Research ArticleCharacteristic Value Method of Well TestAnalysis for Horizontal Gas Well

Xiao-Ping Li1 Ning-Ping Yan12 and Xiao-Hua Tan1

1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum University Xindu Road 8Chengdu 610500 China

2No 1 Gas Production Plant of PetroChina Changqing Oilfield Company Yinchuan 750006 China

Correspondence should be addressed to Xiao-Hua Tan xiaohua-tan163com

Received 16 May 2014 Accepted 28 July 2014 Published 25 September 2014

Academic Editor Kim M Liew

Copyright copy 2014 Xiao-Ping Li 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 presents a study of characteristic value method of well test analysis for horizontal gas well Owing to the complicatedseepage flowmechanism in horizontal gas well and the difficulty in the analysis of transient pressure test data this paper establishesthe mathematical models of well test analysis for horizontal gas well with different inner and outer boundary conditions On thebasis of obtaining the solutions of the mathematical models several type curves are plotted with Stehfest inversion algorithm Forgas reservoir with closed outer boundary in vertical direction and infinite outer boundary in horizontal direction while consideringthe effect of wellbore storage and skin effect the pseudopressure behavior of the horizontal gas well canmanifest four characteristicperiods pure wellbore storage period early vertical radial flow period early linear flow period and late horizontal pseudoradialflow period For gas reservoir with closed outer boundary both in vertical and horizontal directions the pseudopressure behaviorof the horizontal gas well adds the pseudosteady state flow period which appears after the boundary response For gas reservoirwith closed outer boundary in vertical direction and constant pressure outer boundary in horizontal direction the pseudopressurebehavior of the horizontal gas well adds the steady state flow period which appears after the boundary response According to thecharacteristic lines which are manifested by pseudopressure derivative curve of each flow period formulas are developed to obtainhorizontal permeability vertical permeability skin factor reservoir pressure and pore volume of the gas reservoir and thus thecharacteristic value method of well test analysis for horizontal gas well is established Finally the example study verifies that thenew method is reliable Characteristic value method of well test analysis for horizontal gas well makes the well test analysis processmore simple and the results more accurate

1 Introduction

Recent years have seen the ever-growing application of hori-zontal wells technology which aroused considerable interestin the exploration of horizontal well test analysis [1ndash4] Inorder to surmount the challenges in estimating horizontalwell productivity and parameters analytical solutions forinterpreting transient pressure behavior of horizontal wellshave attracted great attention

Numerous studies on the pressure transient analysisof horizontal wells have been documented extensively inthe literature Combined with Newmanrsquos product methodGringarten and Ramey [5] found an access to solve the

unsteady-flow problems in reservoirs by means of the useof source and Greenrsquos function Clonts and Ramey [6]presented an analytical solution for interpreting the tran-sient pressure behavior of horizontal drain holes located inthe heterogeneous reservoir On the basis of finite Fouriertransforms Goode and Thambynayagam [7] addressed asolution for horizontal wells with infinite-conductivity inthe semi-infinite reservoir Ozkan and Rajagopal [8] demon-strated a derivative approach to analyze the pressure-transient behavior of horizontal wells which revealed therelationship between the dimensionless well length andthe horizontal-well pressure responses Odeh and Babu[9] indicated that four significant flow periods could be

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 472728 10 pageshttpdxdoiorg1011552014472728

2 Mathematical Problems in Engineering

observed during the process of horizontal well transientpressure behavior which was further consolidated by thebuildup and drawdown equations Thompson and Temeng[10] introduced the automatic type curve matching methodin analyzing multirate horizontal well pressure transientdata through nonlinear regression analysis techniques Raha-van et al [11] employed a mathematical model to iden-tify the features of pressure responses of a horizontal wellwith multiple fractures Equipped with Laplace transforma-tion and boundary element method Zerzar and Bettam[12] addressed an analytical model for horizontal wellswith finite conductivity vertical fractures By extrapolat-ing the transient pressure data a simplified approach topredict well production was presented by Whittle et al[13]

Owing to the imperfection of common well test anal-ysis methods including the semilog data plotting analysistechnique [14 15] type curve matching analysis method[16 17] and automatic fitting analysis method [18] it isinconvenient to apply those methods during the process ofanalyzing and determining reservoir parameters Thereforethis paper presents the characteristic value method of welltest analysis for horizontal gas well for the sake of over-coming conventional limitations This method involves twosteps The first step is to develop formulas to calculate gasreservoir fluid flow parameters according to the character-istic lines manifested by pseudopressure derivative curvesof each flowing period The next step is to utilize theseformulas to complete the well test analysis for horizontalgas well by means of combining the measured pressurewith the pseudopressure derivative curve The characteristicvalue method of well test analysis for horizontal gas wellenriches and develops the well test analysis theory andmethod

2 Mathematical Models and Solutions of WellTest Analysis for Horizontal Gas Well

The hypothesis the formation thickness is ℎ the initialformation pressure of gas reservoir is 119901

119894and equal every-

where the gas reservoir is anisotropic the horizontal per-meability is 119870

ℎ the vertical permeability is 119870V horizontal

section length is 2119871 and the position of horizontal sec-tion in the gas reservoir which is parallel to the closedtop and bottom boundary is 119911

119908 The surface flow rate

of horizontal gas well is 119902119904119888

and assumed to be constantSingle-phase compressible gas flow obeys Darcy law andthe effect of gravity and capillary pressure is ignored Thephysical model of horizontal gas well seepage is illustrated inFigure 1

Considering the complexity of the seepage flow mech-anism of horizontal gas well and in order to make themathematical modelrsquos solving and calculation more simplethe establishment of mathematical models are divided intotwo parts one is to ignore the effect of wellbore storage andskin effect the other is to consider the effect of wellborestorage and skin effect [19 20]

Re

2L

Zw

h

O

120579

r

z

M(r 120579 z)

Figure 1 Physical model of horizontal gas well seepage

21 The Mathematical Models without Considering the Effectof Wellbore Storage and Skin The diffusivity equation isexpressed by Ozkan and Raghavan [21]

1

119903119863

120597

120597119903119863

(119903119863

120597119898119863

120597119903119863

) + 1198712

119863

1205972119898119863

1205971199112119863

= (ℎ119863119871119863)2 120597119898119863

120597119905119863

(1)

Initial condition is

119898119863(119903119863 0) = 0 (2)

Inner boundary condition is

lim120576rarr0

[ lim119903119863rarr0

int

119911119908119863+1205762

119911119908119863minus1205762

119903119863

120597119898119863

120597119903119863

119889119911119908119863

]

=

0 119911119863

gt (119911119908119863

+120576

2)

minus1

2 (119911

119908119863+

120576

2) ge 119911119863

ge (119911119908119863

minus120576

2)

0 119911119863

lt (119911119908119863

minus120576

2)

(3)

where 120576 is a tiny variableInfinite outer boundary condition in horizontal direction

is

lim119903119863rarrinfin

119898119863(119903119863 119905119863) = 0 (4)

Closed outer boundary condition in horizontal directionis

120597119898119863

120597119903119863

10038161003816100381610038161003816100381610038161003816119903119863=119903119890119863

= 0 (5)

Constant pressure outer boundary condition in horizon-tal direction is

119898119863

10038161003816100381610038161003816119903119863=119903119890119863= 0 (6)

Closed outer boundary conditions in vertical directionare

120597119898119863

120597119911119863

10038161003816100381610038161003816100381610038161003816119911119863=1

= 0120597119898119863

120597119911119863

10038161003816100381610038161003816100381610038161003816119911119863=0

= 0 (7)

Mathematical Problems in Engineering 3

The dimensionless variables are defined as follows

119898119863

=78489119870

ℎℎ

119902119904119888119879

(119898119894minus 119898) 119905

119863=

36119870ℎ119905

1206011205831198881199051199032119908

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119911119863

=119911

ℎ 119911

119903119863= 119911119908119863

+ 119903119908119863

119871119863

119911119908119863

=119911119908

ℎ 119903

119863=

119903

119871

119903119890119863

=119903119890

119871 119903

119908119863=

119903119908

119871

(8)

The defined gas pseudopressure is

119898(119901) = 2int

119901

119901ref

119901

120583 (119901)119885 (119901)119889119901 (9)

22 The Mathematical Model with Considering the Effect ofWellbore Storage and Skin According to Duhamelrsquos principle[22] and the superposition principle while using the defini-tion of dimensionless variables the mathematical model ofhorizontal gas well with considering the effect of wellborestorage and skin is derived as follows

119898119908119863

= 119898119863

+ int

119905119863

0

119862119863

119889119898119908119863

119889120591119863

119889119898119908119863

(119905119863

minus 120591119863)

119889120591119863

119889120591119863

+ (1 minus 119862119863

119889119898119908119863

119889120591119863

)ℎ119863119878

(10)

where

119862119863

=119862

2120587120601119862119905ℎ1198712

(11)

23 The Solutions of the Mathematical Models The solutionsof the mathematical models [23 24] at various outer bound-ary conditions can be obtained by applying source functionand integral transform and taking the Laplace transform to 119904

with respect to 119905119863

For gas reservoir with closed outer boundary in verticaldirection and infinite outer boundary in horizontal directionaccording to (1) (2) (3) (4) and (7) the dimensionlessbottomhole pseudopressure of horizontal gas well in theLaplace space can be obtained This results in

119898119863

=1

2119904int

1

minus1

1198700(radic(119909

119863minus 120572)21205760)119889120572

+ 2

infin

sum

119899=1

int

1

minus1

1198700(radic(119909

119863minus 120572)2120576119899) cos (120573

119899119911119903119863

)

times cos (120573119899119911119908119863

) 119889120572

(12)

where

120573119899= 119899120587

120576119899= radic119904(ℎ

119863119871119863)2+ 1205731198991198712119863

(13)

For gas reservoir with closed outer boundary both invertical and horizontal direction according to (1) (2) (3)(5) and (7) the dimensionless bottomhole pseudopressure ofhorizontal gas well in the Laplace space can be obtainedThisresults in

119898119863

=1

2119904int

1

minus1

1198700(radic(119909

119863minus 120572)21205760)119889120572

+1198701(119903119890119863

1205760)

1198681(119903119890119863

1205760)

int

1

minus1

1198680(radic(119909

119863minus 120572)21205760)119889120572

+ 2

infin

sum

119899=1

[int

1

minus1

1198700(radic(119909

119863minus 120572)2120576119899)119889120572

+1198701(119903119890119863

120576119899)

1198681(119903119890119863

120576119899)

int

1

minus1

1198680(radic(119909

119863minus 120572)2120576119899)119889120572

sdot cos (120573119899119911119903119863

) cos (120573119899119911119908119863

) ]

(14)

For gas reservoir with closed outer boundary in verticaldirection and constant pressure outer boundary in horizontaldirection according to (1) (2) (3) (6) and (7) the dimen-sionless bottomhole pseudopressure of horizontal gas well inLaplace space can be obtained This results in

119898119863

=1

2119904int

1

minus1

1198700(radic(119909

119863minus 120572)21205760)119889120572

minus1198700(119903119890119863

1205760)

1198680(119903119890119863

1205760)

int

1

minus1

1198680(radic(119909

119863minus 120572)21205760)119889120572

+ 2

infin

sum

119899=1

[int

1

minus1

1198700(radic(119909

119863minus 120572)2120576119899)119889120572

minus1198700(119903119890119863

120576119899)

1198680(119903119890119863

120576119899)

int

1

minus1

1198680(radic(119909

119863minus 120572)2120576119899)119889120572

sdot cos (120573119899119911119903119863

) cos (120573119899119911119908119863

) ]

(15)

Making the Laplace transform to 119904 with respect to119905119863119862119863 (10) can be solved for the dimensionless bottomhole

pseudopressure of horizontal gas well considering the effectof wellbore storage and skin in the Laplace spaceThis resultsin

119898119908119863

=119904119898119863

+ ℎ119863119878

119904 + 1199042(119904119898119863

+ ℎ119863119878)

=1

119904 (119904 + 1 (119904119898119863

+ ℎ119863119878))

(16)

4 Mathematical Problems in Engineering

001

01

1

10

hD = 50

hD = 100

hD = 200

tDCD

1Eminus2

1Eminus1

1E0

1E1

1E2

1E3

1E4

1E5

1E6

1E7

1E8

mwDm

998400 wD

I II

III

IV

CDe2S = 104 LD = 10 ZwD = 05

Figure 2 Well test analysis type curve of horizontal well in gasreservoir with infinite outer boundary

3 Type Curves of Well TestAnalysis for Horizontal Gas Well

31 Gas Reservoir with Infinite Outer Boundary in HorizontalDirection For gas reservoir with infinite outer boundary inhorizontal direction according to (12) which is the solution ofhorizontal well seepage mathematical model while combin-ingwith (16) the type curve of well test analysis for horizontalgas well can be plotted with Stehfest inversion algorithm asshown in Figure 2

As seen from Figure 2 for gas reservoir with infiniteouter boundary in horizontal direction the pseudopressurebehavior of horizontal gas well can manifest four character-istic periods pure wellbore storage period (I) early verticalradial flow period (II) early linear flow period (III) and latehorizontal pseudoradial flow period (IV)

311 Pure Wellbore Storage Period The characteristic of purewellbore storage period of horizontal well is the same asvertical well which is manifested as a 45∘ straight linesegment on the log-log plot of 119898

119908119863 1198981015840119908119863

versus 119905119863119862119863 and

the duration of this period is affected by wellbore storage andskin effect

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=119905119863

119862119863

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

119905119863

119862119863

(17)

312 Early Vertical Radial Flow Period The early verticalradial flow period appears after the effect of wellbore storagethe characteristic of this period is manifested as a horizontalstraight line segment on the log-log plot of 119898

1015840

119908119863versus

119905119863119862119863The pseudopressure behavior of this period is affected

by formation thickness horizontal section length and the

Figure 3 The schematic diagram of early vertical radial flow

Figure 4 The schematic diagram of early linear flow

position of horizontal section in the gas reservoir The flowregime of this period is shown in Figure 3

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=1

4119871119863

[ln 225 (119905119863

119862119863

) + ln1198621198631198902119878]

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

4119871119863

(18)

313 Early Linear Flow Period The early linear flow periodappears after the early vertical radial flow period Thecharacteristic of this period is manifested as a straight linesegment with a slope of 05 on the log-log plot of1198981015840

119908119863versus

119905119863119862119863 This characteristic describes the linear flow of fluid

from formation to horizontal section The pseudopressurebehavior of this period is affected by dimensionless formationthickness ℎ

119863 dimensionless horizontal section length 119871

119863

and the position of horizontal section in the gas reservoir 119911119908119863

The flow regime of this period is shown in Figure 4

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

= 2119903119908119863

radic120587119905119863

+ 119878

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)= 119903119908119863

radic120587119905119863

(19)

314 Late Horizontal Pseudoradial Flow Period The late hor-izontal pseudoradial flow period appears after the early linearflow period The characteristic of this period is manifested

Mathematical Problems in Engineering 5

Figure 5 The schematic diagram of late horizontal pseudoradialflow

001

01

1

10

100

tDCD

1Eminus2

1Eminus1

1E0

1E1

1E2

1E3

1E4

1E5

1E6

1E7

1E8

mwDm

998400 wD

CD = 1 S = 5 hD = 30 LD = 5 ZwD = 05

reD = 10

reD = 15

reD = 20

III

IIIIV

V

Figure 6 Well test analysis type curve of horizontal well in gasreservoir with closed outer boundary

as a horizontal straight line segment with the value of 05on the log-log plot of 1198981015840

119908119863versus 119905

119863119862119863 This characteristic

describes the horizontal pseudoradial flow of fluid fromformation horizontal plane in the distance to horizontalsection The flow regime of this period is shown in Figure 5

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=1

2[ln(

119903119908119863

119905119863

119862119863

) + ln1198621198631198902119878]

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

2

(20)

32 Gas Reservoir with Closed Outer Boundary in HorizontalDirection For gas reservoir with closed outer boundary inhorizontal direction according to (14)which is the solution ofhorizontal well seepage mathematical model while combin-ingwith (16) the type curve of well test analysis for horizontalgas well can be plotted with Stehfest inversion algorithm asshown in Figure 6

As seen from Figure 6 for gas reservoir with closed outerboundary in horizontal direction the pseudopressure behav-ior of horizontal gas well can manifest five characteristic

0001

001

01

1

10

1Eminus2

1E0

1E2

1E4

1E6

1E8

1E10

tDCD

reD = 10

reD = 20

reD = 40

CD = 1 S = 10 hD = 30 LD = 10 ZwD = 05

I II

IIIIV

V

Figure 7 Well test analysis type curve of horizontal well in gasreservoir with constant pressure outer boundary

periods The previous four periods of gas reservoir withclosed outer boundary are exactly the same as gas reservoirwith infinite outer boundary but the pseudopressure behav-ior of the horizontal gas well adds the pseudosteady stateflow period (V) which appears after the boundary responseThe characteristic of the pseudosteady state flow period ismanifested as a straight line segment with a slope of 1 onthe log-log plot of 119898

