data The deconvolutionresults are shown in Figure 9 in semilog plot As it isillustrated from Figure 9 the corrected pressure provides aclear straight line starting as early as 15 hours in contrast tothe vague line from the uncorrected pressure response

And the last (field) example is a gas condensatewell buildup test following a variable rate productionhistory The duration of last pressure buildup test is361556 hours (from 1889 to 2250556 hours) The basicreservoir and fluid properties for the last drawdown testare shown in Table 2 However the properties of otherproduction tests are not shown here To use our newtechniques effectively for gas wells we use normalizedpseudopressure and pseudotime functions as defined byMeunier et al [24]The final results of explicit deconvolutionformula and material balance deconvolution methodshow two different slopes (Figure 10) where the earlytimes slope is greater than the last one This identifiesthe radial composite behavior of the reservoir With

the corrected pressures the reservoir parameters can also beestimated

6 Conclusions

In this work we have provided some explicit deconvolutionmethods for restoration of constant rate pressure responsesfrom measured pressure data dominated by wellbore storageeffects without sandface rate measurements Based on ourinvestigations we can state the following specific conclusionsrelated to these techniques

(1) The results obtained from the synthetic and fieldexamples show the practical and computational effi-ciency of our deconvolution methods

(2) Modification of 120573-deconvolution is presented wherethe beta quantity is variable rather than a constant

(3) For situations in which there are no downhole ratemeasurements the explicit deconvolution methods(blind deconvolution approaches) can be used toderive the downhole rate function from the pressuremeasurements and restore the reservoir pressureresponse

(4) We can unveil the early time behavior of a reservoirsystemmasked by wellbore storage effects using thesenew approaches Wellbore geometries and reservoirtypes can be identified from this restored pressuredata Furthermore the conventional methods canbe used to analyze this restored formation pressureto estimate formation parameters So the presentedtechniques can be used as the practical tools toimprove pressure transient test interpretation

(5) Deconvolution in well test analysis is often expressedas an inverse problem This approach can be used asan idea to justify the rate normalization method andthe material balance deconvolution method

Appendices

A Russel Method and Its Modification

A method for correcting pressure buildup pressure data freeof wellbore storage effects at early times has been developedby Russell [14] In this method it is not necessary to measurethe rate of influx after closing in at the surface Instead oneuses a theoretical equation which gives the form that thebottom hole pressure should have fluid accumulates in the

8 Abstract and Applied Analysis

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

250

200

150

100

50

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 7 Deconvolution for a drawdown test in a reservoir with asealing fault

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 8 Deconvolution for a drawdown test in a reservoir with asealing fault

wellbore during buildup This leads to the result that oneshould plot

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

(A1)

versus log (Δ119905) in analyzing pressure buildup data during theearly fill-up period The denominator makes a correction for

104103102101100

Horner time ratio

800

700

600

500

400

300

200

100

0

Dpws

(psi)

Dpws wellbore storageDps explicit deconvolution methodDps material balance deconvolution

Figure 9 Restored pressures for a buildup test in a homogeneousreservoir

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dpa

(psi)

Dpaw wellbore storage explicit deconvolution method material balance deconvolution

DpasDpas

Figure 10 Restored pressures for a buildup test in a radial compositereservoir

the gradually decreasing flow into the wellbore The quantity1198622is obtained by trial and error as the value which makes the

curve straight at early times After obtaining the straight linesection the rest of the analysis is the same as for any otherpressure buildup This method has the advantage or requiresno additional data over that taken routinely during a shut intest

Due to the time consuming and unconfident of trial anderror to determine 119862

2in Russell method we modify this

method Russellrsquos wellbore storage correction is given as

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

= 119891 (Δ119905 = 1 hr) + 119898119904119897log (Δ119905)

(A2)

Abstract and Applied Analysis 9

Defining the following function and using least squarestechnique obtain the final result

119864 (1198622 119891119898)

=

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891(1198622) minus 119898(119862

2) log(Δ119905

119894))

2

(A3)

Tominimize the function119864 it can differentiate119864with respectto 1198622 119891 and 119898 and setting the results equal to zero After

differentiation it reduces to

120597119864

1205971198622

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894) ) = 0

(A4)

120597119864

120597119891

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622) minus 119898 (119862

2) log (Δ119905

119894))

= 0

(A5)120597119864

120597119898

= minus2

119873

sum

119894=1

log (Δ119905119894) (

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus 119898 (1198622) log (Δ119905

119894)) = 0

(A6)

From (A4) we define the following function

119883(1198622)

=

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times(Δ119901119908119904119894

(1minus (11198622Δ119905119894))minus119891 (119862

2) minus119898 (119862

2) log (Δ119905

119894))

= 0

(A7)

The roots of (A7) can be found via the following algorithm

1198622(119894+1)

= 1198622(119894)

minus119883 (1198622)

1198831015840 (1198622) (A8)

where

119883(1198622) =

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894))

1198831015840(1198622) =

119873

sum

119894=1

(minus2Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3)

times (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2

times [Δ119901119908119904119894

(1 minus (11198622Δ119905))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894)])

+ (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (minusΔ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3minus 1198911015840(1198622)

minus1198981015840(1198622) log (Δ119905

119894))

(A9)

where the unknown functions are given by the followingrelations

119891 (1198622) =

120573119906 (1198622) minus 120572V (119862

2)

1205732 minus 119873120572(A10)

whose derivative is

1198911015840(1198622) =

1205731199061015840(1198622) minus 120572V1015840 (119862

2)

1205732 minus 119873120572

119898 (1198622) =

119873119906 (1198622) minus 120573V (119862

2)

119873120572 minus 1205732

(A11)

where its derivative is

1198981015840(1198622) =

1198731199061015840(1198622) minus 120573V1015840 (119862

2)

119873120572 minus 1205732

119906 (1198622) =

119873

sum

119894=1

log (Δ119905119894)

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A12)

10 Abstract and Applied Analysis

and after differentiation it yields

1199061015840(1198622) = minus

119873

sum

119894=1

log (Δ119905

119894)

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2

V (1198622) =

119873

sum

119894=1

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A13)

which has the following derivative

V1015840 (1198622) = minus

119873

sum

119894=1

1

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2 (A14)

finally the values of 120572 and 120573 parameters are given by thefollowing formulas

120572 =

119873

sum

119894=1

(log(Δ119905119894))2

120573 =

119873

sum

119894=1

log (Δ119905119894)

(A15)

B General Formula for Inverse Problem

The convolution of (6) yields

119901119904119863

(119905119863) = int

119905119863

0

119870 (120591) 119901119908119863 (119905119863 minus 120591) 119889120591

= int

119905119863

0

119870(119905119863minus 120591) 119901

119908119863 (120591) 119889120591

(B1)

where

119870(119905119863) = 119871minus1(

1

119904119902 (119904)) (B2)

where 119871minus1 is the inverse Lapalce transform operator 119870(119905119863)

can be computed either from the Laplace transforms of119902119863(119905119863) data a curve-fitted equation of 119902

119863(119905119863) data or using

another technique with a relation between119870(119905119863) and 119902

119863(119905119863)

However it would be time consuming to invert all the 119902119863(119905119863)

data in Laplace space and transform it back to real space inaccordance with (B2) Once an approximation is obtained itwill be easy to compute119870(119905

119863) and integrate (B1) to determine

119901119904119863(119905119863) However finding some approximation functions for

119902119863(119905119863) data may be difficult We provide an approach to

overcome this problemThis idea is based on finding anotherrelation between 119870(119905

119863) and 119902

119863(119905119863) that is

ℓ (1) =1

119904=

1

119904119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B3)or

1 = int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591 = int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591

(B4)

The other relations that can be used are

ℓ (119905119863) =

1

1199042=

1

1199042119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B5)

or

119905119863= int

119905119863

0

119871 (120591) 119902119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

119863(120591) 119889120591

(B6)

ℓ (1) =1

119904=

1

1199042119902119863(119904)

119904119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591)

(B7)

or

1 = int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591

(B8)

where

119871 (119905119863) = ℓminus1(

1

1199042119902 (119904)) (B9)

Equation (B4) can be used simultaneously with (B1) toanalyze the deconvolution problem

Equation (11) can be written as

119901119904119863

(119904) = 119904119901119908119863

(119904)1

1199042119902119863(119904)

(B10)

or

119901119904119863

(119905119863) = int

119905119863

0

119871 (120591) 1199011015840

119908119863(119905119863minus 120591) 119889120591

= int

119905119863

0

119871 (119905119863minus 120591) 119901

1015840

119908119863(120591) 119889120591

(B11)

The applications of (B6) and (B8) are with (B11)For justifying the rate normalizationmethod andmaterial

balance deconvolution it can use (B1) (B4) ((B11) can alsobe used with (B6) or (B8) for this goal) and generalizedmean value theorem for integrals This theorem is expressedas follows

Theorem 1 Let 119891 and 119892 be continuous on [119886 119887] and supposethat 119892 does not change sign on [119886 119887] Then there is a point 119888 in[119886 119887] such that

int

119887

119886

119891 (119905) 119892 (119905) 119889119905 = 119891 (119888) int

119887

119886

119892 (119905) 119889119905 (B12)

Abstract and Applied Analysis 11

By Theorem 1 there are some values 1198881and 1198882between 0

and 119905119863such that

119901119904119863

(119905119863) = 119901119908119863

(1198881) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B13)

1 = 119902119863(1198882) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B14)

By setting 1198881

= 1198882

= 119905119863and dividing the right and left

side of (B13) respectively the rate normalization techniqueis developed However the values 119888

1and 1198882are not equal

to each other and are not equal to 119905119863

in general Thisis the reason of error in rate normalization and materialbalance deconvolution methods 119888

1and 1198882are less than 119905

119863in

these methods for the duration of wellbore storage distortionduring a pressure transient test Since these conditions areapproximately satisfied in material balance deconvolutionthe results of material balance are more accurate and moreconfident than rate normalization

Nomenclature

API API gravity ∘API119861119900 Oil formation volume factor RBSTB

119888119905 Total isothermal compressibility factor

psiminus1

1198622 Arbitrary constant hr

minus1

119862119863 Wellbore storage constant dimensionless

119862119904 Wellbore storage coefficient bblpsi

ℎ Formation thickness ftHTR Horner time ratio dimensionless119896 Formation permeability md119870 Kernel of the convolution integral119871 Kernel of the convolution integral119873119901 Cumulative oil production bbl

119901 Pressure psi119901119886 Normalized pseudopressure psi

119901 Average reservoir pressure psi119901119863 Formation pressure dimensionless

119901119889119901 Dewpoint pressure psi

119901119894 Initial reservoir pressure psi

119901119904119863 Formation pressure including skin

dimensionless119901119904 Formation pressure including skin psi

119901119908119863

Pressure draw down dimensionless119901119908119891 Flowing bottom hole pressure psi

119901119908119904 Shut-in bottom hole pressure psi

119902119863 Sandface flow rate dimensionless

119902119892 Gas flow rate MscfD

119902119900 Oil flow rate STBD

119903119908 Wellbore radius ft

119904 Laplace variable119878 Skin factor119905 Time hr119905119863 Time dimensionless

119905119901 Production time hr

119879 Reservoir temperature ∘F

119879119889119901 Dewpoint temperature ∘F

120572 Explicit deconvolution variabledimensionless

120573 Beta deconvolution variabledimensionless

120574119892 Gas gravity (air = 10)

120582 Interporosity flow coefficient of dualporosity reservoir

120601 Porosity of reservoir rock dimensionless120583119892 Gas viscosity cp

120583119900 Oil viscosity cp

120596 Storativity ratio of dual porosity reservoirminus Laplace transform of a function1015840 Derivative of a function

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Cheng W J Lee and D A McVay ldquoFast-Fourier-transform-based deconvolution for interpretation of pressure-transient-test data dominated by Wellbore storagerdquo SPE Reservoir Eval-uation and Engineering vol 8 no 3 pp 224ndash239 2005

[2] M M Levitan and M R Wilson ldquoDeconvolution of pressureand rate data from gas reservoirs with significant pressuredepletionrdquo SPE Journal vol 17 no 3 pp 727ndash741 2012

[3] Y Liu and R N Horne ldquoInterpreting pressure and flow-ratedata from permanent downhole gauges by use of data-miningapproachesrdquo SPE Journal vol 18 no 1 pp 69ndash82 2013

[4] O Ogunrewo and A C Gringarten ldquoDeconvolution of welltest data in lean and rich gas condensate and volatile oil wellsbelow saturation pressurerdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Document ID 166340-MS New Orleans La USA September-October 2013

[5] R C Earlougher Jr Advances in Well Test Analysis vol 5 ofMonogragh Series SPE Richardson Tex USA 1977

[6] F Kucuk and L Ayestaran ldquoAnalysis of simultaneously mea-sured pressure and sandface flow rate in transient well testingrdquoJournal of PetroleumTechnology vol 37 no 2 pp 323ndash334 1985

[7] M Onur M Cinar D IIk P P Valko T A Blasingame and PHegeman ldquoAn investigation of recent deconvolution methodsfor well-test data analysisrdquo SPE Journal vol 13 no 2 pp 226ndash247 2008

[8] F J KuchukMOnur andFHollaender ldquoPressure transient testdesign and interpretationrdquo Developments in Petroleum Sciencevol 57 pp 303ndash359 2010

[9] A C Gringarten D P Bourdet P A Landel and V J KniazeffldquoA comparison between different skin and wellbore storagetype-curves for early-time transient analysisrdquo in Proceedings ofthe SPE Annual Technical Conference and Exhibition Paper SPE8205 Las Vegas Nev USA 1979

[10] A C Gringarten ldquoFrom straight lines to deconvolution theevolution of the state of the art in well test analysisrdquo SPEReservoir Evaluation and Engineering vol 11 no 1 pp 41ndash622008

[11] J A Cumming D A Wooff T Whittle and A C GringartenldquoMultiple well deconvolutionrdquo in Proceedings of the SPE Annual

12 Abstract and Applied Analysis

Technical Conference and Exhibition Document ID 166458-MS New Orleans La USA September-October 2013

[12] O Bahabanian D Ilk N Hosseinpour-Zonoozi and T ABlasingame ldquoExplicit deconvolution of well test data distortedby wellbore storagerdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Paper SPE 103216 pp 4444ndash4453San Antonio Tex USA September 2006

[13] J Jones and A Chen ldquoSuccessful applications of pressure-ratedeconvolution in theCad-Nik tight gas formations of the BritishColumbia foothillsrdquo Journal of Canadian Petroleum Technologyvol 51 no 3 pp 176ndash192 2012

[14] D G Russell ldquoExtension of pressure buildup analysis methodsrdquoJournal of Petroleum Technology vol 18 no 12 pp 1624ndash16361966

[15] R E Gladfelter G W Tracy and L E Wilsey ldquoSelecting wellswhich will respond to production-stimulation treatmentrdquo inDrilling and Production Practice API Dallas Tex USA 1955

[16] M J Fetkovich and M E Vienot ldquoRate normalization ofbuildup pressure by using afterflow datardquo Journal of PetroleumTechnology vol 36 no 13 pp 2211ndash2224 1984

[17] J L Johnston Variable rate analysis of transient well testdata using semi-analytical methods [MS thesis] Texas AampMUniversity College Station Tex USA 1992

[18] A F van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 6 pp 171ndash176 1953

[19] W Hurst ldquoEstablishment of the skin effect and its impedimentto fluid flow into a well borerdquo The Petroleum Engineer pp B6ndashB16 1953

[20] J A Joseph and L F Koederitz ldquoA simple nonlinear model forrepresentation of field transient responsesrdquo Paper SPE 11435SPE Richardson Tex USA 1982

[21] F J Kuchuk ldquoGladfelter deconvolutionrdquo SPE Formation Evalu-ation vol 5 no 3 pp 285ndash292 1990

[22] A F van Everdingen and W Hurst ldquoThe application of thelaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[23] D Bourdet T M Whittle A A Douglas and Y M Pirard ldquoAnew set of type curves simplifies well test analysisrdquo World Oilpp 95ndash106 1983

[24] D F Meunier C S Kabir and M J Wittmann ldquoGas well testanalysis use of normalized pseudovariablesrdquo SPE FormationEvaluation vol 2 no 4 pp 629ndash636 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dp wellbore storage

Dps explicit deconvolution method(ln

Dps material balance deconvolution

Dps no wellbore storage

ws

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpws

d(Dp )d(ln )) wellbore storagews (HTR

d(Dps)d )) explicit deconvolution method(HTR

d(Dps)d(ln )) no wellbore storage(HTR)

d(Dps)d(ln )) material balance deconvolution (HTR

Figure 3 Deconvolution for a buildup test in a homogeneousreservoir

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

104

103

102

101

ws

ws

and

its lo

garit

hmic

der

ivat

ive (

psi)

Dpws

Dp wellbore storage

Dps explicit deconvolution method

Dps material balance deconvolution

Dps no wellbore storage

d(Dp )d(ln )) wellbore storagews (HTR

d(Dps)d )) explicit deconvolution method(HTR

d(Dps)d(ln )) no wellbore storage(HTR)

d(Dps)d(ln )) material balance deconvolution (HTR

(ln

Figure 4 Deconvolution for a buildup test in a homogeneousreservoir

the analytic solutions (without wellbore storage) whichenables us to estimate reservoir parameters accurately

The second case is an oil well drawdown test in adual porosity reservoir The storativity ratio (120596) and inter-porosity flow coefficient (120582) are assumed to be 01 and10minus6 respectively The constant wellbore storage coefficient

is 119862119904= 015 bblpsi The dimensionless wellbore pressure

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

600

500

400

300

200

100

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 5 Deconvolution for a drawdown test in a dual porosityreservoir

data were obtained from the cylindrical source solution withpseudosteady-state interporosity flow (test design) It is notpossible to identify the dual porosity behavior characteristicof the reservoir from wellbore pressure data because thisbehavior has been masked completely by wellbore storage asindicated in Figures 5 and 6

We deconvolved the pressure and unloading rate andpresent the results in Figures 5 and 6 The deconvolvedresults are reasonably consistent with the analytical solutionAfter deconvolution the dual porosity behavior can beidentified easily from the valley in the logarithmic derivativeof pressure If wellbore storage effects were not removed atearly times it easily could have incorrectly identified thereservoir system as infinite actingThe estimation of reservoirparameters can be obtained now from deconvolved pressure

The third case is an oil well drawdown test in a reservoirwith a sealing fault 250 ft from the well Synthetic pressureresponses were generated by simulation (test design) with aconstant wellbore storage coefficient of 01 bblpsi As we cansee in Figures 7 and 8 the early time infinite acting reservoirbehavior has been masked completely by wellbore storage

To recover the hidden reservoir behavior we performeddeconvolution on the pressure change to remove the wellborestorage effect As in Figures 7 and 8 the deconvolved pressurechange and logarithmic derivative data are almost identicalto the responses without wellbore storageThe deconvolutionresults clearly indicate parallel horizontal lines for the loga-rithmic derivative However without wellbore storage effectselimination this maymislead to identify the reservoir systemas infinite acting So this allows us to correctly identify thereservoir system having a sealing fault

The fourth case is a field example that is taken from theliterature [23] The basic reservoir and fluid properties are

Abstract and Applied Analysis 7

Table 1 Reservoir rock and fluid properties for the first field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119902

119900(STBD) 119861

119900(RBSTB) 119888

119905(psiminus1) 120583

119900(cp) 120601

1533 107 029 174 106 42 times 10minus6 25 025

Table 2 Reservoir rock and fluid properties for the second field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119879 (∘F) 119901 (psi) 119902

119900(STBD) 119902

119892(MscfD) 119888

119891(psiminus1) 119879

119889119901(∘F) 119901

119889119901(psi) ∘API 120574

119892

143111 100 0354 2795 7322 104 13925 3 times 10minus6 212 5445 45375 068

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

10minus1

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 6 Deconvolution for a drawdown test in a dual porosityreservoir

shown in Table 1 In this case we provide the deconvolvedpressure data using the explicit deconvolution formula andthe material balance deconvolution method that is presentedin this work The data are taken from a pressure buildup testand can be reasonably used as field data The deconvolutionresults are shown in Figure 9 in semilog plot As it isillustrated from Figure 9 the corrected pressure provides aclear straight line starting as early as 15 hours in contrast tothe vague line from the uncorrected pressure response

And the last (field) example is a gas condensatewell buildup test following a variable rate productionhistory The duration of last pressure buildup test is361556 hours (from 1889 to 2250556 hours) The basicreservoir and fluid properties for the last drawdown testare shown in Table 2 However the properties of otherproduction tests are not shown here To use our newtechniques effectively for gas wells we use normalizedpseudopressure and pseudotime functions as defined byMeunier et al [24]The final results of explicit deconvolutionformula and material balance deconvolution methodshow two different slopes (Figure 10) where the earlytimes slope is greater than the last one This identifiesthe radial composite behavior of the reservoir With

the corrected pressures the reservoir parameters can also beestimated

6 Conclusions

In this work we have provided some explicit deconvolutionmethods for restoration of constant rate pressure responsesfrom measured pressure data dominated by wellbore storageeffects without sandface rate measurements Based on ourinvestigations we can state the following specific conclusionsrelated to these techniques

(1) The results obtained from the synthetic and fieldexamples show the practical and computational effi-ciency of our deconvolution methods

(2) Modification of 120573-deconvolution is presented wherethe beta quantity is variable rather than a constant

(3) For situations in which there are no downhole ratemeasurements the explicit deconvolution methods(blind deconvolution approaches) can be used toderive the downhole rate function from the pressuremeasurements and restore the reservoir pressureresponse

(4) We can unveil the early time behavior of a reservoirsystemmasked by wellbore storage effects using thesenew approaches Wellbore geometries and reservoirtypes can be identified from this restored pressuredata Furthermore the conventional methods canbe used to analyze this restored formation pressureto estimate formation parameters So the presentedtechniques can be used as the practical tools toimprove pressure transient test interpretation

(5) Deconvolution in well test analysis is often expressedas an inverse problem This approach can be used asan idea to justify the rate normalization method andthe material balance deconvolution method

Appendices

A Russel Method and Its Modification

A method for correcting pressure buildup pressure data freeof wellbore storage effects at early times has been developedby Russell [14] In this method it is not necessary to measurethe rate of influx after closing in at the surface Instead oneuses a theoretical equation which gives the form that thebottom hole pressure should have fluid accumulates in the

8 Abstract and Applied Analysis

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

250

200

150

100

50

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 7 Deconvolution for a drawdown test in a reservoir with asealing fault

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 8 Deconvolution for a drawdown test in a reservoir with asealing fault

wellbore during buildup This leads to the result that oneshould plot

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

(A1)

versus log (Δ119905) in analyzing pressure buildup data during theearly fill-up period The denominator makes a correction for

104103102101100

Horner time ratio

800

700

600

500

400

300

200

100

0

Dpws

(psi)

Dpws wellbore storageDps explicit deconvolution methodDps material balance deconvolution

Figure 9 Restored pressures for a buildup test in a homogeneousreservoir

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dpa

(psi)

Dpaw wellbore storage explicit deconvolution method material balance deconvolution

DpasDpas

Figure 10 Restored pressures for a buildup test in a radial compositereservoir

the gradually decreasing flow into the wellbore The quantity1198622is obtained by trial and error as the value which makes the

curve straight at early times After obtaining the straight linesection the rest of the analysis is the same as for any otherpressure buildup This method has the advantage or requiresno additional data over that taken routinely during a shut intest

Due to the time consuming and unconfident of trial anderror to determine 119862

2in Russell method we modify this

method Russellrsquos wellbore storage correction is given as

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

= 119891 (Δ119905 = 1 hr) + 119898119904119897log (Δ119905)

(A2)

Abstract and Applied Analysis 9

Defining the following function and using least squarestechnique obtain the final result

119864 (1198622 119891119898)

=

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891(1198622) minus 119898(119862

2) log(Δ119905

119894))

2

(A3)

Tominimize the function119864 it can differentiate119864with respectto 1198622 119891 and 119898 and setting the results equal to zero After

differentiation it reduces to

120597119864

1205971198622

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894) ) = 0

(A4)

120597119864

120597119891

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622) minus 119898 (119862

2) log (Δ119905

119894))

= 0

(A5)120597119864

120597119898

= minus2

119873

sum

119894=1

log (Δ119905119894) (

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus 119898 (1198622) log (Δ119905

119894)) = 0

(A6)

From (A4) we define the following function

119883(1198622)

=

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times(Δ119901119908119904119894

(1minus (11198622Δ119905119894))minus119891 (119862

2) minus119898 (119862

2) log (Δ119905

119894))

= 0

(A7)

The roots of (A7) can be found via the following algorithm

1198622(119894+1)

= 1198622(119894)

minus119883 (1198622)

1198831015840 (1198622) (A8)

where

119883(1198622) =

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894))

1198831015840(1198622) =

119873

sum

119894=1

(minus2Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3)

times (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2

times [Δ119901119908119904119894

(1 minus (11198622Δ119905))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894)])

+ (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (minusΔ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3minus 1198911015840(1198622)

minus1198981015840(1198622) log (Δ119905

119894))

(A9)

where the unknown functions are given by the followingrelations

119891 (1198622) =

120573119906 (1198622) minus 120572V (119862

2)

1205732 minus 119873120572(A10)

whose derivative is

1198911015840(1198622) =

1205731199061015840(1198622) minus 120572V1015840 (119862

2)