119908119863 1198981015840119908119863

versus 119905119863119862119863 The greater the

distance of the outer boundary the later the appearance ofthe pseudosteady state flow period The smaller the distanceof the outer boundary the sooner the appearance of thepseudosteady state flow period

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during the pseudosteadystate flow period can be obtained This results in

119898119908119863

= 2120587(119903119908119863

119903119890119863

)

2

119905119863

+ 119878

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)= 2120587(

119903119908119863

119903119890119863

)

2

119905119863

(21)

33 Gas Reservoir with Constant Pressure Outer Boundary inHorizontal Direction For gas reservoir with constant pres-sure outer boundary in horizontal direction according to (15)which is the solution of horizontal well seepagemathematicalmodel while combining with (16) the type curve of well testanalysis for horizontal gas well can be plotted with Stehfestinversion algorithm as shown in Figure 7

As seen from Figure 7 for gas reservoir with constantpressure outer boundary in horizontal direction the pseu-dopressure behavior of horizontal gas well can manifestfive characteristic periods the previous four periods of gasreservoir with constant pressure outer boundary are exactlythe same as gas reservoir with infinite outer boundarybut the pseudopressure behavior of the horizontal gas welladds the steady state flow period (V) which appears afterthe boundary response The occurrence time of the steadystate flow period is affected by the outer boundary distance

6 Mathematical Problems in Engineering

in horizontal direction The smaller the distance of the outerboundary the sooner the appearance of the steady state flowperiod The greater the distance of the outer boundary thelater the appearance of the steady state flow period

34 Characteristic Value Method of Well Test Analysis forHorizontal GasWell The characteristic value method of welltest analysis for horizontal gas well can determine the gasreservoir fluid flowparameters according to the characteristiclines which are manifested by pseudopressure derivativecurve of each flow period on the log-log plot

341 PureWellbore Storage Period The characteristic of purewellbore storage period of horizontal well is manifested asa straight line segment with a slope of 1 on the log-log plotof 119898119908119863

1198981015840119908119863

versus 119905119863119862119863 The expression of dimensionless

bottomhole pseudopressure during this period is

119898119908119863

=119905119863

119862119863

(22)

Equation (22) can be converted to dimensional formand then according to the time and pressure data duringpure wellbore storage period the method to determine thewellbore storage coefficient can be obtained By plottingthe log-log plot of Δ119898 Δ119898

1015840 versus 119905 the wellbore storagecoefficient can be determined by the straight line segmentwith a slope of 1 on the log-log plot

The following can be obtained from the definitions ofdimensionless variables

119905119863

119862119863

=36119870ℎ1199051206011205831198881199051199032

119908

1198622120587120601119888119905ℎ1198712

=72120587119870

ℎℎ1198712

1205831199032119908

119905

119862 (23)

According to (22) (23) and the definition of dimension-less pseudopressure the wellbore storage coefficient can beobtained This results in

119862 =0288119902

119904119888119879

120583

1198712

1199032119908

119905

Δ119898 (24)

where 119905Δ119898 represents the actual value on the log-log plot ofΔ119898 Δ119898

1015840 versus 119905 during the pure wellbore storage period

342 Early Vertical Radial Flow Period

The Determination of Geometric Mean Permeability Thedimensionless bottomhole pseudopressure derivative curveis manifested as a horizontal straight line segment with thevalue of 1(4119871

119863) during the early vertical radial flow period

The expression of dimensionless bottomhole pseudopressurederivative during this period is

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

4119871119863

(25)

According to the definitions of dimensionless variablesthe dimensional form of (25) can be obtained This results in

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)er

=1

4 (119871ℎ)radic119870V119870ℎ (26)

The geometric mean permeability of gas reservoir can bedetermined by (26) This results in

radic119870ℎ119870V =

3185 times 10minus3

119902119904119888119879

119871(119905Δ1198981015840)er (27)

where (119905Δ1198981015840)er represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early vertical radial flow period

The Determination of Skin Factor and Initial Reservoir Pres-sure The expression of dimensionless bottomhole pseudo-pressure during the early vertical radial flow period is

119898119908119863

=1

4119871119863

[ln 225 (119905119863

119862119863

) + ln1198621198631198902119878] (28)

According to the definitions of dimensionless variablesand (28) the skin factor can be obtained This results in

119878 = 05 [Δ119898er

(119905Δ1198981015840)erminus ln

119870ℎ119905er

1206011205831198621199051199032119908

minus 080907] (29)

where Δ119898er and 119905er represent the pseudopressure differenceand time corresponding to the (119905Δ119898

1015840)er respectively

For pressure buildup analysis when Δ119905 rarr infin thelnΔ119905(Δ119905 + 119905

119901) rarr 0 (30) can be obtained through

the use of the definitions of dimensionless variables andpseudopressure difference during the early vertical radial flowperiod

119898119894minus 119898119908119891

119905Δ1198981015840= ln 119905119901119863

+ 080907 + 2119878 (30)

The initial reservoir pseudopressure can be determinedby (30) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)erb

(ln 119905119901119863

+ 080907 + 2119878) (31)

where (119905Δ1198981015840)erb represents the actual value on the pressure

buildup log-log plot of Δ1198981015840 versus 119905 during the early vertical

radial flow period

343 Early Linear Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 05 during theearly linear flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables thefollowing can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)119897=

119903119908

119871radic

36120587119870ℎ119905

1206011205831198621199051199032119908

(32)

The horizontal permeability of gas reservoir can bedetermined by (32) This results in

radic119870ℎ=

428 times 10minus2

119902119904119888119879

119871ℎradic120601120583119862119905

[radic119905

(119905Δ1198981015840)]

119897

(33)

Mathematical Problems in Engineering 7

where (119905Δ1198981015840)119897represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early linear flow period

Combining (27) with (33) the vertical permeability canbe obtained This results in

radic119870V = 744 times 10minus2

ℎradic120601120583119862119905

[(119905Δ1198981015840) radic119905]

119897

(119905Δ1198981015840)er (34)

344 Late Horizontal Pseudoradial Flow Period The dimen-sionless bottomhole pseudopressure derivative curve is man-ifested as a horizontal straight line segment with the valueof 05 during the late horizontal pseudoradial flow periodAccording to the expression of dimensionless bottomholepseudopressure derivative during this period and the defini-tions of dimensionless variables (35) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)lr= 05 (35)

The horizontal permeability of gas reservoir can bedetermined by (35) This results in

119870ℎ=

637 times 10minus3

119902119904119888119879

ℎ(119905Δ1198981015840)lr (36)

where (119905Δ1198981015840)lr represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the late horizontal pseudoradial flow

periodFor pressure buildup analysis when Δ119905 rarr infin the

lnΔ119905(Δ119905 + 119905119901) rarr 0 (37) can be obtained through the use

of the definitions of dimensionless variables and pseudopres-sure difference during the late horizontal pseudoradial flowperiod

119898119894minus 119898119908119891

(119905Δ1198981015840)lrb= ln 119903119908119863

119905119901119863

+ 080907 + 2119878 (37)

The initial reservoir pseudopressure can be determinedby (37) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)lrb

(ln 119903119908119863

119905119901119863

+ 080907 + 2119878) (38)

where (119905Δ1198981015840)lrb represents the actual value on the pressure

buildup log-log plot ofΔ1198981015840 versus 119905 during the late horizontal

pseudoradial flow period

345 Pseudosteady Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 1 during thepseudosteady flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables(39) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)pp

=72120587119870

ℎ119905pp

1199032119890ℎ120601120583119862

119905

(39)

The pore volume of gas reservoir can be determined by(39) This results in

1205871199032

119890ℎ120601 =

0905119902119904119888119879

ℎ120583119862119905

119905pp

(119905Δ1198981015840)pp (40)

01020304050

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

0246810

Gas flow rateWater flow rate

qsc(104m

3d)

qw(m

3d)

Figure 8 The gas flow rate and water flow rate curve of Longping 1well

5

10

15

20

25

5

10

15

20

25

Tubing head pressureCasing head pressure

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

Pwh

(MPa

)

Pch

(MPa

)

Figure 9 The tubing head pressure and casing head pressure curveof Longping 1 well

where (119905Δ1198981015840)pp represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the pseudosteady flow period

4 Example Analysis

The Longping 1 well is a horizontal development well inJingBian gas field the well total depth is 4672m the drilledformation name is Majiagou group the mid-depth of reser-voir is 342563m and the well completion system is screencompletion According to the deliverability test during 26ndash29December 2006 the calculated absolute open flow was 9426times 104m3d The commissioning data of Longping 1 well wasin 12May 2007 the initial formation pressure was 2939MPabefore production and the surface tubing pressure and casingpressure were both 2390MPa The production performancecurves of Longping 1 well are shown in Figures 8 and 9respectively

Longping 1 well has been conducted pressure builduptest during 14 August 2007 and 23 October 2007 The gasflow rate of Longping 1 well was 40 times 104m3d before theshut-in The bottomhole pressure recovered from 2238MPato 2783MPa during the pressure buildup test Physical

8 Mathematical Problems in EngineeringΔmΔ

m998400

Δt (h)10minus3

10minus4

10minus5

10minus6

10minus7

10minus2 10minus1 100 101 102 103 104

III

IIIIV

Figure 10 The pressure buildup log-log plot of Longping 1 well

100 101 102 103 104 105 106 107

(tp + Δt)Δt

m(p

wt)

50

46

42

38

34

30

Figure 11 The pressure buildup semilog plot of Longping 1 well

Table 1 Physical parameters of fluid and reservoir

Parameter ValueInitial formation pressure 119901

119894(MPa) 2939

Formation temperature 119879 (∘C) 9580Formation thickness ℎ (m) 631Porosity 120601 () 777Initial water saturation 119878wi () 1360Well radius 119903

119908(m) 00797

Gas gravity 120574119892

0608Gas deviation factor 119885 09738Gas viscosity 120583

119892(mPasdots) 00222

Table 2 Well test analysis results of Longping 1 well

Parameter Parameter valuesWellbore storage coefficient 119862 (m3MPa) 1229Horizontal permeability 119870

ℎ(mD) 7742

Vertical permeability 119870119907(mD) 0039

Flow capacity 119870ℎℎ (mDsdotm) 48857

Skin factor 119878 minus249

Effective horizontal section length 119871 (m) 19837

Reservoir pressure 119901119877(MPa) 28385

parameters of fluid and reservoir are shown in Table 1 Thepressure buildup log-log plot of Longping 1 well is shown inFigure 10

Δt (h)

m(p

ws)

50

0 300 900 1200 1500

46

42

38

34

30

Figure 12 The pressure history matching plot of Longping 1 well

As seen from the contrast between Figure 10 and well testanalysis type curves of horizontal well the pseudopressurebehavior of Longping 1 well manifests four characteristicperiods during the pressure buildup test pure wellborestorage period (I) early vertical radial flow period (II) earlylinear flow period (III) and late horizontal pseudoradial flowperiod (IV)

Using the above characteristic value method of well testanalysis for horizontal gas well well test analysis results ofLongping 1 well are shown in Table 2 The pressure buildupsemilog plot and pressure history matching plot of Longping1 well are shown in Figures 11 and 12 respectively

5 Summary and Conclusions

The four main conclusions and summary of this study are asfollows

(1) On the basis of establishing the mathematical modelsof well test analysis for horizontal gas well and obtain-ing the solutions of the mathematical models severaltype curves which can be used to identify flow regimehave been plotted and the seepage characteristic ofhorizontal gas well has been analyzed

(2) The expressions of dimensionless bottomhole pseu-dopressure and pseudopressure derivative duringeach characteristic period of horizontal gas well havebeen obtained formulas have been developed tocalculate gas reservoir fluid flow parameters

(3) The example study verifies that the characteristicvalue method of well test analysis for horizontal gaswell is reliable and practical

(4) The characteristic value method of well test analysiswhich has been included in the well test analysissoftware at present has been widely used in verticalwell As long as the characteristic straight line seg-ments which are manifested by pressure derivativecurve appear the reservoir fluid flow parameterscan be calculated by the characteristic value methodof well test analysis for vertical well The proposedcharacteristic value method of well test analysis for

Mathematical Problems in Engineering 9

horizontal gas well enriches and develops the well testanalysis theory and method

Nomenclature

119862 Wellbore storage coefficient m3MPa119862119863 Dimensionless wellbore storage coefficient

119862119905 Total compressibility MPaminus1

ℎ Reservoir thickness mℎ119863 Dimensionless reservoir thickness

119868119899 Modified Bessel function of first kind of order 119899

119870ℎ Horizontal permeability mD

119870119899 Modified Bessel function of second kind of order 119899

119870119881 Vertical permeability mD

119871 Horizontal section length m119871119863 Dimensionless horizontal section length

119898(119901) Pseudopressure MPa2mPasdots119898119863 Dimensionless pseudopressure

119898119894 Initial formation pseudopressure MPa2mPasdots

119898119908119863

Dimensionless bottomhole pseudopressure119898119908119891 Flowing wellbore pseudopressure MPa2mPasdots

1198981015840

119908119863 Derivative of119898

119908119863

119898119863 Laplace transform of 119898

119863

119898119908119863

Laplace transform of 119898119908119863

Δ119898 Pseudopressure difference MPaΔ1198981015840 Derivative of Δ119898

119901119888ℎ Casing head pressure MPa

119901119894 Initial formation pressure MPa

119901119877 Reservoir pressure MPa

119901119908ℎ Tubing head pressure MPa

119901119908119904 Flowing wellbore pressure at shut-in MPa

119902119904119888 Gas flow rate 104m3d

119902119908 Water production rate m3d

119903 Radial distance m119903119863 Dimensionless radial distance

119903119890 Outer boundary distance m

119903119890119863 Dimensionless outer boundary distance

119903119908 Wellbore radius m

119903119908119863

Dimensionless wellbore radius119878 Skin factor119904 Laplace transform variable119878119908119894 Initial water saturation fraction

119905 Time hours119905119863 Dimensionless time

119905119901 Production time hours

119905119901119863

Dimensionless production timeΔ119905 Shut-in time hours119879 Formation temperature ∘C119911 Vertical distance m119911119863 Dimensionless vertical distance

119911119908 Horizontal section position m

119911119908119863

Dimensionless horizontal section position119885 Gas deviation factor120576 Tiny variable120601 Porosity fraction120583 Gas viscosity mPasdots120574119892 Gas gravity

Subscripts

119863 Dimensionlesser Earlyerb Early of buildupℎ Horizontal119894 Initial119897 Linearlr Late horizontal pseudo-radiallrb Late horizontal pseudo-radial of builduppp Pseudosteady flow period119905 TotalV Vertical119908119891 Flowing wellbore119908119904 Shut-in wellbore

Conflict of Interests

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

References

[1] R S Nie J C Guo Y L Jia S Q Zhu Z Rao and C G ZhangldquoNew modelling of transient well test and rate decline analysisfor a horizontal well in a multiple-zone reservoirrdquo Journal ofGeophysics and Engineering vol 8 no 3 pp 464ndash476 2011

[2] D R Yuan H Sun Y C Li and M Zhang ldquoMulti-stagefractured tight gas horizontal well test data interpretationstudyrdquo in Proceedings of the International Petroleum TechnologyConference March 2013

[3] M Mavaddat A Soleimani M R Rasaei Y Mavaddat and AMomeni ldquoWell test analysis ofMultipleHydraulically FracturedHorizontal Wells (MHFHW) in gas condensate reservoirsrdquo inProceedings of the SPEIADC Middle East Drilling TechnologyConference and Exhibition (MEDT rsquo11) pp 39ndash48 October 2011

[4] MMirza P AditamaM AL Raqmi and Z Anwar ldquoHorizontalwell stimulation a pilot test in southern Omanrdquo in Proceedingsof the SPE EOR Conference at Oil and GasWest Asia April 2014

[5] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquo Society of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973

[6] M D Clonts and H J Ramey ldquoPressure transient analysis forwells with horizontal drainholesrdquo in Proceedings of the SPECalifornia Regional Meeting Oakland Calif USA April 1986paper SPE 15116

[7] P A Goode and R K M Thambynayagam ldquoPressure draw-down and buildup analysis of horizontal wells in anisotropicmediardquo SPE Formation Evaluation vol 2 no 4 pp 683ndash6971987

[8] E Ozkan and R Rajagopal ldquoHorizontal well pressure analysisrdquoSPE Formation Evaluation pp 567ndash575 1989

[9] A S Odeh and D K Babu ldquoTransient flow behavior ofhorizontal wells Pressure drawdown and buildup analysisrdquo SPEFormation Evaluation vol 5 no 1 pp 7ndash15 1990

[10] L G Thompson and K O Temeng ldquoAutomatic type-curvematching for horizontal wellsrdquo in Proceedings of the ProductionOperation Symposium Oklahoma City Okla USA 1993 paperSPE 25507

10 Mathematical Problems in Engineering

[11] R S Rahavan C C Chen and B Agarwal ldquoAn analysis ofhorizontal wells intercepted by multiple fracturesrdquo SPE27652(Sep pp 235ndash245 1997

[12] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal wells in closed systemsrdquo in Proceed-ings of the SPE International Improved Oil Recovery Conferencein Asia Pacific (IIORC rsquo03) pp 295ndash307 Alberta CanadaOctober 2003

[13] T Whittle H Jiang S Young and A C Gringarten ldquoWellproduction forecasting by extrapolation of the deconvolutionof the well test pressure transientsrdquo in Proceedings of the SPEEUROPECEAGE Conference Amsterdam The NetherlandJune 2009 SPE 122299 paper

[14] C C Miller A B Dyes and C A Hutchinson ldquoThe estimationof permeability and reservoir pressure from bottom-hole pres-sure buildup Characteristicrdquo Journal of Petroleum Technologypp 91ndash104 1950

[15] D R Horner ldquoPressure buildup in wellsrdquo in Proceedings of the3rd World Petroleum Congress Hague The Netherlands May-June 1951 paper SPE 4135

[16] R Raghavan ldquoThe effect of producing time on type curveanalysisrdquo Journal of Petroleum Technology vol 32 no 6 pp1053ndash1064 1980

[17] D Bourdet ldquoA new set of type curves simplifies well testanalysisrdquoWorld Oil pp 95ndash106 1983

[18] T M Hegre ldquoHydraulically fractured horizontal well simu-lationrdquo in Proceedings of the NPFSPE European ProductionOperations Conference (EPOC rsquo96) pp 59ndash62 April 1996

[19] R G Agarwal R Al-Hussaing and H J Ramey ldquoAn investiga-tion of wellbore storage and skin effect in unsteady liquid flowI Analytical treatmentrdquo Society of Petroleum Engineers Journalvol 10 no 3 pp 291ndash297 1970

[20] H J J Ramey and R G Agarwal ldquoAnnulus unloading ratesas influenced by wellbore storage and skin effectrdquo Society ofPetroleum Engineers Journal vol 12 no 5 pp 453ndash462 1972

[21] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems part 1-analytical considerationsrdquo SPEForma-tion Evaluation 1991

[22] L G Thompson Analysis of variable rate pressure data usingduhamels principle [PhD thesis] University of Tulsa TulsaOkla USA 1985

[23] H Stehfest ldquoNumerical inversion of laplace transformsrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

[24] N Al-Ajmi M Ahmadi E Ozkan and H Kazemi ldquoNumericalinversion of laplace transforms in the solution of transient flowproblemswith discontinuitiesrdquo inProceedings of the SPEAnnualTechnical Conference and Exhibition Paper SPE 116255 pp 21ndash24 Denver Colorado 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

observed during the process of horizontal well transientpressure behavior which was further consolidated by thebuildup and drawdown equations Thompson and Temeng[10] introduced the automatic type curve matching methodin analyzing multirate horizontal well pressure transientdata through nonlinear regression analysis techniques Raha-van et al [11] employed a mathematical model to iden-tify the features of pressure responses of a horizontal wellwith multiple fractures Equipped with Laplace transforma-tion and boundary element method Zerzar and Bettam[12] addressed an analytical model for horizontal wellswith finite conductivity vertical fractures By extrapolat-ing the transient pressure data a simplified approach topredict well production was presented by Whittle et al[13]

Owing to the imperfection of common well test anal-ysis methods including the semilog data plotting analysistechnique [14 15] type curve matching analysis method[16 17] and automatic fitting analysis method [18] it isinconvenient to apply those methods during the process ofanalyzing and determining reservoir parameters Thereforethis paper presents the characteristic value method of welltest analysis for horizontal gas well for the sake of over-coming conventional limitations This method involves twosteps The first step is to develop formulas to calculate gasreservoir fluid flow parameters according to the character-istic lines manifested by pseudopressure derivative curvesof each flowing period The next step is to utilize theseformulas to complete the well test analysis for horizontalgas well by means of combining the measured pressurewith the pseudopressure derivative curve The characteristicvalue method of well test analysis for horizontal gas wellenriches and develops the well test analysis theory andmethod

2 Mathematical Models and Solutions of WellTest Analysis for Horizontal Gas Well

The hypothesis the formation thickness is ℎ the initialformation pressure of gas reservoir is 119901

119894and equal every-

where the gas reservoir is anisotropic the horizontal per-meability is 119870

ℎ the vertical permeability is 119870V horizontal

section length is 2119871 and the position of horizontal sec-tion in the gas reservoir which is parallel to the closedtop and bottom boundary is 119911

119908 The surface flow rate

of horizontal gas well is 119902119904119888

and assumed to be constantSingle-phase compressible gas flow obeys Darcy law andthe effect of gravity and capillary pressure is ignored Thephysical model of horizontal gas well seepage is illustrated inFigure 1

Considering the complexity of the seepage flow mech-anism of horizontal gas well and in order to make themathematical modelrsquos solving and calculation more simplethe establishment of mathematical models are divided intotwo parts one is to ignore the effect of wellbore storage andskin effect the other is to consider the effect of wellborestorage and skin effect [19 20]

Re

2L

Zw

h

O

120579

r

z

M(r 120579 z)

Figure 1 Physical model of horizontal gas well seepage

21 The Mathematical Models without Considering the Effectof Wellbore Storage and Skin The diffusivity equation isexpressed by Ozkan and Raghavan [21]

1

119903119863

120597

120597119903119863

(119903119863

120597119898119863

120597119903119863

) + 1198712

119863

1205972119898119863

1205971199112119863

= (ℎ119863119871119863)2 120597119898119863

120597119905119863

(1)

Initial condition is

119898119863(119903119863 0) = 0 (2)

Inner boundary condition is

lim120576rarr0

[ lim119903119863rarr0

int

119911119908119863+1205762

119911119908119863minus1205762

119903119863

120597119898119863

120597119903119863

119889119911119908119863

]

=

0 119911119863

gt (119911119908119863

+120576

2)

minus1

2 (119911

119908119863+

120576

2) ge 119911119863

ge (119911119908119863

minus120576

2)

0 119911119863

lt (119911119908119863

minus120576

2)

(3)

where 120576 is a tiny variableInfinite outer boundary condition in horizontal direction

is

lim119903119863rarrinfin

119898119863(119903119863 119905119863) = 0 (4)

Closed outer boundary condition in horizontal directionis

120597119898119863

120597119903119863

10038161003816100381610038161003816100381610038161003816119903119863=119903119890119863

= 0 (5)

Constant pressure outer boundary condition in horizon-tal direction is

119898119863

10038161003816100381610038161003816119903119863=119903119890119863= 0 (6)

Closed outer boundary conditions in vertical directionare

120597119898119863

120597119911119863

10038161003816100381610038161003816100381610038161003816119911119863=1

= 0120597119898119863

120597119911119863

10038161003816100381610038161003816100381610038161003816119911119863=0

= 0 (7)

Mathematical Problems in Engineering 3

The dimensionless variables are defined as follows

119898119863

=78489119870

ℎℎ

119902119904119888119879

(119898119894minus 119898) 119905

119863=

36119870ℎ119905

1206011205831198881199051199032119908

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119911119863

=119911

ℎ 119911

119903119863= 119911119908119863

+ 119903119908119863

119871119863

119911119908119863

=119911119908

ℎ 119903

119863=

119903

119871

119903119890119863

=119903119890

119871 119903

119908119863=

119903119908

119871

(8)

The defined gas pseudopressure is

119898(119901) = 2int

119901

119901ref

119901

120583 (119901)119885 (119901)119889119901 (9)

22 The Mathematical Model with Considering the Effect ofWellbore Storage and Skin According to Duhamelrsquos principle[22] and the superposition principle while using the defini-tion of dimensionless variables the mathematical model ofhorizontal gas well with considering the effect of wellborestorage and skin is derived as follows

119898119908119863

= 119898119863

+ int

119905119863

0

119862119863

119889119898119908119863

119889120591119863

119889119898119908119863

(119905119863

minus 120591119863)

119889120591119863

119889120591119863

+ (1 minus 119862119863

119889119898119908119863

119889120591119863

)ℎ119863119878

(10)

where

119862119863

=119862

2120587120601119862119905ℎ1198712

(11)

23 The Solutions of the Mathematical Models The solutionsof the mathematical models [23 24] at various outer bound-ary conditions can be obtained by applying source functionand integral transform and taking the Laplace transform to 119904

with respect to 119905119863

For gas reservoir with closed outer boundary in verticaldirection and infinite outer boundary in horizontal directionaccording to (1) (2) (3) (4) and (7) the dimensionlessbottomhole pseudopressure of horizontal gas well in theLaplace space can be obtained This results in

119898119863

=1

2119904int

1

minus1

1198700(radic(119909

119863minus 120572)21205760)119889120572

+ 2

infin

sum

119899=1

int

1

minus1

1198700(radic(119909

119863minus 120572)2120576119899) cos (120573

119899119911119903119863

)

times cos (120573119899119911119908119863

) 119889120572

(12)

where

120573119899= 119899120587

120576119899= radic119904(ℎ

119863119871119863)2+ 1205731198991198712119863

(13)

For gas reservoir with closed outer boundary both invertical and horizontal direction according to (1) (2) (3)(5) and (7) the dimensionless bottomhole pseudopressure ofhorizontal gas well in the Laplace space can be obtainedThisresults in

119898119863

=1

2119904int

1

minus1

1198700(radic(119909

119863minus 120572)21205760)119889120572

+1198701(119903119890119863

1205760)

1198681(119903119890119863

1205760)

int

1

minus1

1198680(radic(119909

119863minus 120572)21205760)119889120572

+ 2

infin

sum

119899=1

[int

1

minus1

1198700(radic(119909

119863minus 120572)2120576119899)119889120572

+1198701(119903119890119863

120576119899)

1198681(119903119890119863

120576119899)

int

1

minus1

1198680(radic(119909

119863minus 120572)2120576119899)119889120572

sdot cos (120573119899119911119903119863

) cos (120573119899119911119908119863

) ]

(14)

For gas reservoir with closed outer boundary in verticaldirection and constant pressure outer boundary in horizontaldirection according to (1) (2) (3) (6) and (7) the dimen-sionless bottomhole pseudopressure of horizontal gas well inLaplace space can be obtained This results in

119898119863

=1

2119904int

1

minus1

1198700(radic(119909

119863minus 120572)21205760)119889120572

minus1198700(119903119890119863

1205760)

1198680(119903119890119863

1205760)

int

1

minus1

1198680(radic(119909

119863minus 120572)21205760)119889120572

+ 2

infin

sum

119899=1

[int

1

minus1

1198700(radic(119909

119863minus 120572)2120576119899)119889120572

minus1198700(119903119890119863

120576119899)

1198680(119903119890119863

120576119899)

int

1

minus1

1198680(radic(119909

119863minus 120572)2120576119899)119889120572

sdot cos (120573119899119911119903119863

) cos (120573119899119911119908119863

) ]

(15)

Making the Laplace transform to 119904 with respect to119905119863119862119863 (10) can be solved for the dimensionless bottomhole

pseudopressure of horizontal gas well considering the effectof wellbore storage and skin in the Laplace spaceThis resultsin

119898119908119863

=119904119898119863

+ ℎ119863119878

119904 + 1199042(119904119898119863

+ ℎ119863119878)

=1

119904 (119904 + 1 (119904119898119863

+ ℎ119863119878))

(16)

4 Mathematical Problems in Engineering

001

01

1

10

hD = 50

hD = 100

hD = 200

tDCD

1Eminus2

1Eminus1

1E0

1E1

1E2

1E3

1E4

1E5

1E6

1E7

1E8

mwDm

998400 wD

I II

III

IV

CDe2S = 104 LD = 10 ZwD = 05

Figure 2 Well test analysis type curve of horizontal well in gasreservoir with infinite outer boundary

3 Type Curves of Well TestAnalysis for Horizontal Gas Well

31 Gas Reservoir with Infinite Outer Boundary in HorizontalDirection For gas reservoir with infinite outer boundary inhorizontal direction according to (12) which is the solution ofhorizontal well seepage mathematical model while combin-ingwith (16) the type curve of well test analysis for horizontalgas well can be plotted with Stehfest inversion algorithm asshown in Figure 2

As seen from Figure 2 for gas reservoir with infiniteouter boundary in horizontal direction the pseudopressurebehavior of horizontal gas well can manifest four character-istic periods pure wellbore storage period (I) early verticalradial flow period (II) early linear flow period (III) and latehorizontal pseudoradial flow period (IV)

311 Pure Wellbore Storage Period The characteristic of purewellbore storage period of horizontal well is the same asvertical well which is manifested as a 45∘ straight linesegment on the log-log plot of 119898

119908119863 1198981015840119908119863

versus 119905119863119862119863 and

the duration of this period is affected by wellbore storage andskin effect

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=119905119863

119862119863

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

119905119863

119862119863

(17)

312 Early Vertical Radial Flow Period The early verticalradial flow period appears after the effect of wellbore storagethe characteristic of this period is manifested as a horizontalstraight line segment on the log-log plot of 119898

1015840

119908119863versus

119905119863119862119863The pseudopressure behavior of this period is affected

by formation thickness horizontal section length and the

Figure 3 The schematic diagram of early vertical radial flow

Figure 4 The schematic diagram of early linear flow

position of horizontal section in the gas reservoir The flowregime of this period is shown in Figure 3

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=1

4119871119863

[ln 225 (119905119863

119862119863

) + ln1198621198631198902119878]

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

4119871119863

(18)

313 Early Linear Flow Period The early linear flow periodappears after the early vertical radial flow period Thecharacteristic of this period is manifested as a straight linesegment with a slope of 05 on the log-log plot of1198981015840

119908119863versus

119905119863119862119863 This characteristic describes the linear flow of fluid

from formation to horizontal section The pseudopressurebehavior of this period is affected by dimensionless formationthickness ℎ

119863 dimensionless horizontal section length 119871

119863

and the position of horizontal section in the gas reservoir 119911119908119863

The flow regime of this period is shown in Figure 4

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

= 2119903119908119863

radic120587119905119863

+ 119878

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)= 119903119908119863

radic120587119905119863

(19)

314 Late Horizontal Pseudoradial Flow Period The late hor-izontal pseudoradial flow period appears after the early linearflow period The characteristic of this period is manifested

Mathematical Problems in Engineering 5

Figure 5 The schematic diagram of late horizontal pseudoradialflow

001

01

1

10

100

tDCD

1Eminus2

1Eminus1

1E0

1E1

1E2

1E3

1E4

1E5

1E6

1E7

1E8

mwDm

998400 wD

CD = 1 S = 5 hD = 30 LD = 5 ZwD = 05

reD = 10

reD = 15

reD = 20

III

IIIIV

V

Figure 6 Well test analysis type curve of horizontal well in gasreservoir with closed outer boundary

as a horizontal straight line segment with the value of 05on the log-log plot of 1198981015840

119908119863versus 119905

119863119862119863 This characteristic

describes the horizontal pseudoradial flow of fluid fromformation horizontal plane in the distance to horizontalsection The flow regime of this period is shown in Figure 5

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=1

2[ln(

119903119908119863

119905119863

119862119863

) + ln1198621198631198902119878]

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

2

(20)

32 Gas Reservoir with Closed Outer Boundary in HorizontalDirection For gas reservoir with closed outer boundary inhorizontal direction according to (14)which is the solution ofhorizontal well seepage mathematical model while combin-ingwith (16) the type curve of well test analysis for horizontalgas well can be plotted with Stehfest inversion algorithm asshown in Figure 6

As seen from Figure 6 for gas reservoir with closed outerboundary in horizontal direction the pseudopressure behav-ior of horizontal gas well can manifest five characteristic

0001

001

01

1

10

1Eminus2

1E0

1E2

1E4

1E6

1E8

1E10

tDCD

reD = 10

reD = 20

reD = 40

CD = 1 S = 10 hD = 30 LD = 10 ZwD = 05

I II

IIIIV

V

Figure 7 Well test analysis type curve of horizontal well in gasreservoir with constant pressure outer boundary

periods The previous four periods of gas reservoir withclosed outer boundary are exactly the same as gas reservoirwith infinite outer boundary but the pseudopressure behav-ior of the horizontal gas well adds the pseudosteady stateflow period (V) which appears after the boundary responseThe characteristic of the pseudosteady state flow period ismanifested as a straight line segment with a slope of 1 onthe log-log plot of 119898

119908119863 1198981015840119908119863

versus 119905119863119862119863 The greater the

distance of the outer boundary the later the appearance ofthe pseudosteady state flow period The smaller the distanceof the outer boundary the sooner the appearance of thepseudosteady state flow period

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during the pseudosteadystate flow period can be obtained This results in

119898119908119863

= 2120587(119903119908119863

119903119890119863

)

2

119905119863

+ 119878

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)= 2120587(

119903119908119863

119903119890119863

)

2

119905119863

(21)