1205732 minus 119873120572

119898 (1198622) =

119873119906 (1198622) minus 120573V (119862

2)

119873120572 minus 1205732

(A11)

where its derivative is

1198981015840(1198622) =

1198731199061015840(1198622) minus 120573V1015840 (119862

2)

119873120572 minus 1205732

119906 (1198622) =

119873

sum

119894=1

log (Δ119905119894)

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A12)

10 Abstract and Applied Analysis

and after differentiation it yields

1199061015840(1198622) = minus

119873

sum

119894=1

log (Δ119905

119894)

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2

V (1198622) =

119873

sum

119894=1

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A13)

which has the following derivative

V1015840 (1198622) = minus

119873

sum

119894=1

1

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2 (A14)

finally the values of 120572 and 120573 parameters are given by thefollowing formulas

120572 =

119873

sum

119894=1

(log(Δ119905119894))2

120573 =

119873

sum

119894=1

log (Δ119905119894)

(A15)

B General Formula for Inverse Problem

The convolution of (6) yields

119901119904119863

(119905119863) = int

119905119863

0

119870 (120591) 119901119908119863 (119905119863 minus 120591) 119889120591

= int

119905119863

0

119870(119905119863minus 120591) 119901

119908119863 (120591) 119889120591

(B1)

where

119870(119905119863) = 119871minus1(

1

119904119902 (119904)) (B2)

where 119871minus1 is the inverse Lapalce transform operator 119870(119905119863)

can be computed either from the Laplace transforms of119902119863(119905119863) data a curve-fitted equation of 119902

119863(119905119863) data or using

another technique with a relation between119870(119905119863) and 119902

119863(119905119863)

However it would be time consuming to invert all the 119902119863(119905119863)

data in Laplace space and transform it back to real space inaccordance with (B2) Once an approximation is obtained itwill be easy to compute119870(119905

119863) and integrate (B1) to determine

119901119904119863(119905119863) However finding some approximation functions for

119902119863(119905119863) data may be difficult We provide an approach to

overcome this problemThis idea is based on finding anotherrelation between 119870(119905

119863) and 119902

119863(119905119863) that is

ℓ (1) =1

119904=

1

119904119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B3)or

1 = int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591 = int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591

(B4)

The other relations that can be used are

ℓ (119905119863) =

1

1199042=

1

1199042119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B5)

or

119905119863= int

119905119863

0

119871 (120591) 119902119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

119863(120591) 119889120591

(B6)

ℓ (1) =1

119904=

1

1199042119902119863(119904)

119904119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591)

(B7)

or

1 = int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591

(B8)

where

119871 (119905119863) = ℓminus1(

1

1199042119902 (119904)) (B9)

Equation (B4) can be used simultaneously with (B1) toanalyze the deconvolution problem

Equation (11) can be written as

119901119904119863

(119904) = 119904119901119908119863

(119904)1

1199042119902119863(119904)

(B10)

or

119901119904119863

(119905119863) = int

119905119863

0

119871 (120591) 1199011015840

119908119863(119905119863minus 120591) 119889120591

= int

119905119863

0

119871 (119905119863minus 120591) 119901

1015840

119908119863(120591) 119889120591

(B11)

The applications of (B6) and (B8) are with (B11)For justifying the rate normalizationmethod andmaterial

balance deconvolution it can use (B1) (B4) ((B11) can alsobe used with (B6) or (B8) for this goal) and generalizedmean value theorem for integrals This theorem is expressedas follows

Theorem 1 Let 119891 and 119892 be continuous on [119886 119887] and supposethat 119892 does not change sign on [119886 119887] Then there is a point 119888 in[119886 119887] such that

int

119887

119886

119891 (119905) 119892 (119905) 119889119905 = 119891 (119888) int

119887

119886

119892 (119905) 119889119905 (B12)

Abstract and Applied Analysis 11

By Theorem 1 there are some values 1198881and 1198882between 0

and 119905119863such that

119901119904119863

(119905119863) = 119901119908119863

(1198881) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B13)

1 = 119902119863(1198882) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B14)

By setting 1198881

= 1198882

= 119905119863and dividing the right and left

side of (B13) respectively the rate normalization techniqueis developed However the values 119888

1and 1198882are not equal

to each other and are not equal to 119905119863

in general Thisis the reason of error in rate normalization and materialbalance deconvolution methods 119888

1and 1198882are less than 119905

119863in

these methods for the duration of wellbore storage distortionduring a pressure transient test Since these conditions areapproximately satisfied in material balance deconvolutionthe results of material balance are more accurate and moreconfident than rate normalization

Nomenclature

API API gravity ∘API119861119900 Oil formation volume factor RBSTB

119888119905 Total isothermal compressibility factor

psiminus1

1198622 Arbitrary constant hr

minus1

119862119863 Wellbore storage constant dimensionless

119862119904 Wellbore storage coefficient bblpsi

ℎ Formation thickness ftHTR Horner time ratio dimensionless119896 Formation permeability md119870 Kernel of the convolution integral119871 Kernel of the convolution integral119873119901 Cumulative oil production bbl

119901 Pressure psi119901119886 Normalized pseudopressure psi

119901 Average reservoir pressure psi119901119863 Formation pressure dimensionless

119901119889119901 Dewpoint pressure psi

119901119894 Initial reservoir pressure psi

119901119904119863 Formation pressure including skin

dimensionless119901119904 Formation pressure including skin psi

119901119908119863

Pressure draw down dimensionless119901119908119891 Flowing bottom hole pressure psi

119901119908119904 Shut-in bottom hole pressure psi

119902119863 Sandface flow rate dimensionless

119902119892 Gas flow rate MscfD

119902119900 Oil flow rate STBD

119903119908 Wellbore radius ft

119904 Laplace variable119878 Skin factor119905 Time hr119905119863 Time dimensionless

119905119901 Production time hr

119879 Reservoir temperature ∘F

119879119889119901 Dewpoint temperature ∘F

120572 Explicit deconvolution variabledimensionless

120573 Beta deconvolution variabledimensionless

120574119892 Gas gravity (air = 10)

120582 Interporosity flow coefficient of dualporosity reservoir

120601 Porosity of reservoir rock dimensionless120583119892 Gas viscosity cp

120583119900 Oil viscosity cp

120596 Storativity ratio of dual porosity reservoirminus Laplace transform of a function1015840 Derivative of a function

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Cheng W J Lee and D A McVay ldquoFast-Fourier-transform-based deconvolution for interpretation of pressure-transient-test data dominated by Wellbore storagerdquo SPE Reservoir Eval-uation and Engineering vol 8 no 3 pp 224ndash239 2005

[2] M M Levitan and M R Wilson ldquoDeconvolution of pressureand rate data from gas reservoirs with significant pressuredepletionrdquo SPE Journal vol 17 no 3 pp 727ndash741 2012

[3] Y Liu and R N Horne ldquoInterpreting pressure and flow-ratedata from permanent downhole gauges by use of data-miningapproachesrdquo SPE Journal vol 18 no 1 pp 69ndash82 2013

[4] O Ogunrewo and A C Gringarten ldquoDeconvolution of welltest data in lean and rich gas condensate and volatile oil wellsbelow saturation pressurerdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Document ID 166340-MS New Orleans La USA September-October 2013

[5] R C Earlougher Jr Advances in Well Test Analysis vol 5 ofMonogragh Series SPE Richardson Tex USA 1977

[6] F Kucuk and L Ayestaran ldquoAnalysis of simultaneously mea-sured pressure and sandface flow rate in transient well testingrdquoJournal of PetroleumTechnology vol 37 no 2 pp 323ndash334 1985

[7] M Onur M Cinar D IIk P P Valko T A Blasingame and PHegeman ldquoAn investigation of recent deconvolution methodsfor well-test data analysisrdquo SPE Journal vol 13 no 2 pp 226ndash247 2008

[8] F J KuchukMOnur andFHollaender ldquoPressure transient testdesign and interpretationrdquo Developments in Petroleum Sciencevol 57 pp 303ndash359 2010

[9] A C Gringarten D P Bourdet P A Landel and V J KniazeffldquoA comparison between different skin and wellbore storagetype-curves for early-time transient analysisrdquo in Proceedings ofthe SPE Annual Technical Conference and Exhibition Paper SPE8205 Las Vegas Nev USA 1979

[10] A C Gringarten ldquoFrom straight lines to deconvolution theevolution of the state of the art in well test analysisrdquo SPEReservoir Evaluation and Engineering vol 11 no 1 pp 41ndash622008

[11] J A Cumming D A Wooff T Whittle and A C GringartenldquoMultiple well deconvolutionrdquo in Proceedings of the SPE Annual

12 Abstract and Applied Analysis

Technical Conference and Exhibition Document ID 166458-MS New Orleans La USA September-October 2013

[12] O Bahabanian D Ilk N Hosseinpour-Zonoozi and T ABlasingame ldquoExplicit deconvolution of well test data distortedby wellbore storagerdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Paper SPE 103216 pp 4444ndash4453San Antonio Tex USA September 2006

[13] J Jones and A Chen ldquoSuccessful applications of pressure-ratedeconvolution in theCad-Nik tight gas formations of the BritishColumbia foothillsrdquo Journal of Canadian Petroleum Technologyvol 51 no 3 pp 176ndash192 2012

[14] D G Russell ldquoExtension of pressure buildup analysis methodsrdquoJournal of Petroleum Technology vol 18 no 12 pp 1624ndash16361966

[15] R E Gladfelter G W Tracy and L E Wilsey ldquoSelecting wellswhich will respond to production-stimulation treatmentrdquo inDrilling and Production Practice API Dallas Tex USA 1955

[16] M J Fetkovich and M E Vienot ldquoRate normalization ofbuildup pressure by using afterflow datardquo Journal of PetroleumTechnology vol 36 no 13 pp 2211ndash2224 1984

[17] J L Johnston Variable rate analysis of transient well testdata using semi-analytical methods [MS thesis] Texas AampMUniversity College Station Tex USA 1992

[18] A F van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 6 pp 171ndash176 1953

[19] W Hurst ldquoEstablishment of the skin effect and its impedimentto fluid flow into a well borerdquo The Petroleum Engineer pp B6ndashB16 1953

[20] J A Joseph and L F Koederitz ldquoA simple nonlinear model forrepresentation of field transient responsesrdquo Paper SPE 11435SPE Richardson Tex USA 1982

[21] F J Kuchuk ldquoGladfelter deconvolutionrdquo SPE Formation Evalu-ation vol 5 no 3 pp 285ndash292 1990

[22] A F van Everdingen and W Hurst ldquoThe application of thelaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[23] D Bourdet T M Whittle A A Douglas and Y M Pirard ldquoAnew set of type curves simplifies well test analysisrdquo World Oilpp 95ndash106 1983

[24] D F Meunier C S Kabir and M J Wittmann ldquoGas well testanalysis use of normalized pseudovariablesrdquo SPE FormationEvaluation vol 2 no 4 pp 629ndash636 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Table 1 Reservoir rock and fluid properties for the first field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119902

119900(STBD) 119861

119900(RBSTB) 119888

119905(psiminus1) 120583

119900(cp) 120601

1533 107 029 174 106 42 times 10minus6 25 025

Table 2 Reservoir rock and fluid properties for the second field case

119905119901(hr) ℎ (ft) 119903

119908(ft) 119879 (∘F) 119901 (psi) 119902

119900(STBD) 119902

119892(MscfD) 119888

119891(psiminus1) 119879

119889119901(∘F) 119901

119889119901(psi) ∘API 120574

119892

143111 100 0354 2795 7322 104 13925 3 times 10minus6 212 5445 45375 068

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

10minus1

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 6 Deconvolution for a drawdown test in a dual porosityreservoir

shown in Table 1 In this case we provide the deconvolvedpressure data using the explicit deconvolution formula andthe material balance deconvolution method that is presentedin this work The data are taken from a pressure buildup testand can be reasonably used as field data The deconvolutionresults are shown in Figure 9 in semilog plot As it isillustrated from Figure 9 the corrected pressure provides aclear straight line starting as early as 15 hours in contrast tothe vague line from the uncorrected pressure response

And the last (field) example is a gas condensatewell buildup test following a variable rate productionhistory The duration of last pressure buildup test is361556 hours (from 1889 to 2250556 hours) The basicreservoir and fluid properties for the last drawdown testare shown in Table 2 However the properties of otherproduction tests are not shown here To use our newtechniques effectively for gas wells we use normalizedpseudopressure and pseudotime functions as defined byMeunier et al [24]The final results of explicit deconvolutionformula and material balance deconvolution methodshow two different slopes (Figure 10) where the earlytimes slope is greater than the last one This identifiesthe radial composite behavior of the reservoir With

the corrected pressures the reservoir parameters can also beestimated

6 Conclusions

In this work we have provided some explicit deconvolutionmethods for restoration of constant rate pressure responsesfrom measured pressure data dominated by wellbore storageeffects without sandface rate measurements Based on ourinvestigations we can state the following specific conclusionsrelated to these techniques