33 Gas Reservoir with Constant Pressure Outer Boundary inHorizontal Direction For gas reservoir with constant pres-sure outer boundary in horizontal direction according to (15)which is the solution of horizontal well seepagemathematicalmodel while combining with (16) the type curve of well testanalysis for horizontal gas well can be plotted with Stehfestinversion algorithm as shown in Figure 7

As seen from Figure 7 for gas reservoir with constantpressure outer boundary in horizontal direction the pseu-dopressure behavior of horizontal gas well can manifestfive characteristic periods the previous four periods of gasreservoir with constant pressure outer boundary are exactlythe same as gas reservoir with infinite outer boundarybut the pseudopressure behavior of the horizontal gas welladds the steady state flow period (V) which appears afterthe boundary response The occurrence time of the steadystate flow period is affected by the outer boundary distance

6 Mathematical Problems in Engineering

in horizontal direction The smaller the distance of the outerboundary the sooner the appearance of the steady state flowperiod The greater the distance of the outer boundary thelater the appearance of the steady state flow period

34 Characteristic Value Method of Well Test Analysis forHorizontal GasWell The characteristic value method of welltest analysis for horizontal gas well can determine the gasreservoir fluid flowparameters according to the characteristiclines which are manifested by pseudopressure derivativecurve of each flow period on the log-log plot

341 PureWellbore Storage Period The characteristic of purewellbore storage period of horizontal well is manifested asa straight line segment with a slope of 1 on the log-log plotof 119898119908119863

1198981015840119908119863

versus 119905119863119862119863 The expression of dimensionless

bottomhole pseudopressure during this period is

119898119908119863

=119905119863

119862119863

(22)

Equation (22) can be converted to dimensional formand then according to the time and pressure data duringpure wellbore storage period the method to determine thewellbore storage coefficient can be obtained By plottingthe log-log plot of Δ119898 Δ119898

1015840 versus 119905 the wellbore storagecoefficient can be determined by the straight line segmentwith a slope of 1 on the log-log plot

The following can be obtained from the definitions ofdimensionless variables

119905119863

119862119863

=36119870ℎ1199051206011205831198881199051199032

119908

1198622120587120601119888119905ℎ1198712

=72120587119870

ℎℎ1198712

1205831199032119908

119905

119862 (23)

According to (22) (23) and the definition of dimension-less pseudopressure the wellbore storage coefficient can beobtained This results in

119862 =0288119902

119904119888119879

120583

1198712

1199032119908

119905

Δ119898 (24)

where 119905Δ119898 represents the actual value on the log-log plot ofΔ119898 Δ119898

1015840 versus 119905 during the pure wellbore storage period

342 Early Vertical Radial Flow Period

The Determination of Geometric Mean Permeability Thedimensionless bottomhole pseudopressure derivative curveis manifested as a horizontal straight line segment with thevalue of 1(4119871

119863) during the early vertical radial flow period

The expression of dimensionless bottomhole pseudopressurederivative during this period is

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

4119871119863

(25)

According to the definitions of dimensionless variablesthe dimensional form of (25) can be obtained This results in

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)er

=1

4 (119871ℎ)radic119870V119870ℎ (26)

The geometric mean permeability of gas reservoir can bedetermined by (26) This results in

radic119870ℎ119870V =

3185 times 10minus3

119902119904119888119879

119871(119905Δ1198981015840)er (27)

where (119905Δ1198981015840)er represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early vertical radial flow period

The Determination of Skin Factor and Initial Reservoir Pres-sure The expression of dimensionless bottomhole pseudo-pressure during the early vertical radial flow period is

119898119908119863

=1

4119871119863

[ln 225 (119905119863

119862119863

) + ln1198621198631198902119878] (28)

According to the definitions of dimensionless variablesand (28) the skin factor can be obtained This results in

119878 = 05 [Δ119898er

(119905Δ1198981015840)erminus ln

119870ℎ119905er

1206011205831198621199051199032119908

minus 080907] (29)

where Δ119898er and 119905er represent the pseudopressure differenceand time corresponding to the (119905Δ119898

1015840)er respectively

For pressure buildup analysis when Δ119905 rarr infin thelnΔ119905(Δ119905 + 119905

119901) rarr 0 (30) can be obtained through

the use of the definitions of dimensionless variables andpseudopressure difference during the early vertical radial flowperiod

119898119894minus 119898119908119891

119905Δ1198981015840= ln 119905119901119863

+ 080907 + 2119878 (30)

The initial reservoir pseudopressure can be determinedby (30) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)erb

(ln 119905119901119863

+ 080907 + 2119878) (31)

where (119905Δ1198981015840)erb represents the actual value on the pressure

buildup log-log plot of Δ1198981015840 versus 119905 during the early vertical

radial flow period

343 Early Linear Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 05 during theearly linear flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables thefollowing can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)119897=

119903119908

119871radic

36120587119870ℎ119905

1206011205831198621199051199032119908

(32)

The horizontal permeability of gas reservoir can bedetermined by (32) This results in

radic119870ℎ=

428 times 10minus2

119902119904119888119879

119871ℎradic120601120583119862119905

[radic119905

(119905Δ1198981015840)]

119897

(33)

Mathematical Problems in Engineering 7

where (119905Δ1198981015840)119897represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early linear flow period

Combining (27) with (33) the vertical permeability canbe obtained This results in

radic119870V = 744 times 10minus2

ℎradic120601120583119862119905

[(119905Δ1198981015840) radic119905]

119897

(119905Δ1198981015840)er (34)

344 Late Horizontal Pseudoradial Flow Period The dimen-sionless bottomhole pseudopressure derivative curve is man-ifested as a horizontal straight line segment with the valueof 05 during the late horizontal pseudoradial flow periodAccording to the expression of dimensionless bottomholepseudopressure derivative during this period and the defini-tions of dimensionless variables (35) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)lr= 05 (35)

The horizontal permeability of gas reservoir can bedetermined by (35) This results in

119870ℎ=

637 times 10minus3

119902119904119888119879

ℎ(119905Δ1198981015840)lr (36)

where (119905Δ1198981015840)lr represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the late horizontal pseudoradial flow

periodFor pressure buildup analysis when Δ119905 rarr infin the

lnΔ119905(Δ119905 + 119905119901) rarr 0 (37) can be obtained through the use

of the definitions of dimensionless variables and pseudopres-sure difference during the late horizontal pseudoradial flowperiod

119898119894minus 119898119908119891

(119905Δ1198981015840)lrb= ln 119903119908119863

119905119901119863

+ 080907 + 2119878 (37)

The initial reservoir pseudopressure can be determinedby (37) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)lrb

(ln 119903119908119863

119905119901119863

+ 080907 + 2119878) (38)

where (119905Δ1198981015840)lrb represents the actual value on the pressure

buildup log-log plot ofΔ1198981015840 versus 119905 during the late horizontal

pseudoradial flow period

345 Pseudosteady Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 1 during thepseudosteady flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables(39) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)pp

=72120587119870

ℎ119905pp

1199032119890ℎ120601120583119862

119905

(39)

The pore volume of gas reservoir can be determined by(39) This results in

1205871199032

119890ℎ120601 =

0905119902119904119888119879

ℎ120583119862119905

119905pp

(119905Δ1198981015840)pp (40)

01020304050

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

0246810

Gas flow rateWater flow rate

qsc(104m

3d)

qw(m

3d)

Figure 8 The gas flow rate and water flow rate curve of Longping 1well

5

10

15

20

25

5

10

15

20

25

Tubing head pressureCasing head pressure

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

Pwh

(MPa

)

Pch

(MPa

)

Figure 9 The tubing head pressure and casing head pressure curveof Longping 1 well

where (119905Δ1198981015840)pp represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the pseudosteady flow period

4 Example Analysis

The Longping 1 well is a horizontal development well inJingBian gas field the well total depth is 4672m the drilledformation name is Majiagou group the mid-depth of reser-voir is 342563m and the well completion system is screencompletion According to the deliverability test during 26ndash29December 2006 the calculated absolute open flow was 9426times 104m3d The commissioning data of Longping 1 well wasin 12May 2007 the initial formation pressure was 2939MPabefore production and the surface tubing pressure and casingpressure were both 2390MPa The production performancecurves of Longping 1 well are shown in Figures 8 and 9respectively

Longping 1 well has been conducted pressure builduptest during 14 August 2007 and 23 October 2007 The gasflow rate of Longping 1 well was 40 times 104m3d before theshut-in The bottomhole pressure recovered from 2238MPato 2783MPa during the pressure buildup test Physical

8 Mathematical Problems in EngineeringΔmΔ

m998400

Δt (h)10minus3

10minus4

10minus5

10minus6

10minus7

10minus2 10minus1 100 101 102 103 104

III

IIIIV

Figure 10 The pressure buildup log-log plot of Longping 1 well

100 101 102 103 104 105 106 107

(tp + Δt)Δt

m(p

wt)

50

46

42

38

34

30

Figure 11 The pressure buildup semilog plot of Longping 1 well

Table 1 Physical parameters of fluid and reservoir

Parameter ValueInitial formation pressure 119901

119894(MPa) 2939

Formation temperature 119879 (∘C) 9580Formation thickness ℎ (m) 631Porosity 120601 () 777Initial water saturation 119878wi () 1360Well radius 119903

119908(m) 00797

Gas gravity 120574119892

0608Gas deviation factor 119885 09738Gas viscosity 120583

119892(mPasdots) 00222

Table 2 Well test analysis results of Longping 1 well

Parameter Parameter valuesWellbore storage coefficient 119862 (m3MPa) 1229Horizontal permeability 119870

ℎ(mD) 7742

Vertical permeability 119870119907(mD) 0039

Flow capacity 119870ℎℎ (mDsdotm) 48857

Skin factor 119878 minus249

Effective horizontal section length 119871 (m) 19837

Reservoir pressure 119901119877(MPa) 28385

parameters of fluid and reservoir are shown in Table 1 Thepressure buildup log-log plot of Longping 1 well is shown inFigure 10

Δt (h)

m(p

ws)

50

0 300 900 1200 1500

46

42

38

34

30

Figure 12 The pressure history matching plot of Longping 1 well

As seen from the contrast between Figure 10 and well testanalysis type curves of horizontal well the pseudopressurebehavior of Longping 1 well manifests four characteristicperiods during the pressure buildup test pure wellborestorage period (I) early vertical radial flow period (II) earlylinear flow period (III) and late horizontal pseudoradial flowperiod (IV)

Using the above characteristic value method of well testanalysis for horizontal gas well well test analysis results ofLongping 1 well are shown in Table 2 The pressure buildupsemilog plot and pressure history matching plot of Longping1 well are shown in Figures 11 and 12 respectively

5 Summary and Conclusions

The four main conclusions and summary of this study are asfollows

(1) On the basis of establishing the mathematical modelsof well test analysis for horizontal gas well and obtain-ing the solutions of the mathematical models severaltype curves which can be used to identify flow regimehave been plotted and the seepage characteristic ofhorizontal gas well has been analyzed

(2) The expressions of dimensionless bottomhole pseu-dopressure and pseudopressure derivative duringeach characteristic period of horizontal gas well havebeen obtained formulas have been developed tocalculate gas reservoir fluid flow parameters

(3) The example study verifies that the characteristicvalue method of well test analysis for horizontal gaswell is reliable and practical

(4) The characteristic value method of well test analysiswhich has been included in the well test analysissoftware at present has been widely used in verticalwell As long as the characteristic straight line seg-ments which are manifested by pressure derivativecurve appear the reservoir fluid flow parameterscan be calculated by the characteristic value methodof well test analysis for vertical well The proposedcharacteristic value method of well test analysis for

Mathematical Problems in Engineering 9

horizontal gas well enriches and develops the well testanalysis theory and method

Nomenclature

119862 Wellbore storage coefficient m3MPa119862119863 Dimensionless wellbore storage coefficient

119862119905 Total compressibility MPaminus1

ℎ Reservoir thickness mℎ119863 Dimensionless reservoir thickness

119868119899 Modified Bessel function of first kind of order 119899

119870ℎ Horizontal permeability mD

119870119899 Modified Bessel function of second kind of order 119899

119870119881 Vertical permeability mD

119871 Horizontal section length m119871119863 Dimensionless horizontal section length

119898(119901) Pseudopressure MPa2mPasdots119898119863 Dimensionless pseudopressure

119898119894 Initial formation pseudopressure MPa2mPasdots

119898119908119863

Dimensionless bottomhole pseudopressure119898119908119891 Flowing wellbore pseudopressure MPa2mPasdots

1198981015840

119908119863 Derivative of119898

119908119863

119898119863 Laplace transform of 119898

119863

119898119908119863

Laplace transform of 119898119908119863

Δ119898 Pseudopressure difference MPaΔ1198981015840 Derivative of Δ119898

119901119888ℎ Casing head pressure MPa

119901119894 Initial formation pressure MPa

119901119877 Reservoir pressure MPa

119901119908ℎ Tubing head pressure MPa

119901119908119904 Flowing wellbore pressure at shut-in MPa

119902119904119888 Gas flow rate 104m3d

119902119908 Water production rate m3d

119903 Radial distance m119903119863 Dimensionless radial distance

119903119890 Outer boundary distance m

119903119890119863 Dimensionless outer boundary distance

119903119908 Wellbore radius m

119903119908119863

Dimensionless wellbore radius119878 Skin factor119904 Laplace transform variable119878119908119894 Initial water saturation fraction

119905 Time hours119905119863 Dimensionless time

119905119901 Production time hours

119905119901119863

Dimensionless production timeΔ119905 Shut-in time hours119879 Formation temperature ∘C119911 Vertical distance m119911119863 Dimensionless vertical distance

119911119908 Horizontal section position m

119911119908119863

Dimensionless horizontal section position119885 Gas deviation factor120576 Tiny variable120601 Porosity fraction120583 Gas viscosity mPasdots120574119892 Gas gravity

Subscripts

119863 Dimensionlesser Earlyerb Early of buildupℎ Horizontal119894 Initial119897 Linearlr Late horizontal pseudo-radiallrb Late horizontal pseudo-radial of builduppp Pseudosteady flow period119905 TotalV Vertical119908119891 Flowing wellbore119908119904 Shut-in wellbore

Conflict of Interests

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

References

[1] R S Nie J C Guo Y L Jia S Q Zhu Z Rao and C G ZhangldquoNew modelling of transient well test and rate decline analysisfor a horizontal well in a multiple-zone reservoirrdquo Journal ofGeophysics and Engineering vol 8 no 3 pp 464ndash476 2011

[2] D R Yuan H Sun Y C Li and M Zhang ldquoMulti-stagefractured tight gas horizontal well test data interpretationstudyrdquo in Proceedings of the International Petroleum TechnologyConference March 2013

[3] M Mavaddat A Soleimani M R Rasaei Y Mavaddat and AMomeni ldquoWell test analysis ofMultipleHydraulically FracturedHorizontal Wells (MHFHW) in gas condensate reservoirsrdquo inProceedings of the SPEIADC Middle East Drilling TechnologyConference and Exhibition (MEDT rsquo11) pp 39ndash48 October 2011

[4] MMirza P AditamaM AL Raqmi and Z Anwar ldquoHorizontalwell stimulation a pilot test in southern Omanrdquo in Proceedingsof the SPE EOR Conference at Oil and GasWest Asia April 2014

[5] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquo Society of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973

[6] M D Clonts and H J Ramey ldquoPressure transient analysis forwells with horizontal drainholesrdquo in Proceedings of the SPECalifornia Regional Meeting Oakland Calif USA April 1986paper SPE 15116

[7] P A Goode and R K M Thambynayagam ldquoPressure draw-down and buildup analysis of horizontal wells in anisotropicmediardquo SPE Formation Evaluation vol 2 no 4 pp 683ndash6971987

[8] E Ozkan and R Rajagopal ldquoHorizontal well pressure analysisrdquoSPE Formation Evaluation pp 567ndash575 1989

[9] A S Odeh and D K Babu ldquoTransient flow behavior ofhorizontal wells Pressure drawdown and buildup analysisrdquo SPEFormation Evaluation vol 5 no 1 pp 7ndash15 1990

[10] L G Thompson and K O Temeng ldquoAutomatic type-curvematching for horizontal wellsrdquo in Proceedings of the ProductionOperation Symposium Oklahoma City Okla USA 1993 paperSPE 25507

10 Mathematical Problems in Engineering

[11] R S Rahavan C C Chen and B Agarwal ldquoAn analysis ofhorizontal wells intercepted by multiple fracturesrdquo SPE27652(Sep pp 235ndash245 1997

[12] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal wells in closed systemsrdquo in Proceed-ings of the SPE International Improved Oil Recovery Conferencein Asia Pacific (IIORC rsquo03) pp 295ndash307 Alberta CanadaOctober 2003

[13] T Whittle H Jiang S Young and A C Gringarten ldquoWellproduction forecasting by extrapolation of the deconvolutionof the well test pressure transientsrdquo in Proceedings of the SPEEUROPECEAGE Conference Amsterdam The NetherlandJune 2009 SPE 122299 paper

[14] C C Miller A B Dyes and C A Hutchinson ldquoThe estimationof permeability and reservoir pressure from bottom-hole pres-sure buildup Characteristicrdquo Journal of Petroleum Technologypp 91ndash104 1950

[15] D R Horner ldquoPressure buildup in wellsrdquo in Proceedings of the3rd World Petroleum Congress Hague The Netherlands May-June 1951 paper SPE 4135

[16] R Raghavan ldquoThe effect of producing time on type curveanalysisrdquo Journal of Petroleum Technology vol 32 no 6 pp1053ndash1064 1980

[17] D Bourdet ldquoA new set of type curves simplifies well testanalysisrdquoWorld Oil pp 95ndash106 1983

[18] T M Hegre ldquoHydraulically fractured horizontal well simu-lationrdquo in Proceedings of the NPFSPE European ProductionOperations Conference (EPOC rsquo96) pp 59ndash62 April 1996

[19] R G Agarwal R Al-Hussaing and H J Ramey ldquoAn investiga-tion of wellbore storage and skin effect in unsteady liquid flowI Analytical treatmentrdquo Society of Petroleum Engineers Journalvol 10 no 3 pp 291ndash297 1970

[20] H J J Ramey and R G Agarwal ldquoAnnulus unloading ratesas influenced by wellbore storage and skin effectrdquo Society ofPetroleum Engineers Journal vol 12 no 5 pp 453ndash462 1972

[21] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems part 1-analytical considerationsrdquo SPEForma-tion Evaluation 1991

[22] L G Thompson Analysis of variable rate pressure data usingduhamels principle [PhD thesis] University of Tulsa TulsaOkla USA 1985

[23] H Stehfest ldquoNumerical inversion of laplace transformsrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

[24] N Al-Ajmi M Ahmadi E Ozkan and H Kazemi ldquoNumericalinversion of laplace transforms in the solution of transient flowproblemswith discontinuitiesrdquo inProceedings of the SPEAnnualTechnical Conference and Exhibition Paper SPE 116255 pp 21ndash24 Denver Colorado 2008

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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

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Mathematical Problems in Engineering 3

The dimensionless variables are defined as follows

119898119863

=78489119870

ℎℎ

119902119904119888119879

(119898119894minus 119898) 119905

119863=

36119870ℎ119905

1206011205831198881199051199032119908

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119911119863

=119911

ℎ 119911

119903119863= 119911119908119863

+ 119903119908119863

119871119863

119911119908119863

=119911119908

ℎ 119903

119863=

119903

119871

119903119890119863

=119903119890

119871 119903

119908119863=

119903119908

119871

(8)

The defined gas pseudopressure is

119898(119901) = 2int

119901

119901ref

119901

120583 (119901)119885 (119901)119889119901 (9)

22 The Mathematical Model with Considering the Effect ofWellbore Storage and Skin According to Duhamelrsquos principle[22] and the superposition principle while using the defini-tion of dimensionless variables the mathematical model ofhorizontal gas well with considering the effect of wellborestorage and skin is derived as follows

119898119908119863

= 119898119863

+ int

119905119863

0

119862119863

119889119898119908119863

119889120591119863

119889119898119908119863

(119905119863

minus 120591119863)

119889120591119863

119889120591119863

+ (1 minus 119862119863

119889119898119908119863

119889120591119863

)ℎ119863119878

(10)

where

119862119863

=119862

2120587120601119862119905ℎ1198712

(11)

23 The Solutions of the Mathematical Models The solutionsof the mathematical models [23 24] at various outer bound-ary conditions can be obtained by applying source functionand integral transform and taking the Laplace transform to 119904

with respect to 119905119863

For gas reservoir with closed outer boundary in verticaldirection and infinite outer boundary in horizontal directionaccording to (1) (2) (3) (4) and (7) the dimensionlessbottomhole pseudopressure of horizontal gas well in theLaplace space can be obtained This results in

119898119863

=1

2119904int

1

minus1

1198700(radic(119909

119863minus 120572)21205760)119889120572

+ 2

infin

sum

119899=1

int

1

minus1

1198700(radic(119909

119863minus 120572)2120576119899) cos (120573

119899119911119903119863

)

times cos (120573119899119911119908119863

) 119889120572

(12)

where

120573119899= 119899120587

120576119899= radic119904(ℎ

119863119871119863)2+ 1205731198991198712119863

(13)

For gas reservoir with closed outer boundary both invertical and horizontal direction according to (1) (2) (3)(5) and (7) the dimensionless bottomhole pseudopressure ofhorizontal gas well in the Laplace space can be obtainedThisresults in

119898119863

=1

2119904int

1

minus1

1198700(radic(119909

119863minus 120572)21205760)119889120572

+1198701(119903119890119863

1205760)

1198681(119903119890119863

1205760)

int

1

minus1

1198680(radic(119909

119863minus 120572)21205760)119889120572

+ 2

infin

sum

119899=1

[int

1

minus1

1198700(radic(119909

119863minus 120572)2120576119899)119889120572

+1198701(119903119890119863

120576119899)

1198681(119903119890119863

120576119899)

int

1

minus1

1198680(radic(119909

119863minus 120572)2120576119899)119889120572

sdot cos (120573119899119911119903119863

) cos (120573119899119911119908119863

) ]

(14)

For gas reservoir with closed outer boundary in verticaldirection and constant pressure outer boundary in horizontaldirection according to (1) (2) (3) (6) and (7) the dimen-sionless bottomhole pseudopressure of horizontal gas well inLaplace space can be obtained This results in

119898119863

=1

2119904int

1

minus1

1198700(radic(119909

119863minus 120572)21205760)119889120572

minus1198700(119903119890119863

1205760)

1198680(119903119890119863

1205760)

int

1

minus1

1198680(radic(119909

119863minus 120572)21205760)119889120572

+ 2

infin

sum

119899=1

[int

1

minus1

1198700(radic(119909

119863minus 120572)2120576119899)119889120572

minus1198700(119903119890119863

120576119899)

1198680(119903119890119863

120576119899)

int

1

minus1

1198680(radic(119909

119863minus 120572)2120576119899)119889120572

sdot cos (120573119899119911119903119863

) cos (120573119899119911119908119863

) ]

(15)

Making the Laplace transform to 119904 with respect to119905119863119862119863 (10) can be solved for the dimensionless bottomhole

pseudopressure of horizontal gas well considering the effectof wellbore storage and skin in the Laplace spaceThis resultsin

119898119908119863

=119904119898119863

+ ℎ119863119878

119904 + 1199042(119904119898119863

+ ℎ119863119878)

=1

119904 (119904 + 1 (119904119898119863

+ ℎ119863119878))

(16)

4 Mathematical Problems in Engineering

001

01

1

10

hD = 50

hD = 100

hD = 200

tDCD

1Eminus2

1Eminus1

1E0

1E1

1E2

1E3

1E4

1E5

1E6

1E7

1E8

mwDm

998400 wD

I II

III

IV

CDe2S = 104 LD = 10 ZwD = 05

Figure 2 Well test analysis type curve of horizontal well in gasreservoir with infinite outer boundary

3 Type Curves of Well TestAnalysis for Horizontal Gas Well

31 Gas Reservoir with Infinite Outer Boundary in HorizontalDirection For gas reservoir with infinite outer boundary inhorizontal direction according to (12) which is the solution ofhorizontal well seepage mathematical model while combin-ingwith (16) the type curve of well test analysis for horizontalgas well can be plotted with Stehfest inversion algorithm asshown in Figure 2

As seen from Figure 2 for gas reservoir with infiniteouter boundary in horizontal direction the pseudopressurebehavior of horizontal gas well can manifest four character-istic periods pure wellbore storage period (I) early verticalradial flow period (II) early linear flow period (III) and latehorizontal pseudoradial flow period (IV)

311 Pure Wellbore Storage Period The characteristic of purewellbore storage period of horizontal well is the same asvertical well which is manifested as a 45∘ straight linesegment on the log-log plot of 119898

119908119863 1198981015840119908119863

versus 119905119863119862119863 and

the duration of this period is affected by wellbore storage andskin effect

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=119905119863

119862119863

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

119905119863

119862119863

(17)

312 Early Vertical Radial Flow Period The early verticalradial flow period appears after the effect of wellbore storagethe characteristic of this period is manifested as a horizontalstraight line segment on the log-log plot of 119898

1015840

119908119863versus

119905119863119862119863The pseudopressure behavior of this period is affected

by formation thickness horizontal section length and the

Figure 3 The schematic diagram of early vertical radial flow

Figure 4 The schematic diagram of early linear flow

position of horizontal section in the gas reservoir The flowregime of this period is shown in Figure 3

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=1

4119871119863

[ln 225 (119905119863

119862119863

) + ln1198621198631198902119878]

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

4119871119863

(18)

313 Early Linear Flow Period The early linear flow periodappears after the early vertical radial flow period Thecharacteristic of this period is manifested as a straight linesegment with a slope of 05 on the log-log plot of1198981015840

119908119863versus

119905119863119862119863 This characteristic describes the linear flow of fluid

from formation to horizontal section The pseudopressurebehavior of this period is affected by dimensionless formationthickness ℎ

119863 dimensionless horizontal section length 119871

119863

and the position of horizontal section in the gas reservoir 119911119908119863

The flow regime of this period is shown in Figure 4

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

= 2119903119908119863

radic120587119905119863

+ 119878

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)= 119903119908119863

radic120587119905119863

(19)

314 Late Horizontal Pseudoradial Flow Period The late hor-izontal pseudoradial flow period appears after the early linearflow period The characteristic of this period is manifested

Mathematical Problems in Engineering 5

Figure 5 The schematic diagram of late horizontal pseudoradialflow

001

01

1

10

100

tDCD

1Eminus2

1Eminus1

1E0

1E1

1E2

1E3

1E4

1E5

1E6

1E7

1E8

mwDm

998400 wD

CD = 1 S = 5 hD = 30 LD = 5 ZwD = 05

reD = 10

reD = 15

reD = 20

III

IIIIV

V

Figure 6 Well test analysis type curve of horizontal well in gasreservoir with closed outer boundary

as a horizontal straight line segment with the value of 05on the log-log plot of 1198981015840

119908119863versus 119905

119863119862119863 This characteristic

describes the horizontal pseudoradial flow of fluid fromformation horizontal plane in the distance to horizontalsection The flow regime of this period is shown in Figure 5

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=1

2[ln(

119903119908119863

119905119863

119862119863

) + ln1198621198631198902119878]

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

2

(20)

32 Gas Reservoir with Closed Outer Boundary in HorizontalDirection For gas reservoir with closed outer boundary inhorizontal direction according to (14)which is the solution ofhorizontal well seepage mathematical model while combin-ingwith (16) the type curve of well test analysis for horizontalgas well can be plotted with Stehfest inversion algorithm asshown in Figure 6

As seen from Figure 6 for gas reservoir with closed outerboundary in horizontal direction the pseudopressure behav-ior of horizontal gas well can manifest five characteristic

0001

001

01

1

10

1Eminus2

1E0

1E2

1E4

1E6

1E8

1E10

tDCD

reD = 10

reD = 20

reD = 40

CD = 1 S = 10 hD = 30 LD = 10 ZwD = 05

I II

IIIIV

V

Figure 7 Well test analysis type curve of horizontal well in gasreservoir with constant pressure outer boundary

periods The previous four periods of gas reservoir withclosed outer boundary are exactly the same as gas reservoirwith infinite outer boundary but the pseudopressure behav-ior of the horizontal gas well adds the pseudosteady stateflow period (V) which appears after the boundary responseThe characteristic of the pseudosteady state flow period ismanifested as a straight line segment with a slope of 1 onthe log-log plot of 119898

119908119863 1198981015840119908119863

versus 119905119863119862119863 The greater the

distance of the outer boundary the later the appearance ofthe pseudosteady state flow period The smaller the distanceof the outer boundary the sooner the appearance of thepseudosteady state flow period

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during the pseudosteadystate flow period can be obtained This results in

119898119908119863

= 2120587(119903119908119863

119903119890119863

)

2

119905119863

+ 119878

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)= 2120587(

119903119908119863

119903119890119863

)

2

119905119863

(21)

33 Gas Reservoir with Constant Pressure Outer Boundary inHorizontal Direction For gas reservoir with constant pres-sure outer boundary in horizontal direction according to (15)which is the solution of horizontal well seepagemathematicalmodel while combining with (16) the type curve of well testanalysis for horizontal gas well can be plotted with Stehfestinversion algorithm as shown in Figure 7

As seen from Figure 7 for gas reservoir with constantpressure outer boundary in horizontal direction the pseu-dopressure behavior of horizontal gas well can manifestfive characteristic periods the previous four periods of gasreservoir with constant pressure outer boundary are exactlythe same as gas reservoir with infinite outer boundarybut the pseudopressure behavior of the horizontal gas welladds the steady state flow period (V) which appears afterthe boundary response The occurrence time of the steadystate flow period is affected by the outer boundary distance

6 Mathematical Problems in Engineering

in horizontal direction The smaller the distance of the outerboundary the sooner the appearance of the steady state flowperiod The greater the distance of the outer boundary thelater the appearance of the steady state flow period

34 Characteristic Value Method of Well Test Analysis forHorizontal GasWell The characteristic value method of welltest analysis for horizontal gas well can determine the gasreservoir fluid flowparameters according to the characteristiclines which are manifested by pseudopressure derivativecurve of each flow period on the log-log plot

341 PureWellbore Storage Period The characteristic of purewellbore storage period of horizontal well is manifested asa straight line segment with a slope of 1 on the log-log plotof 119898119908119863

1198981015840119908119863

versus 119905119863119862119863 The expression of dimensionless

bottomhole pseudopressure during this period is

119898119908119863

=119905119863

119862119863

(22)

Equation (22) can be converted to dimensional formand then according to the time and pressure data duringpure wellbore storage period the method to determine thewellbore storage coefficient can be obtained By plottingthe log-log plot of Δ119898 Δ119898

1015840 versus 119905 the wellbore storagecoefficient can be determined by the straight line segmentwith a slope of 1 on the log-log plot

The following can be obtained from the definitions ofdimensionless variables

119905119863

119862119863

=36119870ℎ1199051206011205831198881199051199032

119908

1198622120587120601119888119905ℎ1198712

=72120587119870

ℎℎ1198712

1205831199032119908

119905

119862 (23)

According to (22) (23) and the definition of dimension-less pseudopressure the wellbore storage coefficient can beobtained This results in

119862 =0288119902

119904119888119879

120583

1198712

1199032119908

119905

Δ119898 (24)

where 119905Δ119898 represents the actual value on the log-log plot ofΔ119898 Δ119898

1015840 versus 119905 during the pure wellbore storage period

342 Early Vertical Radial Flow Period

The Determination of Geometric Mean Permeability Thedimensionless bottomhole pseudopressure derivative curveis manifested as a horizontal straight line segment with thevalue of 1(4119871

119863) during the early vertical radial flow period

The expression of dimensionless bottomhole pseudopressurederivative during this period is

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

4119871119863

(25)

According to the definitions of dimensionless variablesthe dimensional form of (25) can be obtained This results in

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)er

=1

4 (119871ℎ)radic119870V119870ℎ (26)

The geometric mean permeability of gas reservoir can bedetermined by (26) This results in

radic119870ℎ119870V =

3185 times 10minus3

119902119904119888119879

119871(119905Δ1198981015840)er (27)

where (119905Δ1198981015840)er represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early vertical radial flow period

The Determination of Skin Factor and Initial Reservoir Pres-sure The expression of dimensionless bottomhole pseudo-pressure during the early vertical radial flow period is

119898119908119863

=1

4119871119863

[ln 225 (119905119863

119862119863

) + ln1198621198631198902119878] (28)

According to the definitions of dimensionless variablesand (28) the skin factor can be obtained This results in

119878 = 05 [Δ119898er

(119905Δ1198981015840)erminus ln

119870ℎ119905er

1206011205831198621199051199032119908

minus 080907] (29)

where Δ119898er and 119905er represent the pseudopressure differenceand time corresponding to the (119905Δ119898

1015840)er respectively

For pressure buildup analysis when Δ119905 rarr infin thelnΔ119905(Δ119905 + 119905

119901) rarr 0 (30) can be obtained through

the use of the definitions of dimensionless variables andpseudopressure difference during the early vertical radial flowperiod

119898119894minus 119898119908119891

119905Δ1198981015840= ln 119905119901119863

+ 080907 + 2119878 (30)

The initial reservoir pseudopressure can be determinedby (30) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)erb

(ln 119905119901119863

+ 080907 + 2119878) (31)

where (119905Δ1198981015840)erb represents the actual value on the pressure

buildup log-log plot of Δ1198981015840 versus 119905 during the early vertical

radial flow period

343 Early Linear Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 05 during theearly linear flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables thefollowing can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)119897=

119903119908

119871radic

36120587119870ℎ119905

1206011205831198621199051199032119908

(32)