(1) The results obtained from the synthetic and fieldexamples show the practical and computational effi-ciency of our deconvolution methods

(2) Modification of 120573-deconvolution is presented wherethe beta quantity is variable rather than a constant

(3) For situations in which there are no downhole ratemeasurements the explicit deconvolution methods(blind deconvolution approaches) can be used toderive the downhole rate function from the pressuremeasurements and restore the reservoir pressureresponse

(4) We can unveil the early time behavior of a reservoirsystemmasked by wellbore storage effects using thesenew approaches Wellbore geometries and reservoirtypes can be identified from this restored pressuredata Furthermore the conventional methods canbe used to analyze this restored formation pressureto estimate formation parameters So the presentedtechniques can be used as the practical tools toimprove pressure transient test interpretation

(5) Deconvolution in well test analysis is often expressedas an inverse problem This approach can be used asan idea to justify the rate normalization method andthe material balance deconvolution method

Appendices

A Russel Method and Its Modification

A method for correcting pressure buildup pressure data freeof wellbore storage effects at early times has been developedby Russell [14] In this method it is not necessary to measurethe rate of influx after closing in at the surface Instead oneuses a theoretical equation which gives the form that thebottom hole pressure should have fluid accumulates in the

8 Abstract and Applied Analysis

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

250

200

150

100

50

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 7 Deconvolution for a drawdown test in a reservoir with asealing fault

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 8 Deconvolution for a drawdown test in a reservoir with asealing fault

wellbore during buildup This leads to the result that oneshould plot

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

(A1)

versus log (Δ119905) in analyzing pressure buildup data during theearly fill-up period The denominator makes a correction for

104103102101100

Horner time ratio

800

700

600

500

400

300

200

100

0

Dpws

(psi)

Dpws wellbore storageDps explicit deconvolution methodDps material balance deconvolution

Figure 9 Restored pressures for a buildup test in a homogeneousreservoir

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dpa

(psi)

Dpaw wellbore storage explicit deconvolution method material balance deconvolution

DpasDpas

Figure 10 Restored pressures for a buildup test in a radial compositereservoir

the gradually decreasing flow into the wellbore The quantity1198622is obtained by trial and error as the value which makes the

curve straight at early times After obtaining the straight linesection the rest of the analysis is the same as for any otherpressure buildup This method has the advantage or requiresno additional data over that taken routinely during a shut intest

Due to the time consuming and unconfident of trial anderror to determine 119862

2in Russell method we modify this

method Russellrsquos wellbore storage correction is given as

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

= 119891 (Δ119905 = 1 hr) + 119898119904119897log (Δ119905)

(A2)

Abstract and Applied Analysis 9

Defining the following function and using least squarestechnique obtain the final result

119864 (1198622 119891119898)

=

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891(1198622) minus 119898(119862

2) log(Δ119905

119894))

2

(A3)

Tominimize the function119864 it can differentiate119864with respectto 1198622 119891 and 119898 and setting the results equal to zero After

differentiation it reduces to

120597119864

1205971198622

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894) ) = 0

(A4)

120597119864

120597119891

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622) minus 119898 (119862

2) log (Δ119905

119894))

= 0

(A5)120597119864

120597119898

= minus2

119873

sum

119894=1

log (Δ119905119894) (

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus 119898 (1198622) log (Δ119905

119894)) = 0

(A6)

From (A4) we define the following function

119883(1198622)

=

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times(Δ119901119908119904119894

(1minus (11198622Δ119905119894))minus119891 (119862

2) minus119898 (119862

2) log (Δ119905

119894))

= 0

(A7)

The roots of (A7) can be found via the following algorithm

1198622(119894+1)

= 1198622(119894)

minus119883 (1198622)

1198831015840 (1198622) (A8)

where

119883(1198622) =

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894))

1198831015840(1198622) =

119873

sum

119894=1

(minus2Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3)

times (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2

times [Δ119901119908119904119894

(1 minus (11198622Δ119905))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894)])

+ (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (minusΔ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3minus 1198911015840(1198622)

minus1198981015840(1198622) log (Δ119905

119894))

(A9)

where the unknown functions are given by the followingrelations

119891 (1198622) =

120573119906 (1198622) minus 120572V (119862

2)

1205732 minus 119873120572(A10)

whose derivative is

1198911015840(1198622) =

1205731199061015840(1198622) minus 120572V1015840 (119862

2)

1205732 minus 119873120572

119898 (1198622) =

119873119906 (1198622) minus 120573V (119862

2)

119873120572 minus 1205732

(A11)

where its derivative is

1198981015840(1198622) =

1198731199061015840(1198622) minus 120573V1015840 (119862

2)

119873120572 minus 1205732

119906 (1198622) =

119873

sum

119894=1

log (Δ119905119894)

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A12)

10 Abstract and Applied Analysis

and after differentiation it yields

1199061015840(1198622) = minus

119873

sum

119894=1

log (Δ119905

119894)

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2

V (1198622) =

119873

sum

119894=1

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A13)

which has the following derivative

V1015840 (1198622) = minus

119873

sum

119894=1

1

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2 (A14)

finally the values of 120572 and 120573 parameters are given by thefollowing formulas

120572 =

119873

sum

119894=1

(log(Δ119905119894))2

120573 =

119873

sum

119894=1

log (Δ119905119894)

(A15)

B General Formula for Inverse Problem

The convolution of (6) yields

119901119904119863

(119905119863) = int

119905119863

0

119870 (120591) 119901119908119863 (119905119863 minus 120591) 119889120591

= int

119905119863

0

119870(119905119863minus 120591) 119901

119908119863 (120591) 119889120591

(B1)

where

119870(119905119863) = 119871minus1(

1

119904119902 (119904)) (B2)

where 119871minus1 is the inverse Lapalce transform operator 119870(119905119863)

can be computed either from the Laplace transforms of119902119863(119905119863) data a curve-fitted equation of 119902

119863(119905119863) data or using

another technique with a relation between119870(119905119863) and 119902

119863(119905119863)

However it would be time consuming to invert all the 119902119863(119905119863)

data in Laplace space and transform it back to real space inaccordance with (B2) Once an approximation is obtained itwill be easy to compute119870(119905

119863) and integrate (B1) to determine

119901119904119863(119905119863) However finding some approximation functions for

119902119863(119905119863) data may be difficult We provide an approach to

overcome this problemThis idea is based on finding anotherrelation between 119870(119905

119863) and 119902

119863(119905119863) that is

ℓ (1) =1

119904=

1

119904119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B3)or

1 = int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591 = int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591

(B4)

The other relations that can be used are

ℓ (119905119863) =

1

1199042=

1

1199042119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B5)

or

119905119863= int

119905119863

0

119871 (120591) 119902119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

119863(120591) 119889120591

(B6)

ℓ (1) =1

119904=

1

1199042119902119863(119904)

119904119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591)

(B7)

or

1 = int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591

(B8)

where

119871 (119905119863) = ℓminus1(

1

1199042119902 (119904)) (B9)

Equation (B4) can be used simultaneously with (B1) toanalyze the deconvolution problem

Equation (11) can be written as

119901119904119863

(119904) = 119904119901119908119863

(119904)1

1199042119902119863(119904)

(B10)

or

119901119904119863

(119905119863) = int

119905119863

0

119871 (120591) 1199011015840

119908119863(119905119863minus 120591) 119889120591

= int

119905119863

0

119871 (119905119863minus 120591) 119901

1015840

119908119863(120591) 119889120591

(B11)

The applications of (B6) and (B8) are with (B11)For justifying the rate normalizationmethod andmaterial

balance deconvolution it can use (B1) (B4) ((B11) can alsobe used with (B6) or (B8) for this goal) and generalizedmean value theorem for integrals This theorem is expressedas follows

Theorem 1 Let 119891 and 119892 be continuous on [119886 119887] and supposethat 119892 does not change sign on [119886 119887] Then there is a point 119888 in[119886 119887] such that

int

119887

119886

119891 (119905) 119892 (119905) 119889119905 = 119891 (119888) int

119887

119886

119892 (119905) 119889119905 (B12)

Abstract and Applied Analysis 11

By Theorem 1 there are some values 1198881and 1198882between 0

and 119905119863such that

119901119904119863

(119905119863) = 119901119908119863

(1198881) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B13)

1 = 119902119863(1198882) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B14)

By setting 1198881

= 1198882

= 119905119863and dividing the right and left

side of (B13) respectively the rate normalization techniqueis developed However the values 119888

1and 1198882are not equal

to each other and are not equal to 119905119863

in general Thisis the reason of error in rate normalization and materialbalance deconvolution methods 119888

1and 1198882are less than 119905

119863in

these methods for the duration of wellbore storage distortionduring a pressure transient test Since these conditions areapproximately satisfied in material balance deconvolutionthe results of material balance are more accurate and moreconfident than rate normalization

Nomenclature

API API gravity ∘API119861119900 Oil formation volume factor RBSTB

119888119905 Total isothermal compressibility factor

psiminus1

1198622 Arbitrary constant hr

minus1

119862119863 Wellbore storage constant dimensionless

119862119904 Wellbore storage coefficient bblpsi

ℎ Formation thickness ftHTR Horner time ratio dimensionless119896 Formation permeability md119870 Kernel of the convolution integral119871 Kernel of the convolution integral119873119901 Cumulative oil production bbl

119901 Pressure psi119901119886 Normalized pseudopressure psi

119901 Average reservoir pressure psi119901119863 Formation pressure dimensionless

119901119889119901 Dewpoint pressure psi

119901119894 Initial reservoir pressure psi

119901119904119863 Formation pressure including skin

dimensionless119901119904 Formation pressure including skin psi

119901119908119863

Pressure draw down dimensionless119901119908119891 Flowing bottom hole pressure psi

119901119908119904 Shut-in bottom hole pressure psi

119902119863 Sandface flow rate dimensionless

119902119892 Gas flow rate MscfD

119902119900 Oil flow rate STBD

119903119908 Wellbore radius ft

119904 Laplace variable119878 Skin factor119905 Time hr119905119863 Time dimensionless

119905119901 Production time hr

119879 Reservoir temperature ∘F

119879119889119901 Dewpoint temperature ∘F

120572 Explicit deconvolution variabledimensionless

120573 Beta deconvolution variabledimensionless

120574119892 Gas gravity (air = 10)

120582 Interporosity flow coefficient of dualporosity reservoir

120601 Porosity of reservoir rock dimensionless120583119892 Gas viscosity cp

120583119900 Oil viscosity cp

120596 Storativity ratio of dual porosity reservoirminus Laplace transform of a function1015840 Derivative of a function

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Cheng W J Lee and D A McVay ldquoFast-Fourier-transform-based deconvolution for interpretation of pressure-transient-test data dominated by Wellbore storagerdquo SPE Reservoir Eval-uation and Engineering vol 8 no 3 pp 224ndash239 2005

[2] M M Levitan and M R Wilson ldquoDeconvolution of pressureand rate data from gas reservoirs with significant pressuredepletionrdquo SPE Journal vol 17 no 3 pp 727ndash741 2012

[3] Y Liu and R N Horne ldquoInterpreting pressure and flow-ratedata from permanent downhole gauges by use of data-miningapproachesrdquo SPE Journal vol 18 no 1 pp 69ndash82 2013

[4] O Ogunrewo and A C Gringarten ldquoDeconvolution of welltest data in lean and rich gas condensate and volatile oil wellsbelow saturation pressurerdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Document ID 166340-MS New Orleans La USA September-October 2013

[5] R C Earlougher Jr Advances in Well Test Analysis vol 5 ofMonogragh Series SPE Richardson Tex USA 1977

[6] F Kucuk and L Ayestaran ldquoAnalysis of simultaneously mea-sured pressure and sandface flow rate in transient well testingrdquoJournal of PetroleumTechnology vol 37 no 2 pp 323ndash334 1985

[7] M Onur M Cinar D IIk P P Valko T A Blasingame and PHegeman ldquoAn investigation of recent deconvolution methodsfor well-test data analysisrdquo SPE Journal vol 13 no 2 pp 226ndash247 2008

[8] F J KuchukMOnur andFHollaender ldquoPressure transient testdesign and interpretationrdquo Developments in Petroleum Sciencevol 57 pp 303ndash359 2010

[9] A C Gringarten D P Bourdet P A Landel and V J KniazeffldquoA comparison between different skin and wellbore storagetype-curves for early-time transient analysisrdquo in Proceedings ofthe SPE Annual Technical Conference and Exhibition Paper SPE8205 Las Vegas Nev USA 1979