The horizontal permeability of gas reservoir can bedetermined by (32) This results in

radic119870ℎ=

428 times 10minus2

119902119904119888119879

119871ℎradic120601120583119862119905

[radic119905

(119905Δ1198981015840)]

119897

(33)

Mathematical Problems in Engineering 7

where (119905Δ1198981015840)119897represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early linear flow period

Combining (27) with (33) the vertical permeability canbe obtained This results in

radic119870V = 744 times 10minus2

ℎradic120601120583119862119905

[(119905Δ1198981015840) radic119905]

119897

(119905Δ1198981015840)er (34)

344 Late Horizontal Pseudoradial Flow Period The dimen-sionless bottomhole pseudopressure derivative curve is man-ifested as a horizontal straight line segment with the valueof 05 during the late horizontal pseudoradial flow periodAccording to the expression of dimensionless bottomholepseudopressure derivative during this period and the defini-tions of dimensionless variables (35) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)lr= 05 (35)

The horizontal permeability of gas reservoir can bedetermined by (35) This results in

119870ℎ=

637 times 10minus3

119902119904119888119879

ℎ(119905Δ1198981015840)lr (36)

where (119905Δ1198981015840)lr represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the late horizontal pseudoradial flow

periodFor pressure buildup analysis when Δ119905 rarr infin the

lnΔ119905(Δ119905 + 119905119901) rarr 0 (37) can be obtained through the use

of the definitions of dimensionless variables and pseudopres-sure difference during the late horizontal pseudoradial flowperiod

119898119894minus 119898119908119891

(119905Δ1198981015840)lrb= ln 119903119908119863

119905119901119863

+ 080907 + 2119878 (37)

The initial reservoir pseudopressure can be determinedby (37) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)lrb

(ln 119903119908119863

119905119901119863

+ 080907 + 2119878) (38)

where (119905Δ1198981015840)lrb represents the actual value on the pressure

buildup log-log plot ofΔ1198981015840 versus 119905 during the late horizontal

pseudoradial flow period

345 Pseudosteady Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 1 during thepseudosteady flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables(39) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)pp

=72120587119870

ℎ119905pp

1199032119890ℎ120601120583119862

119905

(39)

The pore volume of gas reservoir can be determined by(39) This results in

1205871199032

119890ℎ120601 =

0905119902119904119888119879

ℎ120583119862119905

119905pp

(119905Δ1198981015840)pp (40)

01020304050

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

0246810

Gas flow rateWater flow rate

qsc(104m

3d)

qw(m

3d)

Figure 8 The gas flow rate and water flow rate curve of Longping 1well

5

10

15

20

25

5

10

15

20

25

Tubing head pressureCasing head pressure

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

Pwh

(MPa

)

Pch

(MPa

)

Figure 9 The tubing head pressure and casing head pressure curveof Longping 1 well

where (119905Δ1198981015840)pp represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the pseudosteady flow period

4 Example Analysis

The Longping 1 well is a horizontal development well inJingBian gas field the well total depth is 4672m the drilledformation name is Majiagou group the mid-depth of reser-voir is 342563m and the well completion system is screencompletion According to the deliverability test during 26ndash29December 2006 the calculated absolute open flow was 9426times 104m3d The commissioning data of Longping 1 well wasin 12May 2007 the initial formation pressure was 2939MPabefore production and the surface tubing pressure and casingpressure were both 2390MPa The production performancecurves of Longping 1 well are shown in Figures 8 and 9respectively

Longping 1 well has been conducted pressure builduptest during 14 August 2007 and 23 October 2007 The gasflow rate of Longping 1 well was 40 times 104m3d before theshut-in The bottomhole pressure recovered from 2238MPato 2783MPa during the pressure buildup test Physical

8 Mathematical Problems in EngineeringΔmΔ

m998400

Δt (h)10minus3

10minus4

10minus5

10minus6

10minus7

10minus2 10minus1 100 101 102 103 104

III

IIIIV

Figure 10 The pressure buildup log-log plot of Longping 1 well

100 101 102 103 104 105 106 107

(tp + Δt)Δt

m(p

wt)

50

46

42

38

34

30

Figure 11 The pressure buildup semilog plot of Longping 1 well

Table 1 Physical parameters of fluid and reservoir

Parameter ValueInitial formation pressure 119901

119894(MPa) 2939

Formation temperature 119879 (∘C) 9580Formation thickness ℎ (m) 631Porosity 120601 () 777Initial water saturation 119878wi () 1360Well radius 119903

119908(m) 00797

Gas gravity 120574119892

0608Gas deviation factor 119885 09738Gas viscosity 120583

119892(mPasdots) 00222

Table 2 Well test analysis results of Longping 1 well

Parameter Parameter valuesWellbore storage coefficient 119862 (m3MPa) 1229Horizontal permeability 119870

ℎ(mD) 7742

Vertical permeability 119870119907(mD) 0039

Flow capacity 119870ℎℎ (mDsdotm) 48857

Skin factor 119878 minus249

Effective horizontal section length 119871 (m) 19837

Reservoir pressure 119901119877(MPa) 28385

parameters of fluid and reservoir are shown in Table 1 Thepressure buildup log-log plot of Longping 1 well is shown inFigure 10

Δt (h)

m(p

ws)

50

0 300 900 1200 1500

46

42

38

34

30

Figure 12 The pressure history matching plot of Longping 1 well

As seen from the contrast between Figure 10 and well testanalysis type curves of horizontal well the pseudopressurebehavior of Longping 1 well manifests four characteristicperiods during the pressure buildup test pure wellborestorage period (I) early vertical radial flow period (II) earlylinear flow period (III) and late horizontal pseudoradial flowperiod (IV)

Using the above characteristic value method of well testanalysis for horizontal gas well well test analysis results ofLongping 1 well are shown in Table 2 The pressure buildupsemilog plot and pressure history matching plot of Longping1 well are shown in Figures 11 and 12 respectively

5 Summary and Conclusions

The four main conclusions and summary of this study are asfollows

(1) On the basis of establishing the mathematical modelsof well test analysis for horizontal gas well and obtain-ing the solutions of the mathematical models severaltype curves which can be used to identify flow regimehave been plotted and the seepage characteristic ofhorizontal gas well has been analyzed

(2) The expressions of dimensionless bottomhole pseu-dopressure and pseudopressure derivative duringeach characteristic period of horizontal gas well havebeen obtained formulas have been developed tocalculate gas reservoir fluid flow parameters

(3) The example study verifies that the characteristicvalue method of well test analysis for horizontal gaswell is reliable and practical

(4) The characteristic value method of well test analysiswhich has been included in the well test analysissoftware at present has been widely used in verticalwell As long as the characteristic straight line seg-ments which are manifested by pressure derivativecurve appear the reservoir fluid flow parameterscan be calculated by the characteristic value methodof well test analysis for vertical well The proposedcharacteristic value method of well test analysis for

Mathematical Problems in Engineering 9

horizontal gas well enriches and develops the well testanalysis theory and method

Nomenclature

119862 Wellbore storage coefficient m3MPa119862119863 Dimensionless wellbore storage coefficient

119862119905 Total compressibility MPaminus1

ℎ Reservoir thickness mℎ119863 Dimensionless reservoir thickness

119868119899 Modified Bessel function of first kind of order 119899

119870ℎ Horizontal permeability mD

119870119899 Modified Bessel function of second kind of order 119899

119870119881 Vertical permeability mD

119871 Horizontal section length m119871119863 Dimensionless horizontal section length

119898(119901) Pseudopressure MPa2mPasdots119898119863 Dimensionless pseudopressure

119898119894 Initial formation pseudopressure MPa2mPasdots

119898119908119863

Dimensionless bottomhole pseudopressure119898119908119891 Flowing wellbore pseudopressure MPa2mPasdots

1198981015840

119908119863 Derivative of119898

119908119863

119898119863 Laplace transform of 119898

119863

119898119908119863

Laplace transform of 119898119908119863

Δ119898 Pseudopressure difference MPaΔ1198981015840 Derivative of Δ119898

119901119888ℎ Casing head pressure MPa

119901119894 Initial formation pressure MPa

119901119877 Reservoir pressure MPa

119901119908ℎ Tubing head pressure MPa

119901119908119904 Flowing wellbore pressure at shut-in MPa

119902119904119888 Gas flow rate 104m3d

119902119908 Water production rate m3d

119903 Radial distance m119903119863 Dimensionless radial distance

119903119890 Outer boundary distance m

119903119890119863 Dimensionless outer boundary distance

119903119908 Wellbore radius m

119903119908119863

Dimensionless wellbore radius119878 Skin factor119904 Laplace transform variable119878119908119894 Initial water saturation fraction

119905 Time hours119905119863 Dimensionless time

119905119901 Production time hours

119905119901119863

Dimensionless production timeΔ119905 Shut-in time hours119879 Formation temperature ∘C119911 Vertical distance m119911119863 Dimensionless vertical distance

119911119908 Horizontal section position m

119911119908119863

Dimensionless horizontal section position119885 Gas deviation factor120576 Tiny variable120601 Porosity fraction120583 Gas viscosity mPasdots120574119892 Gas gravity

Subscripts

119863 Dimensionlesser Earlyerb Early of buildupℎ Horizontal119894 Initial119897 Linearlr Late horizontal pseudo-radiallrb Late horizontal pseudo-radial of builduppp Pseudosteady flow period119905 TotalV Vertical119908119891 Flowing wellbore119908119904 Shut-in wellbore

Conflict of Interests

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

References

[1] R S Nie J C Guo Y L Jia S Q Zhu Z Rao and C G ZhangldquoNew modelling of transient well test and rate decline analysisfor a horizontal well in a multiple-zone reservoirrdquo Journal ofGeophysics and Engineering vol 8 no 3 pp 464ndash476 2011

[2] D R Yuan H Sun Y C Li and M Zhang ldquoMulti-stagefractured tight gas horizontal well test data interpretationstudyrdquo in Proceedings of the International Petroleum TechnologyConference March 2013

[3] M Mavaddat A Soleimani M R Rasaei Y Mavaddat and AMomeni ldquoWell test analysis ofMultipleHydraulically FracturedHorizontal Wells (MHFHW) in gas condensate reservoirsrdquo inProceedings of the SPEIADC Middle East Drilling TechnologyConference and Exhibition (MEDT rsquo11) pp 39ndash48 October 2011

[4] MMirza P AditamaM AL Raqmi and Z Anwar ldquoHorizontalwell stimulation a pilot test in southern Omanrdquo in Proceedingsof the SPE EOR Conference at Oil and GasWest Asia April 2014

[5] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquo Society of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973

[6] M D Clonts and H J Ramey ldquoPressure transient analysis forwells with horizontal drainholesrdquo in Proceedings of the SPECalifornia Regional Meeting Oakland Calif USA April 1986paper SPE 15116

[7] P A Goode and R K M Thambynayagam ldquoPressure draw-down and buildup analysis of horizontal wells in anisotropicmediardquo SPE Formation Evaluation vol 2 no 4 pp 683ndash6971987

[8] E Ozkan and R Rajagopal ldquoHorizontal well pressure analysisrdquoSPE Formation Evaluation pp 567ndash575 1989

[9] A S Odeh and D K Babu ldquoTransient flow behavior ofhorizontal wells Pressure drawdown and buildup analysisrdquo SPEFormation Evaluation vol 5 no 1 pp 7ndash15 1990

[10] L G Thompson and K O Temeng ldquoAutomatic type-curvematching for horizontal wellsrdquo in Proceedings of the ProductionOperation Symposium Oklahoma City Okla USA 1993 paperSPE 25507

10 Mathematical Problems in Engineering

[11] R S Rahavan C C Chen and B Agarwal ldquoAn analysis ofhorizontal wells intercepted by multiple fracturesrdquo SPE27652(Sep pp 235ndash245 1997

[12] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal wells in closed systemsrdquo in Proceed-ings of the SPE International Improved Oil Recovery Conferencein Asia Pacific (IIORC rsquo03) pp 295ndash307 Alberta CanadaOctober 2003

[13] T Whittle H Jiang S Young and A C Gringarten ldquoWellproduction forecasting by extrapolation of the deconvolutionof the well test pressure transientsrdquo in Proceedings of the SPEEUROPECEAGE Conference Amsterdam The NetherlandJune 2009 SPE 122299 paper

[14] C C Miller A B Dyes and C A Hutchinson ldquoThe estimationof permeability and reservoir pressure from bottom-hole pres-sure buildup Characteristicrdquo Journal of Petroleum Technologypp 91ndash104 1950

[15] D R Horner ldquoPressure buildup in wellsrdquo in Proceedings of the3rd World Petroleum Congress Hague The Netherlands May-June 1951 paper SPE 4135

[16] R Raghavan ldquoThe effect of producing time on type curveanalysisrdquo Journal of Petroleum Technology vol 32 no 6 pp1053ndash1064 1980

[17] D Bourdet ldquoA new set of type curves simplifies well testanalysisrdquoWorld Oil pp 95ndash106 1983

[18] T M Hegre ldquoHydraulically fractured horizontal well simu-lationrdquo in Proceedings of the NPFSPE European ProductionOperations Conference (EPOC rsquo96) pp 59ndash62 April 1996

[19] R G Agarwal R Al-Hussaing and H J Ramey ldquoAn investiga-tion of wellbore storage and skin effect in unsteady liquid flowI Analytical treatmentrdquo Society of Petroleum Engineers Journalvol 10 no 3 pp 291ndash297 1970

[20] H J J Ramey and R G Agarwal ldquoAnnulus unloading ratesas influenced by wellbore storage and skin effectrdquo Society ofPetroleum Engineers Journal vol 12 no 5 pp 453ndash462 1972

[21] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems part 1-analytical considerationsrdquo SPEForma-tion Evaluation 1991

[22] L G Thompson Analysis of variable rate pressure data usingduhamels principle [PhD thesis] University of Tulsa TulsaOkla USA 1985

[23] H Stehfest ldquoNumerical inversion of laplace transformsrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

[24] N Al-Ajmi M Ahmadi E Ozkan and H Kazemi ldquoNumericalinversion of laplace transforms in the solution of transient flowproblemswith discontinuitiesrdquo inProceedings of the SPEAnnualTechnical Conference and Exhibition Paper SPE 116255 pp 21ndash24 Denver Colorado 2008

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

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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 Mathematical Problems in Engineering

001

01

1

10

hD = 50

hD = 100

hD = 200

tDCD

1Eminus2

1Eminus1

1E0

1E1

1E2

1E3

1E4

1E5

1E6

1E7

1E8

mwDm

998400 wD

I II

III

IV

CDe2S = 104 LD = 10 ZwD = 05

Figure 2 Well test analysis type curve of horizontal well in gasreservoir with infinite outer boundary

3 Type Curves of Well TestAnalysis for Horizontal Gas Well

31 Gas Reservoir with Infinite Outer Boundary in HorizontalDirection For gas reservoir with infinite outer boundary inhorizontal direction according to (12) which is the solution ofhorizontal well seepage mathematical model while combin-ingwith (16) the type curve of well test analysis for horizontalgas well can be plotted with Stehfest inversion algorithm asshown in Figure 2

As seen from Figure 2 for gas reservoir with infiniteouter boundary in horizontal direction the pseudopressurebehavior of horizontal gas well can manifest four character-istic periods pure wellbore storage period (I) early verticalradial flow period (II) early linear flow period (III) and latehorizontal pseudoradial flow period (IV)

311 Pure Wellbore Storage Period The characteristic of purewellbore storage period of horizontal well is the same asvertical well which is manifested as a 45∘ straight linesegment on the log-log plot of 119898

119908119863 1198981015840119908119863

versus 119905119863119862119863 and

the duration of this period is affected by wellbore storage andskin effect

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=119905119863

119862119863

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

119905119863

119862119863

(17)

312 Early Vertical Radial Flow Period The early verticalradial flow period appears after the effect of wellbore storagethe characteristic of this period is manifested as a horizontalstraight line segment on the log-log plot of 119898

1015840

119908119863versus

119905119863119862119863The pseudopressure behavior of this period is affected

by formation thickness horizontal section length and the

Figure 3 The schematic diagram of early vertical radial flow

Figure 4 The schematic diagram of early linear flow

position of horizontal section in the gas reservoir The flowregime of this period is shown in Figure 3

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=1

4119871119863

[ln 225 (119905119863

119862119863

) + ln1198621198631198902119878]

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

4119871119863

(18)

313 Early Linear Flow Period The early linear flow periodappears after the early vertical radial flow period Thecharacteristic of this period is manifested as a straight linesegment with a slope of 05 on the log-log plot of1198981015840

119908119863versus

119905119863119862119863 This characteristic describes the linear flow of fluid

from formation to horizontal section The pseudopressurebehavior of this period is affected by dimensionless formationthickness ℎ

119863 dimensionless horizontal section length 119871

119863

and the position of horizontal section in the gas reservoir 119911119908119863

The flow regime of this period is shown in Figure 4

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

= 2119903119908119863

radic120587119905119863

+ 119878

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)= 119903119908119863

radic120587119905119863

(19)

314 Late Horizontal Pseudoradial Flow Period The late hor-izontal pseudoradial flow period appears after the early linearflow period The characteristic of this period is manifested

Mathematical Problems in Engineering 5

Figure 5 The schematic diagram of late horizontal pseudoradialflow

001

01

1

10

100

tDCD

1Eminus2

1Eminus1

1E0

1E1

1E2

1E3

1E4

1E5

1E6

1E7

1E8

mwDm

998400 wD

CD = 1 S = 5 hD = 30 LD = 5 ZwD = 05

reD = 10

reD = 15

reD = 20

III

IIIIV

V

Figure 6 Well test analysis type curve of horizontal well in gasreservoir with closed outer boundary

as a horizontal straight line segment with the value of 05on the log-log plot of 1198981015840

119908119863versus 119905

119863119862119863 This characteristic

describes the horizontal pseudoradial flow of fluid fromformation horizontal plane in the distance to horizontalsection The flow regime of this period is shown in Figure 5

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=1

2[ln(

119903119908119863

119905119863

119862119863

) + ln1198621198631198902119878]

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

2

(20)

32 Gas Reservoir with Closed Outer Boundary in HorizontalDirection For gas reservoir with closed outer boundary inhorizontal direction according to (14)which is the solution ofhorizontal well seepage mathematical model while combin-ingwith (16) the type curve of well test analysis for horizontalgas well can be plotted with Stehfest inversion algorithm asshown in Figure 6

As seen from Figure 6 for gas reservoir with closed outerboundary in horizontal direction the pseudopressure behav-ior of horizontal gas well can manifest five characteristic

0001

001

01

1

10

1Eminus2

1E0

1E2

1E4

1E6

1E8

1E10

tDCD

reD = 10

reD = 20

reD = 40

CD = 1 S = 10 hD = 30 LD = 10 ZwD = 05

I II

IIIIV

V

Figure 7 Well test analysis type curve of horizontal well in gasreservoir with constant pressure outer boundary

periods The previous four periods of gas reservoir withclosed outer boundary are exactly the same as gas reservoirwith infinite outer boundary but the pseudopressure behav-ior of the horizontal gas well adds the pseudosteady stateflow period (V) which appears after the boundary responseThe characteristic of the pseudosteady state flow period ismanifested as a straight line segment with a slope of 1 onthe log-log plot of 119898

119908119863 1198981015840119908119863

versus 119905119863119862119863 The greater the

distance of the outer boundary the later the appearance ofthe pseudosteady state flow period The smaller the distanceof the outer boundary the sooner the appearance of thepseudosteady state flow period

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during the pseudosteadystate flow period can be obtained This results in

119898119908119863

= 2120587(119903119908119863

119903119890119863

)

2

119905119863

+ 119878

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)= 2120587(

119903119908119863

119903119890119863

)

2

119905119863

(21)

33 Gas Reservoir with Constant Pressure Outer Boundary inHorizontal Direction For gas reservoir with constant pres-sure outer boundary in horizontal direction according to (15)which is the solution of horizontal well seepagemathematicalmodel while combining with (16) the type curve of well testanalysis for horizontal gas well can be plotted with Stehfestinversion algorithm as shown in Figure 7

As seen from Figure 7 for gas reservoir with constantpressure outer boundary in horizontal direction the pseu-dopressure behavior of horizontal gas well can manifestfive characteristic periods the previous four periods of gasreservoir with constant pressure outer boundary are exactlythe same as gas reservoir with infinite outer boundarybut the pseudopressure behavior of the horizontal gas welladds the steady state flow period (V) which appears afterthe boundary response The occurrence time of the steadystate flow period is affected by the outer boundary distance

6 Mathematical Problems in Engineering

in horizontal direction The smaller the distance of the outerboundary the sooner the appearance of the steady state flowperiod The greater the distance of the outer boundary thelater the appearance of the steady state flow period

34 Characteristic Value Method of Well Test Analysis forHorizontal GasWell The characteristic value method of welltest analysis for horizontal gas well can determine the gasreservoir fluid flowparameters according to the characteristiclines which are manifested by pseudopressure derivativecurve of each flow period on the log-log plot

341 PureWellbore Storage Period The characteristic of purewellbore storage period of horizontal well is manifested asa straight line segment with a slope of 1 on the log-log plotof 119898119908119863

1198981015840119908119863

versus 119905119863119862119863 The expression of dimensionless

bottomhole pseudopressure during this period is

119898119908119863

=119905119863

119862119863

(22)

Equation (22) can be converted to dimensional formand then according to the time and pressure data duringpure wellbore storage period the method to determine thewellbore storage coefficient can be obtained By plottingthe log-log plot of Δ119898 Δ119898

1015840 versus 119905 the wellbore storagecoefficient can be determined by the straight line segmentwith a slope of 1 on the log-log plot

The following can be obtained from the definitions ofdimensionless variables

119905119863

119862119863

=36119870ℎ1199051206011205831198881199051199032

119908

1198622120587120601119888119905ℎ1198712

=72120587119870

ℎℎ1198712

1205831199032119908

119905

119862 (23)

According to (22) (23) and the definition of dimension-less pseudopressure the wellbore storage coefficient can beobtained This results in

119862 =0288119902

119904119888119879

120583

1198712

1199032119908

119905

Δ119898 (24)

where 119905Δ119898 represents the actual value on the log-log plot ofΔ119898 Δ119898

1015840 versus 119905 during the pure wellbore storage period

342 Early Vertical Radial Flow Period

The Determination of Geometric Mean Permeability Thedimensionless bottomhole pseudopressure derivative curveis manifested as a horizontal straight line segment with thevalue of 1(4119871

119863) during the early vertical radial flow period

The expression of dimensionless bottomhole pseudopressurederivative during this period is

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

4119871119863

(25)

According to the definitions of dimensionless variablesthe dimensional form of (25) can be obtained This results in

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)er

=1

4 (119871ℎ)radic119870V119870ℎ (26)

The geometric mean permeability of gas reservoir can bedetermined by (26) This results in

radic119870ℎ119870V =

3185 times 10minus3

119902119904119888119879

119871(119905Δ1198981015840)er (27)

where (119905Δ1198981015840)er represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early vertical radial flow period

The Determination of Skin Factor and Initial Reservoir Pres-sure The expression of dimensionless bottomhole pseudo-pressure during the early vertical radial flow period is

119898119908119863

=1

4119871119863

[ln 225 (119905119863

119862119863

) + ln1198621198631198902119878] (28)

According to the definitions of dimensionless variablesand (28) the skin factor can be obtained This results in

119878 = 05 [Δ119898er

(119905Δ1198981015840)erminus ln

119870ℎ119905er

1206011205831198621199051199032119908

minus 080907] (29)

where Δ119898er and 119905er represent the pseudopressure differenceand time corresponding to the (119905Δ119898

1015840)er respectively

For pressure buildup analysis when Δ119905 rarr infin thelnΔ119905(Δ119905 + 119905

119901) rarr 0 (30) can be obtained through

the use of the definitions of dimensionless variables andpseudopressure difference during the early vertical radial flowperiod

119898119894minus 119898119908119891

119905Δ1198981015840= ln 119905119901119863

+ 080907 + 2119878 (30)

The initial reservoir pseudopressure can be determinedby (30) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)erb

(ln 119905119901119863

+ 080907 + 2119878) (31)

where (119905Δ1198981015840)erb represents the actual value on the pressure

buildup log-log plot of Δ1198981015840 versus 119905 during the early vertical

radial flow period

343 Early Linear Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 05 during theearly linear flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables thefollowing can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)119897=

119903119908

119871radic

36120587119870ℎ119905

1206011205831198621199051199032119908

(32)

The horizontal permeability of gas reservoir can bedetermined by (32) This results in

radic119870ℎ=

428 times 10minus2

119902119904119888119879

119871ℎradic120601120583119862119905

[radic119905

(119905Δ1198981015840)]

119897

(33)

Mathematical Problems in Engineering 7

where (119905Δ1198981015840)119897represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early linear flow period

Combining (27) with (33) the vertical permeability canbe obtained This results in

radic119870V = 744 times 10minus2

ℎradic120601120583119862119905

[(119905Δ1198981015840) radic119905]

119897

(119905Δ1198981015840)er (34)

344 Late Horizontal Pseudoradial Flow Period The dimen-sionless bottomhole pseudopressure derivative curve is man-ifested as a horizontal straight line segment with the valueof 05 during the late horizontal pseudoradial flow periodAccording to the expression of dimensionless bottomholepseudopressure derivative during this period and the defini-tions of dimensionless variables (35) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)lr= 05 (35)

The horizontal permeability of gas reservoir can bedetermined by (35) This results in

119870ℎ=

637 times 10minus3

119902119904119888119879

ℎ(119905Δ1198981015840)lr (36)

where (119905Δ1198981015840)lr represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the late horizontal pseudoradial flow

periodFor pressure buildup analysis when Δ119905 rarr infin the

lnΔ119905(Δ119905 + 119905119901) rarr 0 (37) can be obtained through the use

of the definitions of dimensionless variables and pseudopres-sure difference during the late horizontal pseudoradial flowperiod

119898119894minus 119898119908119891

(119905Δ1198981015840)lrb= ln 119903119908119863

119905119901119863

+ 080907 + 2119878 (37)

The initial reservoir pseudopressure can be determinedby (37) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)lrb

(ln 119903119908119863

119905119901119863

+ 080907 + 2119878) (38)

where (119905Δ1198981015840)lrb represents the actual value on the pressure

buildup log-log plot ofΔ1198981015840 versus 119905 during the late horizontal

pseudoradial flow period

345 Pseudosteady Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 1 during thepseudosteady flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables(39) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)pp

=72120587119870

ℎ119905pp

1199032119890ℎ120601120583119862

119905

(39)

The pore volume of gas reservoir can be determined by(39) This results in

1205871199032

119890ℎ120601 =

0905119902119904119888119879

ℎ120583119862119905

119905pp

(119905Δ1198981015840)pp (40)

01020304050

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

0246810

Gas flow rateWater flow rate

qsc(104m

3d)

qw(m

3d)

Figure 8 The gas flow rate and water flow rate curve of Longping 1well

5

10

15

20

25

5

10

15

20

25

Tubing head pressureCasing head pressure

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

Pwh

(MPa

)

Pch

(MPa

)

Figure 9 The tubing head pressure and casing head pressure curveof Longping 1 well

where (119905Δ1198981015840)pp represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the pseudosteady flow period

4 Example Analysis

The Longping 1 well is a horizontal development well inJingBian gas field the well total depth is 4672m the drilledformation name is Majiagou group the mid-depth of reser-voir is 342563m and the well completion system is screencompletion According to the deliverability test during 26ndash29December 2006 the calculated absolute open flow was 9426times 104m3d The commissioning data of Longping 1 well wasin 12May 2007 the initial formation pressure was 2939MPabefore production and the surface tubing pressure and casingpressure were both 2390MPa The production performancecurves of Longping 1 well are shown in Figures 8 and 9respectively

Longping 1 well has been conducted pressure builduptest during 14 August 2007 and 23 October 2007 The gasflow rate of Longping 1 well was 40 times 104m3d before theshut-in The bottomhole pressure recovered from 2238MPato 2783MPa during the pressure buildup test Physical

8 Mathematical Problems in EngineeringΔmΔ

m998400

Δt (h)10minus3

10minus4

10minus5

10minus6

10minus7

10minus2 10minus1 100 101 102 103 104

III

IIIIV

Figure 10 The pressure buildup log-log plot of Longping 1 well

100 101 102 103 104 105 106 107

(tp + Δt)Δt

m(p

wt)

50

46

42

38

34

30

Figure 11 The pressure buildup semilog plot of Longping 1 well

Table 1 Physical parameters of fluid and reservoir

Parameter ValueInitial formation pressure 119901

119894(MPa) 2939

Formation temperature 119879 (∘C) 9580Formation thickness ℎ (m) 631Porosity 120601 () 777Initial water saturation 119878wi () 1360Well radius 119903

119908(m) 00797

Gas gravity 120574119892

0608Gas deviation factor 119885 09738Gas viscosity 120583

119892(mPasdots) 00222

Table 2 Well test analysis results of Longping 1 well

Parameter Parameter valuesWellbore storage coefficient 119862 (m3MPa) 1229Horizontal permeability 119870

ℎ(mD) 7742

Vertical permeability 119870119907(mD) 0039

Flow capacity 119870ℎℎ (mDsdotm) 48857

Skin factor 119878 minus249

Effective horizontal section length 119871 (m) 19837

Reservoir pressure 119901119877(MPa) 28385

parameters of fluid and reservoir are shown in Table 1 Thepressure buildup log-log plot of Longping 1 well is shown inFigure 10

Δt (h)

m(p

ws)

50

0 300 900 1200 1500

46

42

38

34

30

Figure 12 The pressure history matching plot of Longping 1 well

As seen from the contrast between Figure 10 and well testanalysis type curves of horizontal well the pseudopressurebehavior of Longping 1 well manifests four characteristicperiods during the pressure buildup test pure wellborestorage period (I) early vertical radial flow period (II) earlylinear flow period (III) and late horizontal pseudoradial flowperiod (IV)

Using the above characteristic value method of well testanalysis for horizontal gas well well test analysis results ofLongping 1 well are shown in Table 2 The pressure buildupsemilog plot and pressure history matching plot of Longping1 well are shown in Figures 11 and 12 respectively

5 Summary and Conclusions

The four main conclusions and summary of this study are asfollows

(1) On the basis of establishing the mathematical modelsof well test analysis for horizontal gas well and obtain-ing the solutions of the mathematical models severaltype curves which can be used to identify flow regimehave been plotted and the seepage characteristic ofhorizontal gas well has been analyzed

(2) The expressions of dimensionless bottomhole pseu-dopressure and pseudopressure derivative duringeach characteristic period of horizontal gas well havebeen obtained formulas have been developed tocalculate gas reservoir fluid flow parameters

(3) The example study verifies that the characteristicvalue method of well test analysis for horizontal gaswell is reliable and practical

(4) The characteristic value method of well test analysiswhich has been included in the well test analysissoftware at present has been widely used in verticalwell As long as the characteristic straight line seg-ments which are manifested by pressure derivativecurve appear the reservoir fluid flow parameterscan be calculated by the characteristic value methodof well test analysis for vertical well The proposedcharacteristic value method of well test analysis for

Mathematical Problems in Engineering 9

horizontal gas well enriches and develops the well testanalysis theory and method

Nomenclature

119862 Wellbore storage coefficient m3MPa119862119863 Dimensionless wellbore storage coefficient

119862119905 Total compressibility MPaminus1

ℎ Reservoir thickness mℎ119863 Dimensionless reservoir thickness

119868119899 Modified Bessel function of first kind of order 119899

119870ℎ Horizontal permeability mD

119870119899 Modified Bessel function of second kind of order 119899

119870119881 Vertical permeability mD

119871 Horizontal section length m119871119863 Dimensionless horizontal section length

119898(119901) Pseudopressure MPa2mPasdots119898119863 Dimensionless pseudopressure

119898119894 Initial formation pseudopressure MPa2mPasdots

119898119908119863

Dimensionless bottomhole pseudopressure119898119908119891 Flowing wellbore pseudopressure MPa2mPasdots

1198981015840

119908119863 Derivative of119898

119908119863

119898119863 Laplace transform of 119898

119863

119898119908119863

Laplace transform of 119898119908119863

Δ119898 Pseudopressure difference MPaΔ1198981015840 Derivative of Δ119898

119901119888ℎ Casing head pressure MPa

119901119894 Initial formation pressure MPa

119901119877 Reservoir pressure MPa

119901119908ℎ Tubing head pressure MPa

119901119908119904 Flowing wellbore pressure at shut-in MPa

119902119904119888 Gas flow rate 104m3d

119902119908 Water production rate m3d

119903 Radial distance m119903119863 Dimensionless radial distance

119903119890 Outer boundary distance m

119903119890119863 Dimensionless outer boundary distance

119903119908 Wellbore radius m

119903119908119863

Dimensionless wellbore radius119878 Skin factor119904 Laplace transform variable119878119908119894 Initial water saturation fraction

119905 Time hours119905119863 Dimensionless time

119905119901 Production time hours

119905119901119863

Dimensionless production timeΔ119905 Shut-in time hours119879 Formation temperature ∘C119911 Vertical distance m119911119863 Dimensionless vertical distance

119911119908 Horizontal section position m

119911119908119863

Dimensionless horizontal section position119885 Gas deviation factor120576 Tiny variable120601 Porosity fraction120583 Gas viscosity mPasdots120574119892 Gas gravity

Subscripts

119863 Dimensionlesser Earlyerb Early of buildupℎ Horizontal119894 Initial119897 Linearlr Late horizontal pseudo-radiallrb Late horizontal pseudo-radial of builduppp Pseudosteady flow period119905 TotalV Vertical119908119891 Flowing wellbore119908119904 Shut-in wellbore