[10] A C Gringarten ldquoFrom straight lines to deconvolution theevolution of the state of the art in well test analysisrdquo SPEReservoir Evaluation and Engineering vol 11 no 1 pp 41ndash622008

[11] J A Cumming D A Wooff T Whittle and A C GringartenldquoMultiple well deconvolutionrdquo in Proceedings of the SPE Annual

12 Abstract and Applied Analysis

Technical Conference and Exhibition Document ID 166458-MS New Orleans La USA September-October 2013

[12] O Bahabanian D Ilk N Hosseinpour-Zonoozi and T ABlasingame ldquoExplicit deconvolution of well test data distortedby wellbore storagerdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Paper SPE 103216 pp 4444ndash4453San Antonio Tex USA September 2006

[13] J Jones and A Chen ldquoSuccessful applications of pressure-ratedeconvolution in theCad-Nik tight gas formations of the BritishColumbia foothillsrdquo Journal of Canadian Petroleum Technologyvol 51 no 3 pp 176ndash192 2012

[14] D G Russell ldquoExtension of pressure buildup analysis methodsrdquoJournal of Petroleum Technology vol 18 no 12 pp 1624ndash16361966

[15] R E Gladfelter G W Tracy and L E Wilsey ldquoSelecting wellswhich will respond to production-stimulation treatmentrdquo inDrilling and Production Practice API Dallas Tex USA 1955

[16] M J Fetkovich and M E Vienot ldquoRate normalization ofbuildup pressure by using afterflow datardquo Journal of PetroleumTechnology vol 36 no 13 pp 2211ndash2224 1984

[17] J L Johnston Variable rate analysis of transient well testdata using semi-analytical methods [MS thesis] Texas AampMUniversity College Station Tex USA 1992

[18] A F van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 6 pp 171ndash176 1953

[19] W Hurst ldquoEstablishment of the skin effect and its impedimentto fluid flow into a well borerdquo The Petroleum Engineer pp B6ndashB16 1953

[20] J A Joseph and L F Koederitz ldquoA simple nonlinear model forrepresentation of field transient responsesrdquo Paper SPE 11435SPE Richardson Tex USA 1982

[21] F J Kuchuk ldquoGladfelter deconvolutionrdquo SPE Formation Evalu-ation vol 5 no 3 pp 285ndash292 1990

[22] A F van Everdingen and W Hurst ldquoThe application of thelaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[23] D Bourdet T M Whittle A A Douglas and Y M Pirard ldquoAnew set of type curves simplifies well test analysisrdquo World Oilpp 95ndash106 1983

[24] D F Meunier C S Kabir and M J Wittmann ldquoGas well testanalysis use of normalized pseudovariablesrdquo SPE FormationEvaluation vol 2 no 4 pp 629ndash636 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

250

200

150

100

50

0

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 7 Deconvolution for a drawdown test in a reservoir with asealing fault

10minus3 10minus2 10minus1 100 101 102 103

Time (hr)

103

102

101

100

Dpwf wellbore storaged(Dpwf)d(ln(t)) wellbore storageDps explicit deconvolution methodd(Dps)d(ln(t)) explicit deconvolution methodDps material balance deconvolutiond(Dps)d(ln(t)) material balance deconvolutionDps no wellbore storaged(Dps)d(ln(t)) no wellbore storage

Dpwf

and

its lo

garit

hmic

der

ivat

ive (

psi)

Figure 8 Deconvolution for a drawdown test in a reservoir with asealing fault

wellbore during buildup This leads to the result that oneshould plot

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

(A1)

versus log (Δ119905) in analyzing pressure buildup data during theearly fill-up period The denominator makes a correction for

104103102101100

Horner time ratio

800

700

600

500

400

300

200

100

0

Dpws

(psi)

Dpws wellbore storageDps explicit deconvolution methodDps material balance deconvolution

Figure 9 Restored pressures for a buildup test in a homogeneousreservoir

0

500

1000

1500

2000

2500

3000

3500

4000

105104103102101100

Horner time ratio

Dpa

(psi)

Dpaw wellbore storage explicit deconvolution method material balance deconvolution

DpasDpas

Figure 10 Restored pressures for a buildup test in a radial compositereservoir

the gradually decreasing flow into the wellbore The quantity1198622is obtained by trial and error as the value which makes the

curve straight at early times After obtaining the straight linesection the rest of the analysis is the same as for any otherpressure buildup This method has the advantage or requiresno additional data over that taken routinely during a shut intest

Due to the time consuming and unconfident of trial anderror to determine 119862

2in Russell method we modify this

method Russellrsquos wellbore storage correction is given as

[119901119908119904 (Δ119905) minus 119901

119908119891 (Δ119905 = 0)]

[1 minus (11198622Δ119905)]

= 119891 (Δ119905 = 1 hr) + 119898119904119897log (Δ119905)

(A2)

Abstract and Applied Analysis 9

Defining the following function and using least squarestechnique obtain the final result

119864 (1198622 119891119898)

=

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891(1198622) minus 119898(119862

2) log(Δ119905

119894))

2

(A3)

Tominimize the function119864 it can differentiate119864with respectto 1198622 119891 and 119898 and setting the results equal to zero After

differentiation it reduces to

120597119864

1205971198622

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894) ) = 0

(A4)

120597119864

120597119891

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622) minus 119898 (119862

2) log (Δ119905

119894))

= 0

(A5)120597119864

120597119898

= minus2

119873

sum

119894=1

log (Δ119905119894) (

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus 119898 (1198622) log (Δ119905

119894)) = 0

(A6)

From (A4) we define the following function

119883(1198622)

=

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times(Δ119901119908119904119894

(1minus (11198622Δ119905119894))minus119891 (119862

2) minus119898 (119862

2) log (Δ119905

119894))

= 0

(A7)

The roots of (A7) can be found via the following algorithm

1198622(119894+1)

= 1198622(119894)

minus119883 (1198622)

1198831015840 (1198622) (A8)

where

119883(1198622) =

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894))

1198831015840(1198622) =

119873

sum

119894=1

(minus2Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3)

times (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2

times [Δ119901119908119904119894

(1 minus (11198622Δ119905))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894)])

+ (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (minusΔ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3minus 1198911015840(1198622)

minus1198981015840(1198622) log (Δ119905

119894))

(A9)

where the unknown functions are given by the followingrelations

119891 (1198622) =

120573119906 (1198622) minus 120572V (119862

2)

1205732 minus 119873120572(A10)

whose derivative is

1198911015840(1198622) =

1205731199061015840(1198622) minus 120572V1015840 (119862

2)

1205732 minus 119873120572

119898 (1198622) =

119873119906 (1198622) minus 120573V (119862

2)

119873120572 minus 1205732

(A11)

where its derivative is

1198981015840(1198622) =

1198731199061015840(1198622) minus 120573V1015840 (119862

2)

119873120572 minus 1205732

119906 (1198622) =

119873

sum

119894=1

log (Δ119905119894)

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A12)

10 Abstract and Applied Analysis

and after differentiation it yields

1199061015840(1198622) = minus

119873

sum

119894=1

log (Δ119905

119894)

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2

V (1198622) =

119873

sum

119894=1

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A13)

which has the following derivative

V1015840 (1198622) = minus

119873

sum

119894=1

1

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2 (A14)

finally the values of 120572 and 120573 parameters are given by thefollowing formulas

120572 =

119873

sum

119894=1

(log(Δ119905119894))2

120573 =

119873

sum

119894=1

log (Δ119905119894)

(A15)

B General Formula for Inverse Problem

The convolution of (6) yields

119901119904119863

(119905119863) = int

119905119863

0

119870 (120591) 119901119908119863 (119905119863 minus 120591) 119889120591

= int

119905119863

0

119870(119905119863minus 120591) 119901

119908119863 (120591) 119889120591

(B1)

where

119870(119905119863) = 119871minus1(

1

119904119902 (119904)) (B2)

where 119871minus1 is the inverse Lapalce transform operator 119870(119905119863)

can be computed either from the Laplace transforms of119902119863(119905119863) data a curve-fitted equation of 119902

119863(119905119863) data or using

another technique with a relation between119870(119905119863) and 119902

119863(119905119863)

However it would be time consuming to invert all the 119902119863(119905119863)

data in Laplace space and transform it back to real space inaccordance with (B2) Once an approximation is obtained itwill be easy to compute119870(119905

119863) and integrate (B1) to determine

119901119904119863(119905119863) However finding some approximation functions for

119902119863(119905119863) data may be difficult We provide an approach to

overcome this problemThis idea is based on finding anotherrelation between 119870(119905

119863) and 119902

119863(119905119863) that is

ℓ (1) =1

119904=

1

119904119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B3)or

1 = int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591 = int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591

(B4)

The other relations that can be used are

ℓ (119905119863) =

1

1199042=

1

1199042119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B5)

or

119905119863= int

119905119863

0

119871 (120591) 119902119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

119863(120591) 119889120591

(B6)

ℓ (1) =1

119904=

1

1199042119902119863(119904)

119904119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591)

(B7)

or

1 = int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591

(B8)

where

119871 (119905119863) = ℓminus1(

1

1199042119902 (119904)) (B9)

Equation (B4) can be used simultaneously with (B1) toanalyze the deconvolution problem

Equation (11) can be written as

119901119904119863

(119904) = 119904119901119908119863

(119904)1

1199042119902119863(119904)

(B10)

or

119901119904119863

(119905119863) = int

119905119863

0

119871 (120591) 1199011015840

119908119863(119905119863minus 120591) 119889120591

= int

119905119863

0

119871 (119905119863minus 120591) 119901

1015840

119908119863(120591) 119889120591

(B11)

The applications of (B6) and (B8) are with (B11)For justifying the rate normalizationmethod andmaterial

balance deconvolution it can use (B1) (B4) ((B11) can alsobe used with (B6) or (B8) for this goal) and generalizedmean value theorem for integrals This theorem is expressedas follows

Theorem 1 Let 119891 and 119892 be continuous on [119886 119887] and supposethat 119892 does not change sign on [119886 119887] Then there is a point 119888 in[119886 119887] such that

int

119887

119886

119891 (119905) 119892 (119905) 119889119905 = 119891 (119888) int

119887

119886

119892 (119905) 119889119905 (B12)

Abstract and Applied Analysis 11

By Theorem 1 there are some values 1198881and 1198882between 0

and 119905119863such that

119901119904119863

(119905119863) = 119901119908119863

(1198881) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B13)

1 = 119902119863(1198882) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B14)

By setting 1198881

= 1198882

= 119905119863and dividing the right and left

side of (B13) respectively the rate normalization techniqueis developed However the values 119888

1and 1198882are not equal

to each other and are not equal to 119905119863

in general Thisis the reason of error in rate normalization and materialbalance deconvolution methods 119888

1and 1198882are less than 119905

119863in

these methods for the duration of wellbore storage distortionduring a pressure transient test Since these conditions areapproximately satisfied in material balance deconvolutionthe results of material balance are more accurate and moreconfident than rate normalization

Nomenclature

API API gravity ∘API119861119900 Oil formation volume factor RBSTB

119888119905 Total isothermal compressibility factor

psiminus1

1198622 Arbitrary constant hr

minus1

119862119863 Wellbore storage constant dimensionless

119862119904 Wellbore storage coefficient bblpsi

ℎ Formation thickness ftHTR Horner time ratio dimensionless119896 Formation permeability md119870 Kernel of the convolution integral119871 Kernel of the convolution integral119873119901 Cumulative oil production bbl

119901 Pressure psi119901119886 Normalized pseudopressure psi

119901 Average reservoir pressure psi119901119863 Formation pressure dimensionless

119901119889119901 Dewpoint pressure psi

119901119894 Initial reservoir pressure psi

119901119904119863 Formation pressure including skin

dimensionless119901119904 Formation pressure including skin psi

119901119908119863

Pressure draw down dimensionless119901119908119891 Flowing bottom hole pressure psi

119901119908119904 Shut-in bottom hole pressure psi

119902119863 Sandface flow rate dimensionless

119902119892 Gas flow rate MscfD

119902119900 Oil flow rate STBD

119903119908 Wellbore radius ft

119904 Laplace variable119878 Skin factor119905 Time hr119905119863 Time dimensionless

119905119901 Production time hr

119879 Reservoir temperature ∘F

119879119889119901 Dewpoint temperature ∘F

120572 Explicit deconvolution variabledimensionless

120573 Beta deconvolution variabledimensionless

120574119892 Gas gravity (air = 10)

120582 Interporosity flow coefficient of dualporosity reservoir

120601 Porosity of reservoir rock dimensionless120583119892 Gas viscosity cp

120583119900 Oil viscosity cp

120596 Storativity ratio of dual porosity reservoirminus Laplace transform of a function1015840 Derivative of a function

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Cheng W J Lee and D A McVay ldquoFast-Fourier-transform-based deconvolution for interpretation of pressure-transient-test data dominated by Wellbore storagerdquo SPE Reservoir Eval-uation and Engineering vol 8 no 3 pp 224ndash239 2005

[2] M M Levitan and M R Wilson ldquoDeconvolution of pressureand rate data from gas reservoirs with significant pressuredepletionrdquo SPE Journal vol 17 no 3 pp 727ndash741 2012

[3] Y Liu and R N Horne ldquoInterpreting pressure and flow-ratedata from permanent downhole gauges by use of data-miningapproachesrdquo SPE Journal vol 18 no 1 pp 69ndash82 2013

[4] O Ogunrewo and A C Gringarten ldquoDeconvolution of welltest data in lean and rich gas condensate and volatile oil wellsbelow saturation pressurerdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Document ID 166340-MS New Orleans La USA September-October 2013

[5] R C Earlougher Jr Advances in Well Test Analysis vol 5 ofMonogragh Series SPE Richardson Tex USA 1977

[6] F Kucuk and L Ayestaran ldquoAnalysis of simultaneously mea-sured pressure and sandface flow rate in transient well testingrdquoJournal of PetroleumTechnology vol 37 no 2 pp 323ndash334 1985

[7] M Onur M Cinar D IIk P P Valko T A Blasingame and PHegeman ldquoAn investigation of recent deconvolution methodsfor well-test data analysisrdquo SPE Journal vol 13 no 2 pp 226ndash247 2008

[8] F J KuchukMOnur andFHollaender ldquoPressure transient testdesign and interpretationrdquo Developments in Petroleum Sciencevol 57 pp 303ndash359 2010

[9] A C Gringarten D P Bourdet P A Landel and V J KniazeffldquoA comparison between different skin and wellbore storagetype-curves for early-time transient analysisrdquo in Proceedings ofthe SPE Annual Technical Conference and Exhibition Paper SPE8205 Las Vegas Nev USA 1979

[10] A C Gringarten ldquoFrom straight lines to deconvolution theevolution of the state of the art in well test analysisrdquo SPEReservoir Evaluation and Engineering vol 11 no 1 pp 41ndash622008

[11] J A Cumming D A Wooff T Whittle and A C GringartenldquoMultiple well deconvolutionrdquo in Proceedings of the SPE Annual

12 Abstract and Applied Analysis

Technical Conference and Exhibition Document ID 166458-MS New Orleans La USA September-October 2013

[12] O Bahabanian D Ilk N Hosseinpour-Zonoozi and T ABlasingame ldquoExplicit deconvolution of well test data distortedby wellbore storagerdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Paper SPE 103216 pp 4444ndash4453San Antonio Tex USA September 2006

[13] J Jones and A Chen ldquoSuccessful applications of pressure-ratedeconvolution in theCad-Nik tight gas formations of the BritishColumbia foothillsrdquo Journal of Canadian Petroleum Technologyvol 51 no 3 pp 176ndash192 2012

[14] D G Russell ldquoExtension of pressure buildup analysis methodsrdquoJournal of Petroleum Technology vol 18 no 12 pp 1624ndash16361966

[15] R E Gladfelter G W Tracy and L E Wilsey ldquoSelecting wellswhich will respond to production-stimulation treatmentrdquo inDrilling and Production Practice API Dallas Tex USA 1955

[16] M J Fetkovich and M E Vienot ldquoRate normalization ofbuildup pressure by using afterflow datardquo Journal of PetroleumTechnology vol 36 no 13 pp 2211ndash2224 1984

[17] J L Johnston Variable rate analysis of transient well testdata using semi-analytical methods [MS thesis] Texas AampMUniversity College Station Tex USA 1992

[18] A F van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 6 pp 171ndash176 1953

[19] W Hurst ldquoEstablishment of the skin effect and its impedimentto fluid flow into a well borerdquo The Petroleum Engineer pp B6ndashB16 1953

[20] J A Joseph and L F Koederitz ldquoA simple nonlinear model forrepresentation of field transient responsesrdquo Paper SPE 11435SPE Richardson Tex USA 1982

[21] F J Kuchuk ldquoGladfelter deconvolutionrdquo SPE Formation Evalu-ation vol 5 no 3 pp 285ndash292 1990

[22] A F van Everdingen and W Hurst ldquoThe application of thelaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[23] D Bourdet T M Whittle A A Douglas and Y M Pirard ldquoAnew set of type curves simplifies well test analysisrdquo World Oilpp 95ndash106 1983

[24] D F Meunier C S Kabir and M J Wittmann ldquoGas well testanalysis use of normalized pseudovariablesrdquo SPE FormationEvaluation vol 2 no 4 pp 629ndash636 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

Defining the following function and using least squarestechnique obtain the final result

119864 (1198622 119891119898)

=

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891(1198622) minus 119898(119862

2) log(Δ119905

119894))

2

(A3)

Tominimize the function119864 it can differentiate119864with respectto 1198622 119891 and 119898 and setting the results equal to zero After

differentiation it reduces to

120597119864

1205971198622

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894) ) = 0

(A4)

120597119864

120597119891

= minus2

119873

sum

119894=1

(Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622) minus 119898 (119862

2) log (Δ119905

119894))

= 0

(A5)120597119864

120597119898

= minus2

119873

sum

119894=1

log (Δ119905119894) (

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus 119898 (1198622) log (Δ119905

119894)) = 0

(A6)

From (A4) we define the following function

119883(1198622)

=

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times(Δ119901119908119904119894

(1minus (11198622Δ119905119894))minus119891 (119862

2) minus119898 (119862

2) log (Δ119905

119894))

= 0

(A7)

The roots of (A7) can be found via the following algorithm

1198622(119894+1)

= 1198622(119894)

minus119883 (1198622)

1198831015840 (1198622) (A8)

where

119883(1198622) =

119873

sum

119894=1

(Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894))

1198831015840(1198622) =

119873

sum

119894=1

(minus2Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3)

times (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2

times [Δ119901119908119904119894

(1 minus (11198622Δ119905))

minus 119891 (1198622)

minus119898 (1198622) log (Δ119905

119894)])

+ (Δ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))2)

times (minusΔ119901119908119904119894

Δ119905119894(1198622minus (1Δ119905

119894))3minus 1198911015840(1198622)

minus1198981015840(1198622) log (Δ119905

119894))

(A9)

where the unknown functions are given by the followingrelations

119891 (1198622) =

120573119906 (1198622) minus 120572V (119862

2)

1205732 minus 119873120572(A10)

whose derivative is

1198911015840(1198622) =

1205731199061015840(1198622) minus 120572V1015840 (119862

2)

1205732 minus 119873120572

119898 (1198622) =

119873119906 (1198622) minus 120573V (119862

2)

119873120572 minus 1205732

(A11)

where its derivative is

1198981015840(1198622) =

1198731199061015840(1198622) minus 120573V1015840 (119862

2)

119873120572 minus 1205732

119906 (1198622) =

119873

sum

119894=1

log (Δ119905119894)

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A12)

10 Abstract and Applied Analysis

and after differentiation it yields

1199061015840(1198622) = minus

119873

sum

119894=1

log (Δ119905

119894)

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2

V (1198622) =

119873

sum

119894=1

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A13)

which has the following derivative

V1015840 (1198622) = minus

119873

sum

119894=1

1

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2 (A14)

finally the values of 120572 and 120573 parameters are given by thefollowing formulas

120572 =

119873

sum

119894=1

(log(Δ119905119894))2

120573 =

119873

sum

119894=1

log (Δ119905119894)

(A15)

B General Formula for Inverse Problem

The convolution of (6) yields

119901119904119863

(119905119863) = int

119905119863

0

119870 (120591) 119901119908119863 (119905119863 minus 120591) 119889120591

= int

119905119863

0

119870(119905119863minus 120591) 119901

119908119863 (120591) 119889120591

(B1)

where

119870(119905119863) = 119871minus1(

1

119904119902 (119904)) (B2)

where 119871minus1 is the inverse Lapalce transform operator 119870(119905119863)

can be computed either from the Laplace transforms of119902119863(119905119863) data a curve-fitted equation of 119902

119863(119905119863) data or using

another technique with a relation between119870(119905119863) and 119902

119863(119905119863)

However it would be time consuming to invert all the 119902119863(119905119863)

data in Laplace space and transform it back to real space inaccordance with (B2) Once an approximation is obtained itwill be easy to compute119870(119905

119863) and integrate (B1) to determine

119901119904119863(119905119863) However finding some approximation functions for

119902119863(119905119863) data may be difficult We provide an approach to

overcome this problemThis idea is based on finding anotherrelation between 119870(119905

119863) and 119902

119863(119905119863) that is

ℓ (1) =1

119904=

1

119904119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B3)or

1 = int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591 = int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591

(B4)

The other relations that can be used are

ℓ (119905119863) =

1

1199042=

1

1199042119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B5)

or

119905119863= int

119905119863

0

119871 (120591) 119902119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

119863(120591) 119889120591

(B6)

ℓ (1) =1

119904=

1

1199042119902119863(119904)

119904119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591)

(B7)

or

1 = int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591

(B8)

where

119871 (119905119863) = ℓminus1(

1

1199042119902 (119904)) (B9)

Equation (B4) can be used simultaneously with (B1) toanalyze the deconvolution problem

Equation (11) can be written as

119901119904119863

(119904) = 119904119901119908119863

(119904)1

1199042119902119863(119904)

(B10)

or

119901119904119863

(119905119863) = int

119905119863

0

119871 (120591) 1199011015840

119908119863(119905119863minus 120591) 119889120591

= int

119905119863

0

119871 (119905119863minus 120591) 119901

1015840

119908119863(120591) 119889120591

(B11)

The applications of (B6) and (B8) are with (B11)For justifying the rate normalizationmethod andmaterial

balance deconvolution it can use (B1) (B4) ((B11) can alsobe used with (B6) or (B8) for this goal) and generalizedmean value theorem for integrals This theorem is expressedas follows

Theorem 1 Let 119891 and 119892 be continuous on [119886 119887] and supposethat 119892 does not change sign on [119886 119887] Then there is a point 119888 in[119886 119887] such that

int

119887

119886

119891 (119905) 119892 (119905) 119889119905 = 119891 (119888) int

119887

119886

119892 (119905) 119889119905 (B12)

Abstract and Applied Analysis 11

By Theorem 1 there are some values 1198881and 1198882between 0

and 119905119863such that

119901119904119863

(119905119863) = 119901119908119863

(1198881) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B13)

1 = 119902119863(1198882) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B14)

By setting 1198881

= 1198882

= 119905119863and dividing the right and left

side of (B13) respectively the rate normalization techniqueis developed However the values 119888

1and 1198882are not equal

to each other and are not equal to 119905119863

in general Thisis the reason of error in rate normalization and materialbalance deconvolution methods 119888

1and 1198882are less than 119905

119863in

these methods for the duration of wellbore storage distortionduring a pressure transient test Since these conditions areapproximately satisfied in material balance deconvolutionthe results of material balance are more accurate and moreconfident than rate normalization

Nomenclature

API API gravity ∘API119861119900 Oil formation volume factor RBSTB

119888119905 Total isothermal compressibility factor

psiminus1

1198622 Arbitrary constant hr

minus1

119862119863 Wellbore storage constant dimensionless

119862119904 Wellbore storage coefficient bblpsi

ℎ Formation thickness ftHTR Horner time ratio dimensionless119896 Formation permeability md119870 Kernel of the convolution integral119871 Kernel of the convolution integral119873119901 Cumulative oil production bbl

119901 Pressure psi119901119886 Normalized pseudopressure psi

119901 Average reservoir pressure psi119901119863 Formation pressure dimensionless

119901119889119901 Dewpoint pressure psi

119901119894 Initial reservoir pressure psi

119901119904119863 Formation pressure including skin

dimensionless119901119904 Formation pressure including skin psi

119901119908119863

Pressure draw down dimensionless119901119908119891 Flowing bottom hole pressure psi

119901119908119904 Shut-in bottom hole pressure psi

119902119863 Sandface flow rate dimensionless

119902119892 Gas flow rate MscfD

119902119900 Oil flow rate STBD

119903119908 Wellbore radius ft

119904 Laplace variable119878 Skin factor119905 Time hr119905119863 Time dimensionless

119905119901 Production time hr

119879 Reservoir temperature ∘F

119879119889119901 Dewpoint temperature ∘F

120572 Explicit deconvolution variabledimensionless

120573 Beta deconvolution variabledimensionless

120574119892 Gas gravity (air = 10)

120582 Interporosity flow coefficient of dualporosity reservoir

120601 Porosity of reservoir rock dimensionless120583119892 Gas viscosity cp

120583119900 Oil viscosity cp

120596 Storativity ratio of dual porosity reservoirminus Laplace transform of a function1015840 Derivative of a function

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Cheng W J Lee and D A McVay ldquoFast-Fourier-transform-based deconvolution for interpretation of pressure-transient-test data dominated by Wellbore storagerdquo SPE Reservoir Eval-uation and Engineering vol 8 no 3 pp 224ndash239 2005

[2] M M Levitan and M R Wilson ldquoDeconvolution of pressureand rate data from gas reservoirs with significant pressuredepletionrdquo SPE Journal vol 17 no 3 pp 727ndash741 2012

[3] Y Liu and R N Horne ldquoInterpreting pressure and flow-ratedata from permanent downhole gauges by use of data-miningapproachesrdquo SPE Journal vol 18 no 1 pp 69ndash82 2013

[4] O Ogunrewo and A C Gringarten ldquoDeconvolution of welltest data in lean and rich gas condensate and volatile oil wellsbelow saturation pressurerdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Document ID 166340-MS New Orleans La USA September-October 2013

[5] R C Earlougher Jr Advances in Well Test Analysis vol 5 ofMonogragh Series SPE Richardson Tex USA 1977

[6] F Kucuk and L Ayestaran ldquoAnalysis of simultaneously mea-sured pressure and sandface flow rate in transient well testingrdquoJournal of PetroleumTechnology vol 37 no 2 pp 323ndash334 1985

[7] M Onur M Cinar D IIk P P Valko T A Blasingame and PHegeman ldquoAn investigation of recent deconvolution methodsfor well-test data analysisrdquo SPE Journal vol 13 no 2 pp 226ndash247 2008

[8] F J KuchukMOnur andFHollaender ldquoPressure transient testdesign and interpretationrdquo Developments in Petroleum Sciencevol 57 pp 303ndash359 2010

[9] A C Gringarten D P Bourdet P A Landel and V J KniazeffldquoA comparison between different skin and wellbore storagetype-curves for early-time transient analysisrdquo in Proceedings ofthe SPE Annual Technical Conference and Exhibition Paper SPE8205 Las Vegas Nev USA 1979

[10] A C Gringarten ldquoFrom straight lines to deconvolution theevolution of the state of the art in well test analysisrdquo SPEReservoir Evaluation and Engineering vol 11 no 1 pp 41ndash622008

[11] J A Cumming D A Wooff T Whittle and A C GringartenldquoMultiple well deconvolutionrdquo in Proceedings of the SPE Annual

12 Abstract and Applied Analysis

Technical Conference and Exhibition Document ID 166458-MS New Orleans La USA September-October 2013

[12] O Bahabanian D Ilk N Hosseinpour-Zonoozi and T ABlasingame ldquoExplicit deconvolution of well test data distortedby wellbore storagerdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Paper SPE 103216 pp 4444ndash4453San Antonio Tex USA September 2006

[13] J Jones and A Chen ldquoSuccessful applications of pressure-ratedeconvolution in theCad-Nik tight gas formations of the BritishColumbia foothillsrdquo Journal of Canadian Petroleum Technologyvol 51 no 3 pp 176ndash192 2012

[14] D G Russell ldquoExtension of pressure buildup analysis methodsrdquoJournal of Petroleum Technology vol 18 no 12 pp 1624ndash16361966

[15] R E Gladfelter G W Tracy and L E Wilsey ldquoSelecting wellswhich will respond to production-stimulation treatmentrdquo inDrilling and Production Practice API Dallas Tex USA 1955

[16] M J Fetkovich and M E Vienot ldquoRate normalization ofbuildup pressure by using afterflow datardquo Journal of PetroleumTechnology vol 36 no 13 pp 2211ndash2224 1984

[17] J L Johnston Variable rate analysis of transient well testdata using semi-analytical methods [MS thesis] Texas AampMUniversity College Station Tex USA 1992

[18] A F van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 6 pp 171ndash176 1953

[19] W Hurst ldquoEstablishment of the skin effect and its impedimentto fluid flow into a well borerdquo The Petroleum Engineer pp B6ndashB16 1953

[20] J A Joseph and L F Koederitz ldquoA simple nonlinear model forrepresentation of field transient responsesrdquo Paper SPE 11435SPE Richardson Tex USA 1982

[21] F J Kuchuk ldquoGladfelter deconvolutionrdquo SPE Formation Evalu-ation vol 5 no 3 pp 285ndash292 1990

[22] A F van Everdingen and W Hurst ldquoThe application of thelaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[23] D Bourdet T M Whittle A A Douglas and Y M Pirard ldquoAnew set of type curves simplifies well test analysisrdquo World Oilpp 95ndash106 1983

[24] D F Meunier C S Kabir and M J Wittmann ldquoGas well testanalysis use of normalized pseudovariablesrdquo SPE FormationEvaluation vol 2 no 4 pp 629ndash636 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

and after differentiation it yields

1199061015840(1198622) = minus

119873

sum

119894=1

log (Δ119905

119894)

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2

V (1198622) =

119873

sum

119894=1

Δ119901119908119904119894

(1 minus (11198622Δ119905119894))

(A13)

which has the following derivative

V1015840 (1198622) = minus

119873

sum

119894=1

1

Δ119905119894

Δ119901119908119904119894

(1198622minus (1Δ119905

119894))2 (A14)

finally the values of 120572 and 120573 parameters are given by thefollowing formulas

120572 =

119873

sum

119894=1

(log(Δ119905119894))2

120573 =

119873

sum

119894=1

log (Δ119905119894)

(A15)

B General Formula for Inverse Problem

The convolution of (6) yields

119901119904119863

(119905119863) = int

119905119863

0

119870 (120591) 119901119908119863 (119905119863 minus 120591) 119889120591

= int

119905119863

0

119870(119905119863minus 120591) 119901

119908119863 (120591) 119889120591

(B1)

where

119870(119905119863) = 119871minus1(

1

119904119902 (119904)) (B2)

where 119871minus1 is the inverse Lapalce transform operator 119870(119905119863)

can be computed either from the Laplace transforms of119902119863(119905119863) data a curve-fitted equation of 119902

119863(119905119863) data or using

another technique with a relation between119870(119905119863) and 119902

119863(119905119863)

However it would be time consuming to invert all the 119902119863(119905119863)

data in Laplace space and transform it back to real space inaccordance with (B2) Once an approximation is obtained itwill be easy to compute119870(119905

119863) and integrate (B1) to determine

119901119904119863(119905119863) However finding some approximation functions for

119902119863(119905119863) data may be difficult We provide an approach to

overcome this problemThis idea is based on finding anotherrelation between 119870(119905

119863) and 119902

119863(119905119863) that is

ℓ (1) =1

119904=

1

119904119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B3)or

1 = int

119905119863

0

119870 (120591) 119902119863 (119905119863 minus 120591) 119889120591 = int

119905119863

0

119870(119905119863minus 120591) 119902

119863 (120591) 119889120591

(B4)

The other relations that can be used are

ℓ (119905119863) =

1

1199042=

1

1199042119902119863(119904)

119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 119902119863 (119905119863 minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

119863 (120591) 119889120591)

(B5)

or

119905119863= int

119905119863

0

119871 (120591) 119902119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

119863(120591) 119889120591

(B6)

ℓ (1) =1

119904=

1

1199042119902119863(119904)

119904119902119863(119904) = ℓ (int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591)

= ℓ(int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591)

(B7)

or

1 = int

119905119863

0

119871 (120591) 1199021015840

119863(119905119863minus 120591) 119889120591 = int

119905119863

0

119871 (119905119863minus 120591) 119902

1015840

119863(120591) 119889120591

(B8)

where

119871 (119905119863) = ℓminus1(

1

1199042119902 (119904)) (B9)

Equation (B4) can be used simultaneously with (B1) toanalyze the deconvolution problem

Equation (11) can be written as

119901119904119863

(119904) = 119904119901119908119863

(119904)1

1199042119902119863(119904)

(B10)

or

119901119904119863

(119905119863) = int

119905119863

0

119871 (120591) 1199011015840

119908119863(119905119863minus 120591) 119889120591

= int

119905119863

0

119871 (119905119863minus 120591) 119901

1015840

119908119863(120591) 119889120591

(B11)

The applications of (B6) and (B8) are with (B11)For justifying the rate normalizationmethod andmaterial

balance deconvolution it can use (B1) (B4) ((B11) can alsobe used with (B6) or (B8) for this goal) and generalizedmean value theorem for integrals This theorem is expressedas follows

Theorem 1 Let 119891 and 119892 be continuous on [119886 119887] and supposethat 119892 does not change sign on [119886 119887] Then there is a point 119888 in[119886 119887] such that

int

119887

119886

119891 (119905) 119892 (119905) 119889119905 = 119891 (119888) int

119887

119886

119892 (119905) 119889119905 (B12)

Abstract and Applied Analysis 11

By Theorem 1 there are some values 1198881and 1198882between 0

and 119905119863such that

119901119904119863

(119905119863) = 119901119908119863

(1198881) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B13)

1 = 119902119863(1198882) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B14)

By setting 1198881

= 1198882

= 119905119863and dividing the right and left

side of (B13) respectively the rate normalization techniqueis developed However the values 119888

1and 1198882are not equal

to each other and are not equal to 119905119863

in general Thisis the reason of error in rate normalization and materialbalance deconvolution methods 119888

1and 1198882are less than 119905

119863in

these methods for the duration of wellbore storage distortionduring a pressure transient test Since these conditions areapproximately satisfied in material balance deconvolutionthe results of material balance are more accurate and moreconfident than rate normalization

Nomenclature

API API gravity ∘API119861119900 Oil formation volume factor RBSTB

119888119905 Total isothermal compressibility factor

psiminus1

1198622 Arbitrary constant hr

minus1

119862119863 Wellbore storage constant dimensionless

119862119904 Wellbore storage coefficient bblpsi

ℎ Formation thickness ftHTR Horner time ratio dimensionless119896 Formation permeability md119870 Kernel of the convolution integral119871 Kernel of the convolution integral119873119901 Cumulative oil production bbl

119901 Pressure psi119901119886 Normalized pseudopressure psi

119901 Average reservoir pressure psi119901119863 Formation pressure dimensionless

119901119889119901 Dewpoint pressure psi

119901119894 Initial reservoir pressure psi

119901119904119863 Formation pressure including skin

dimensionless119901119904 Formation pressure including skin psi

119901119908119863

Pressure draw down dimensionless119901119908119891 Flowing bottom hole pressure psi

119901119908119904 Shut-in bottom hole pressure psi

119902119863 Sandface flow rate dimensionless

119902119892 Gas flow rate MscfD

119902119900 Oil flow rate STBD

119903119908 Wellbore radius ft

119904 Laplace variable119878 Skin factor119905 Time hr119905119863 Time dimensionless

119905119901 Production time hr

119879 Reservoir temperature ∘F

119879119889119901 Dewpoint temperature ∘F

120572 Explicit deconvolution variabledimensionless

120573 Beta deconvolution variabledimensionless

120574119892 Gas gravity (air = 10)

120582 Interporosity flow coefficient of dualporosity reservoir

120601 Porosity of reservoir rock dimensionless120583119892 Gas viscosity cp

120583119900 Oil viscosity cp

120596 Storativity ratio of dual porosity reservoirminus Laplace transform of a function1015840 Derivative of a function

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Cheng W J Lee and D A McVay ldquoFast-Fourier-transform-based deconvolution for interpretation of pressure-transient-test data dominated by Wellbore storagerdquo SPE Reservoir Eval-uation and Engineering vol 8 no 3 pp 224ndash239 2005

[2] M M Levitan and M R Wilson ldquoDeconvolution of pressureand rate data from gas reservoirs with significant pressuredepletionrdquo SPE Journal vol 17 no 3 pp 727ndash741 2012

[3] Y Liu and R N Horne ldquoInterpreting pressure and flow-ratedata from permanent downhole gauges by use of data-miningapproachesrdquo SPE Journal vol 18 no 1 pp 69ndash82 2013

[4] O Ogunrewo and A C Gringarten ldquoDeconvolution of welltest data in lean and rich gas condensate and volatile oil wellsbelow saturation pressurerdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Document ID 166340-MS New Orleans La USA September-October 2013

[5] R C Earlougher Jr Advances in Well Test Analysis vol 5 ofMonogragh Series SPE Richardson Tex USA 1977

[6] F Kucuk and L Ayestaran ldquoAnalysis of simultaneously mea-sured pressure and sandface flow rate in transient well testingrdquoJournal of PetroleumTechnology vol 37 no 2 pp 323ndash334 1985

[7] M Onur M Cinar D IIk P P Valko T A Blasingame and PHegeman ldquoAn investigation of recent deconvolution methodsfor well-test data analysisrdquo SPE Journal vol 13 no 2 pp 226ndash247 2008

[8] F J KuchukMOnur andFHollaender ldquoPressure transient testdesign and interpretationrdquo Developments in Petroleum Sciencevol 57 pp 303ndash359 2010

[9] A C Gringarten D P Bourdet P A Landel and V J KniazeffldquoA comparison between different skin and wellbore storagetype-curves for early-time transient analysisrdquo in Proceedings ofthe SPE Annual Technical Conference and Exhibition Paper SPE8205 Las Vegas Nev USA 1979

[10] A C Gringarten ldquoFrom straight lines to deconvolution theevolution of the state of the art in well test analysisrdquo SPEReservoir Evaluation and Engineering vol 11 no 1 pp 41ndash622008

[11] J A Cumming D A Wooff T Whittle and A C GringartenldquoMultiple well deconvolutionrdquo in Proceedings of the SPE Annual

12 Abstract and Applied Analysis

Technical Conference and Exhibition Document ID 166458-MS New Orleans La USA September-October 2013

[12] O Bahabanian D Ilk N Hosseinpour-Zonoozi and T ABlasingame ldquoExplicit deconvolution of well test data distortedby wellbore storagerdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Paper SPE 103216 pp 4444ndash4453San Antonio Tex USA September 2006

[13] J Jones and A Chen ldquoSuccessful applications of pressure-ratedeconvolution in theCad-Nik tight gas formations of the BritishColumbia foothillsrdquo Journal of Canadian Petroleum Technologyvol 51 no 3 pp 176ndash192 2012

[14] D G Russell ldquoExtension of pressure buildup analysis methodsrdquoJournal of Petroleum Technology vol 18 no 12 pp 1624ndash16361966

[15] R E Gladfelter G W Tracy and L E Wilsey ldquoSelecting wellswhich will respond to production-stimulation treatmentrdquo inDrilling and Production Practice API Dallas Tex USA 1955

[16] M J Fetkovich and M E Vienot ldquoRate normalization ofbuildup pressure by using afterflow datardquo Journal of PetroleumTechnology vol 36 no 13 pp 2211ndash2224 1984

[17] J L Johnston Variable rate analysis of transient well testdata using semi-analytical methods [MS thesis] Texas AampMUniversity College Station Tex USA 1992

[18] A F van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 6 pp 171ndash176 1953

[19] W Hurst ldquoEstablishment of the skin effect and its impedimentto fluid flow into a well borerdquo The Petroleum Engineer pp B6ndashB16 1953

[20] J A Joseph and L F Koederitz ldquoA simple nonlinear model forrepresentation of field transient responsesrdquo Paper SPE 11435SPE Richardson Tex USA 1982

[21] F J Kuchuk ldquoGladfelter deconvolutionrdquo SPE Formation Evalu-ation vol 5 no 3 pp 285ndash292 1990

[22] A F van Everdingen and W Hurst ldquoThe application of thelaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[23] D Bourdet T M Whittle A A Douglas and Y M Pirard ldquoAnew set of type curves simplifies well test analysisrdquo World Oilpp 95ndash106 1983

[24] D F Meunier C S Kabir and M J Wittmann ldquoGas well testanalysis use of normalized pseudovariablesrdquo SPE FormationEvaluation vol 2 no 4 pp 629ndash636 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

By Theorem 1 there are some values 1198881and 1198882between 0

and 119905119863such that

119901119904119863

(119905119863) = 119901119908119863

(1198881) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B13)

1 = 119902119863(1198882) int

119905119863

0

119870(119905119863minus 120591) 119889120591 (B14)

By setting 1198881

= 1198882

= 119905119863and dividing the right and left

side of (B13) respectively the rate normalization techniqueis developed However the values 119888

1and 1198882are not equal

to each other and are not equal to 119905119863

in general Thisis the reason of error in rate normalization and materialbalance deconvolution methods 119888

1and 1198882are less than 119905

119863in

these methods for the duration of wellbore storage distortionduring a pressure transient test Since these conditions areapproximately satisfied in material balance deconvolutionthe results of material balance are more accurate and moreconfident than rate normalization

Nomenclature

API API gravity ∘API119861119900 Oil formation volume factor RBSTB

119888119905 Total isothermal compressibility factor

psiminus1

1198622 Arbitrary constant hr

minus1

119862119863 Wellbore storage constant dimensionless

119862119904 Wellbore storage coefficient bblpsi

ℎ Formation thickness ftHTR Horner time ratio dimensionless119896 Formation permeability md119870 Kernel of the convolution integral119871 Kernel of the convolution integral119873119901 Cumulative oil production bbl

119901 Pressure psi119901119886 Normalized pseudopressure psi

119901 Average reservoir pressure psi119901119863 Formation pressure dimensionless

119901119889119901 Dewpoint pressure psi

119901119894 Initial reservoir pressure psi

119901119904119863 Formation pressure including skin

dimensionless119901119904 Formation pressure including skin psi

119901119908119863

Pressure draw down dimensionless119901119908119891 Flowing bottom hole pressure psi

119901119908119904 Shut-in bottom hole pressure psi

119902119863 Sandface flow rate dimensionless

119902119892 Gas flow rate MscfD

119902119900 Oil flow rate STBD

119903119908 Wellbore radius ft

119904 Laplace variable119878 Skin factor119905 Time hr119905119863 Time dimensionless

119905119901 Production time hr

119879 Reservoir temperature ∘F

119879119889119901 Dewpoint temperature ∘F

120572 Explicit deconvolution variabledimensionless

120573 Beta deconvolution variabledimensionless

120574119892 Gas gravity (air = 10)

120582 Interporosity flow coefficient of dualporosity reservoir

120601 Porosity of reservoir rock dimensionless120583119892 Gas viscosity cp

120583119900 Oil viscosity cp

120596 Storativity ratio of dual porosity reservoirminus Laplace transform of a function1015840 Derivative of a function

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Cheng W J Lee and D A McVay ldquoFast-Fourier-transform-based deconvolution for interpretation of pressure-transient-test data dominated by Wellbore storagerdquo SPE Reservoir Eval-uation and Engineering vol 8 no 3 pp 224ndash239 2005

[2] M M Levitan and M R Wilson ldquoDeconvolution of pressureand rate data from gas reservoirs with significant pressuredepletionrdquo SPE Journal vol 17 no 3 pp 727ndash741 2012

[3] Y Liu and R N Horne ldquoInterpreting pressure and flow-ratedata from permanent downhole gauges by use of data-miningapproachesrdquo SPE Journal vol 18 no 1 pp 69ndash82 2013

[4] O Ogunrewo and A C Gringarten ldquoDeconvolution of welltest data in lean and rich gas condensate and volatile oil wellsbelow saturation pressurerdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Document ID 166340-MS New Orleans La USA September-October 2013

[5] R C Earlougher Jr Advances in Well Test Analysis vol 5 ofMonogragh Series SPE Richardson Tex USA 1977

[6] F Kucuk and L Ayestaran ldquoAnalysis of simultaneously mea-sured pressure and sandface flow rate in transient well testingrdquoJournal of PetroleumTechnology vol 37 no 2 pp 323ndash334 1985

[7] M Onur M Cinar D IIk P P Valko T A Blasingame and PHegeman ldquoAn investigation of recent deconvolution methodsfor well-test data analysisrdquo SPE Journal vol 13 no 2 pp 226ndash247 2008

[8] F J KuchukMOnur andFHollaender ldquoPressure transient testdesign and interpretationrdquo Developments in Petroleum Sciencevol 57 pp 303ndash359 2010

[9] A C Gringarten D P Bourdet P A Landel and V J KniazeffldquoA comparison between different skin and wellbore storagetype-curves for early-time transient analysisrdquo in Proceedings ofthe SPE Annual Technical Conference and Exhibition Paper SPE8205 Las Vegas Nev USA 1979

[10] A C Gringarten ldquoFrom straight lines to deconvolution theevolution of the state of the art in well test analysisrdquo SPEReservoir Evaluation and Engineering vol 11 no 1 pp 41ndash622008

[11] J A Cumming D A Wooff T Whittle and A C GringartenldquoMultiple well deconvolutionrdquo in Proceedings of the SPE Annual

12 Abstract and Applied Analysis

Technical Conference and Exhibition Document ID 166458-MS New Orleans La USA September-October 2013

[12] O Bahabanian D Ilk N Hosseinpour-Zonoozi and T ABlasingame ldquoExplicit deconvolution of well test data distortedby wellbore storagerdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Paper SPE 103216 pp 4444ndash4453San Antonio Tex USA September 2006

[13] J Jones and A Chen ldquoSuccessful applications of pressure-ratedeconvolution in theCad-Nik tight gas formations of the BritishColumbia foothillsrdquo Journal of Canadian Petroleum Technologyvol 51 no 3 pp 176ndash192 2012

[14] D G Russell ldquoExtension of pressure buildup analysis methodsrdquoJournal of Petroleum Technology vol 18 no 12 pp 1624ndash16361966

[15] R E Gladfelter G W Tracy and L E Wilsey ldquoSelecting wellswhich will respond to production-stimulation treatmentrdquo inDrilling and Production Practice API Dallas Tex USA 1955

[16] M J Fetkovich and M E Vienot ldquoRate normalization ofbuildup pressure by using afterflow datardquo Journal of PetroleumTechnology vol 36 no 13 pp 2211ndash2224 1984

[17] J L Johnston Variable rate analysis of transient well testdata using semi-analytical methods [MS thesis] Texas AampMUniversity College Station Tex USA 1992

[18] A F van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 6 pp 171ndash176 1953

[19] W Hurst ldquoEstablishment of the skin effect and its impedimentto fluid flow into a well borerdquo The Petroleum Engineer pp B6ndashB16 1953

[20] J A Joseph and L F Koederitz ldquoA simple nonlinear model forrepresentation of field transient responsesrdquo Paper SPE 11435SPE Richardson Tex USA 1982

[21] F J Kuchuk ldquoGladfelter deconvolutionrdquo SPE Formation Evalu-ation vol 5 no 3 pp 285ndash292 1990

[22] A F van Everdingen and W Hurst ldquoThe application of thelaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[23] D Bourdet T M Whittle A A Douglas and Y M Pirard ldquoAnew set of type curves simplifies well test analysisrdquo World Oilpp 95ndash106 1983

[24] D F Meunier C S Kabir and M J Wittmann ldquoGas well testanalysis use of normalized pseudovariablesrdquo SPE FormationEvaluation vol 2 no 4 pp 629ndash636 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Abstract and Applied Analysis

Technical Conference and Exhibition Document ID 166458-MS New Orleans La USA September-October 2013

[12] O Bahabanian D Ilk N Hosseinpour-Zonoozi and T ABlasingame ldquoExplicit deconvolution of well test data distortedby wellbore storagerdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Paper SPE 103216 pp 4444ndash4453San Antonio Tex USA September 2006

[13] J Jones and A Chen ldquoSuccessful applications of pressure-ratedeconvolution in theCad-Nik tight gas formations of the BritishColumbia foothillsrdquo Journal of Canadian Petroleum Technologyvol 51 no 3 pp 176ndash192 2012

[14] D G Russell ldquoExtension of pressure buildup analysis methodsrdquoJournal of Petroleum Technology vol 18 no 12 pp 1624ndash16361966

[15] R E Gladfelter G W Tracy and L E Wilsey ldquoSelecting wellswhich will respond to production-stimulation treatmentrdquo inDrilling and Production Practice API Dallas Tex USA 1955

[16] M J Fetkovich and M E Vienot ldquoRate normalization ofbuildup pressure by using afterflow datardquo Journal of PetroleumTechnology vol 36 no 13 pp 2211ndash2224 1984

[17] J L Johnston Variable rate analysis of transient well testdata using semi-analytical methods [MS thesis] Texas AampMUniversity College Station Tex USA 1992

[18] A F van Everdingen ldquoThe skin effect and its influence on theproductive capacity of a wellrdquo Journal of Petroleum Technologyvol 5 no 6 pp 171ndash176 1953

[19] W Hurst ldquoEstablishment of the skin effect and its impedimentto fluid flow into a well borerdquo The Petroleum Engineer pp B6ndashB16 1953

[20] J A Joseph and L F Koederitz ldquoA simple nonlinear model forrepresentation of field transient responsesrdquo Paper SPE 11435SPE Richardson Tex USA 1982

[21] F J Kuchuk ldquoGladfelter deconvolutionrdquo SPE Formation Evalu-ation vol 5 no 3 pp 285ndash292 1990

[22] A F van Everdingen and W Hurst ldquoThe application of thelaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[23] D Bourdet T M Whittle A A Douglas and Y M Pirard ldquoAnew set of type curves simplifies well test analysisrdquo World Oilpp 95ndash106 1983

[24] D F Meunier C S Kabir and M J Wittmann ldquoGas well testanalysis use of normalized pseudovariablesrdquo SPE FormationEvaluation vol 2 no 4 pp 629ndash636 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of