Conflict of Interests

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

References

[1] R S Nie J C Guo Y L Jia S Q Zhu Z Rao and C G ZhangldquoNew modelling of transient well test and rate decline analysisfor a horizontal well in a multiple-zone reservoirrdquo Journal ofGeophysics and Engineering vol 8 no 3 pp 464ndash476 2011

[2] D R Yuan H Sun Y C Li and M Zhang ldquoMulti-stagefractured tight gas horizontal well test data interpretationstudyrdquo in Proceedings of the International Petroleum TechnologyConference March 2013

[3] M Mavaddat A Soleimani M R Rasaei Y Mavaddat and AMomeni ldquoWell test analysis ofMultipleHydraulically FracturedHorizontal Wells (MHFHW) in gas condensate reservoirsrdquo inProceedings of the SPEIADC Middle East Drilling TechnologyConference and Exhibition (MEDT rsquo11) pp 39ndash48 October 2011

[4] MMirza P AditamaM AL Raqmi and Z Anwar ldquoHorizontalwell stimulation a pilot test in southern Omanrdquo in Proceedingsof the SPE EOR Conference at Oil and GasWest Asia April 2014

[5] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquo Society of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973

[6] M D Clonts and H J Ramey ldquoPressure transient analysis forwells with horizontal drainholesrdquo in Proceedings of the SPECalifornia Regional Meeting Oakland Calif USA April 1986paper SPE 15116

[7] P A Goode and R K M Thambynayagam ldquoPressure draw-down and buildup analysis of horizontal wells in anisotropicmediardquo SPE Formation Evaluation vol 2 no 4 pp 683ndash6971987

[8] E Ozkan and R Rajagopal ldquoHorizontal well pressure analysisrdquoSPE Formation Evaluation pp 567ndash575 1989

[9] A S Odeh and D K Babu ldquoTransient flow behavior ofhorizontal wells Pressure drawdown and buildup analysisrdquo SPEFormation Evaluation vol 5 no 1 pp 7ndash15 1990

[10] L G Thompson and K O Temeng ldquoAutomatic type-curvematching for horizontal wellsrdquo in Proceedings of the ProductionOperation Symposium Oklahoma City Okla USA 1993 paperSPE 25507

10 Mathematical Problems in Engineering

[11] R S Rahavan C C Chen and B Agarwal ldquoAn analysis ofhorizontal wells intercepted by multiple fracturesrdquo SPE27652(Sep pp 235ndash245 1997

[12] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal wells in closed systemsrdquo in Proceed-ings of the SPE International Improved Oil Recovery Conferencein Asia Pacific (IIORC rsquo03) pp 295ndash307 Alberta CanadaOctober 2003

[13] T Whittle H Jiang S Young and A C Gringarten ldquoWellproduction forecasting by extrapolation of the deconvolutionof the well test pressure transientsrdquo in Proceedings of the SPEEUROPECEAGE Conference Amsterdam The NetherlandJune 2009 SPE 122299 paper

[14] C C Miller A B Dyes and C A Hutchinson ldquoThe estimationof permeability and reservoir pressure from bottom-hole pres-sure buildup Characteristicrdquo Journal of Petroleum Technologypp 91ndash104 1950

[15] D R Horner ldquoPressure buildup in wellsrdquo in Proceedings of the3rd World Petroleum Congress Hague The Netherlands May-June 1951 paper SPE 4135

[16] R Raghavan ldquoThe effect of producing time on type curveanalysisrdquo Journal of Petroleum Technology vol 32 no 6 pp1053ndash1064 1980

[17] D Bourdet ldquoA new set of type curves simplifies well testanalysisrdquoWorld Oil pp 95ndash106 1983

[18] T M Hegre ldquoHydraulically fractured horizontal well simu-lationrdquo in Proceedings of the NPFSPE European ProductionOperations Conference (EPOC rsquo96) pp 59ndash62 April 1996

[19] R G Agarwal R Al-Hussaing and H J Ramey ldquoAn investiga-tion of wellbore storage and skin effect in unsteady liquid flowI Analytical treatmentrdquo Society of Petroleum Engineers Journalvol 10 no 3 pp 291ndash297 1970

[20] H J J Ramey and R G Agarwal ldquoAnnulus unloading ratesas influenced by wellbore storage and skin effectrdquo Society ofPetroleum Engineers Journal vol 12 no 5 pp 453ndash462 1972

[21] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems part 1-analytical considerationsrdquo SPEForma-tion Evaluation 1991

[22] L G Thompson Analysis of variable rate pressure data usingduhamels principle [PhD thesis] University of Tulsa TulsaOkla USA 1985

[23] H Stehfest ldquoNumerical inversion of laplace transformsrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

[24] N Al-Ajmi M Ahmadi E Ozkan and H Kazemi ldquoNumericalinversion of laplace transforms in the solution of transient flowproblemswith discontinuitiesrdquo inProceedings of the SPEAnnualTechnical Conference and Exhibition Paper SPE 116255 pp 21ndash24 Denver Colorado 2008

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

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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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

Mathematical Problems in Engineering 5

Figure 5 The schematic diagram of late horizontal pseudoradialflow

001

01

1

10

100

tDCD

1Eminus2

1Eminus1

1E0

1E1

1E2

1E3

1E4

1E5

1E6

1E7

1E8

mwDm

998400 wD

CD = 1 S = 5 hD = 30 LD = 5 ZwD = 05

reD = 10

reD = 15

reD = 20

III

IIIIV

V

Figure 6 Well test analysis type curve of horizontal well in gasreservoir with closed outer boundary

as a horizontal straight line segment with the value of 05on the log-log plot of 1198981015840

119908119863versus 119905

119863119862119863 This characteristic

describes the horizontal pseudoradial flow of fluid fromformation horizontal plane in the distance to horizontalsection The flow regime of this period is shown in Figure 5

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during this period can beobtained This results in

119898119908119863

=1

2[ln(

119903119908119863

119905119863

119862119863

) + ln1198621198631198902119878]

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

2

(20)

32 Gas Reservoir with Closed Outer Boundary in HorizontalDirection For gas reservoir with closed outer boundary inhorizontal direction according to (14)which is the solution ofhorizontal well seepage mathematical model while combin-ingwith (16) the type curve of well test analysis for horizontalgas well can be plotted with Stehfest inversion algorithm asshown in Figure 6

As seen from Figure 6 for gas reservoir with closed outerboundary in horizontal direction the pseudopressure behav-ior of horizontal gas well can manifest five characteristic

0001

001

01

1

10

1Eminus2

1E0

1E2

1E4

1E6

1E8

1E10

tDCD

reD = 10

reD = 20

reD = 40

CD = 1 S = 10 hD = 30 LD = 10 ZwD = 05

I II

IIIIV

V

Figure 7 Well test analysis type curve of horizontal well in gasreservoir with constant pressure outer boundary

periods The previous four periods of gas reservoir withclosed outer boundary are exactly the same as gas reservoirwith infinite outer boundary but the pseudopressure behav-ior of the horizontal gas well adds the pseudosteady stateflow period (V) which appears after the boundary responseThe characteristic of the pseudosteady state flow period ismanifested as a straight line segment with a slope of 1 onthe log-log plot of 119898

119908119863 1198981015840119908119863

versus 119905119863119862119863 The greater the

distance of the outer boundary the later the appearance ofthe pseudosteady state flow period The smaller the distanceof the outer boundary the sooner the appearance of thepseudosteady state flow period

Expressions of dimensionless bottomhole pseudopres-sure and pseudopressure derivative during the pseudosteadystate flow period can be obtained This results in

119898119908119863

= 2120587(119903119908119863

119903119890119863

)

2

119905119863

+ 119878

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)= 2120587(

119903119908119863

119903119890119863

)

2

119905119863

(21)

33 Gas Reservoir with Constant Pressure Outer Boundary inHorizontal Direction For gas reservoir with constant pres-sure outer boundary in horizontal direction according to (15)which is the solution of horizontal well seepagemathematicalmodel while combining with (16) the type curve of well testanalysis for horizontal gas well can be plotted with Stehfestinversion algorithm as shown in Figure 7

As seen from Figure 7 for gas reservoir with constantpressure outer boundary in horizontal direction the pseu-dopressure behavior of horizontal gas well can manifestfive characteristic periods the previous four periods of gasreservoir with constant pressure outer boundary are exactlythe same as gas reservoir with infinite outer boundarybut the pseudopressure behavior of the horizontal gas welladds the steady state flow period (V) which appears afterthe boundary response The occurrence time of the steadystate flow period is affected by the outer boundary distance

6 Mathematical Problems in Engineering

in horizontal direction The smaller the distance of the outerboundary the sooner the appearance of the steady state flowperiod The greater the distance of the outer boundary thelater the appearance of the steady state flow period

34 Characteristic Value Method of Well Test Analysis forHorizontal GasWell The characteristic value method of welltest analysis for horizontal gas well can determine the gasreservoir fluid flowparameters according to the characteristiclines which are manifested by pseudopressure derivativecurve of each flow period on the log-log plot

341 PureWellbore Storage Period The characteristic of purewellbore storage period of horizontal well is manifested asa straight line segment with a slope of 1 on the log-log plotof 119898119908119863

1198981015840119908119863

versus 119905119863119862119863 The expression of dimensionless

bottomhole pseudopressure during this period is

119898119908119863

=119905119863

119862119863

(22)

Equation (22) can be converted to dimensional formand then according to the time and pressure data duringpure wellbore storage period the method to determine thewellbore storage coefficient can be obtained By plottingthe log-log plot of Δ119898 Δ119898

1015840 versus 119905 the wellbore storagecoefficient can be determined by the straight line segmentwith a slope of 1 on the log-log plot

The following can be obtained from the definitions ofdimensionless variables

119905119863

119862119863

=36119870ℎ1199051206011205831198881199051199032

119908

1198622120587120601119888119905ℎ1198712

=72120587119870

ℎℎ1198712

1205831199032119908

119905

119862 (23)

According to (22) (23) and the definition of dimension-less pseudopressure the wellbore storage coefficient can beobtained This results in

119862 =0288119902

119904119888119879

120583

1198712

1199032119908

119905

Δ119898 (24)

where 119905Δ119898 represents the actual value on the log-log plot ofΔ119898 Δ119898

1015840 versus 119905 during the pure wellbore storage period

342 Early Vertical Radial Flow Period

The Determination of Geometric Mean Permeability Thedimensionless bottomhole pseudopressure derivative curveis manifested as a horizontal straight line segment with thevalue of 1(4119871

119863) during the early vertical radial flow period

The expression of dimensionless bottomhole pseudopressurederivative during this period is

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

4119871119863

(25)

According to the definitions of dimensionless variablesthe dimensional form of (25) can be obtained This results in

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)er

=1

4 (119871ℎ)radic119870V119870ℎ (26)

The geometric mean permeability of gas reservoir can bedetermined by (26) This results in

radic119870ℎ119870V =

3185 times 10minus3

119902119904119888119879

119871(119905Δ1198981015840)er (27)

where (119905Δ1198981015840)er represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early vertical radial flow period

The Determination of Skin Factor and Initial Reservoir Pres-sure The expression of dimensionless bottomhole pseudo-pressure during the early vertical radial flow period is

119898119908119863

=1

4119871119863

[ln 225 (119905119863

119862119863

) + ln1198621198631198902119878] (28)

According to the definitions of dimensionless variablesand (28) the skin factor can be obtained This results in

119878 = 05 [Δ119898er

(119905Δ1198981015840)erminus ln

119870ℎ119905er

1206011205831198621199051199032119908

minus 080907] (29)

where Δ119898er and 119905er represent the pseudopressure differenceand time corresponding to the (119905Δ119898

1015840)er respectively

For pressure buildup analysis when Δ119905 rarr infin thelnΔ119905(Δ119905 + 119905

119901) rarr 0 (30) can be obtained through

the use of the definitions of dimensionless variables andpseudopressure difference during the early vertical radial flowperiod

119898119894minus 119898119908119891

119905Δ1198981015840= ln 119905119901119863

+ 080907 + 2119878 (30)

The initial reservoir pseudopressure can be determinedby (30) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)erb

(ln 119905119901119863

+ 080907 + 2119878) (31)

where (119905Δ1198981015840)erb represents the actual value on the pressure

buildup log-log plot of Δ1198981015840 versus 119905 during the early vertical

radial flow period

343 Early Linear Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 05 during theearly linear flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables thefollowing can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)119897=

119903119908

119871radic

36120587119870ℎ119905

1206011205831198621199051199032119908

(32)

The horizontal permeability of gas reservoir can bedetermined by (32) This results in

radic119870ℎ=

428 times 10minus2

119902119904119888119879

119871ℎradic120601120583119862119905

[radic119905

(119905Δ1198981015840)]

119897

(33)

Mathematical Problems in Engineering 7

where (119905Δ1198981015840)119897represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early linear flow period

Combining (27) with (33) the vertical permeability canbe obtained This results in

radic119870V = 744 times 10minus2

ℎradic120601120583119862119905

[(119905Δ1198981015840) radic119905]

119897

(119905Δ1198981015840)er (34)

344 Late Horizontal Pseudoradial Flow Period The dimen-sionless bottomhole pseudopressure derivative curve is man-ifested as a horizontal straight line segment with the valueof 05 during the late horizontal pseudoradial flow periodAccording to the expression of dimensionless bottomholepseudopressure derivative during this period and the defini-tions of dimensionless variables (35) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)lr= 05 (35)

The horizontal permeability of gas reservoir can bedetermined by (35) This results in

119870ℎ=

637 times 10minus3

119902119904119888119879

ℎ(119905Δ1198981015840)lr (36)

where (119905Δ1198981015840)lr represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the late horizontal pseudoradial flow

periodFor pressure buildup analysis when Δ119905 rarr infin the

lnΔ119905(Δ119905 + 119905119901) rarr 0 (37) can be obtained through the use

of the definitions of dimensionless variables and pseudopres-sure difference during the late horizontal pseudoradial flowperiod

119898119894minus 119898119908119891

(119905Δ1198981015840)lrb= ln 119903119908119863

119905119901119863

+ 080907 + 2119878 (37)

The initial reservoir pseudopressure can be determinedby (37) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)lrb

(ln 119903119908119863

119905119901119863

+ 080907 + 2119878) (38)

where (119905Δ1198981015840)lrb represents the actual value on the pressure

buildup log-log plot ofΔ1198981015840 versus 119905 during the late horizontal

pseudoradial flow period

345 Pseudosteady Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 1 during thepseudosteady flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables(39) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)pp

=72120587119870

ℎ119905pp

1199032119890ℎ120601120583119862

119905

(39)

The pore volume of gas reservoir can be determined by(39) This results in

1205871199032

119890ℎ120601 =

0905119902119904119888119879

ℎ120583119862119905

119905pp

(119905Δ1198981015840)pp (40)

01020304050

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

0246810

Gas flow rateWater flow rate

qsc(104m

3d)

qw(m

3d)

Figure 8 The gas flow rate and water flow rate curve of Longping 1well

5

10

15

20

25

5

10

15

20

25

Tubing head pressureCasing head pressure

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

Pwh

(MPa

)

Pch

(MPa

)

Figure 9 The tubing head pressure and casing head pressure curveof Longping 1 well

where (119905Δ1198981015840)pp represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the pseudosteady flow period

4 Example Analysis

The Longping 1 well is a horizontal development well inJingBian gas field the well total depth is 4672m the drilledformation name is Majiagou group the mid-depth of reser-voir is 342563m and the well completion system is screencompletion According to the deliverability test during 26ndash29December 2006 the calculated absolute open flow was 9426times 104m3d The commissioning data of Longping 1 well wasin 12May 2007 the initial formation pressure was 2939MPabefore production and the surface tubing pressure and casingpressure were both 2390MPa The production performancecurves of Longping 1 well are shown in Figures 8 and 9respectively

Longping 1 well has been conducted pressure builduptest during 14 August 2007 and 23 October 2007 The gasflow rate of Longping 1 well was 40 times 104m3d before theshut-in The bottomhole pressure recovered from 2238MPato 2783MPa during the pressure buildup test Physical

8 Mathematical Problems in EngineeringΔmΔ

m998400

Δt (h)10minus3

10minus4

10minus5

10minus6

10minus7

10minus2 10minus1 100 101 102 103 104

III

IIIIV

Figure 10 The pressure buildup log-log plot of Longping 1 well

100 101 102 103 104 105 106 107

(tp + Δt)Δt

m(p

wt)

50

46

42

38

34

30

Figure 11 The pressure buildup semilog plot of Longping 1 well

Table 1 Physical parameters of fluid and reservoir

Parameter ValueInitial formation pressure 119901

119894(MPa) 2939

Formation temperature 119879 (∘C) 9580Formation thickness ℎ (m) 631Porosity 120601 () 777Initial water saturation 119878wi () 1360Well radius 119903

119908(m) 00797

Gas gravity 120574119892

0608Gas deviation factor 119885 09738Gas viscosity 120583

119892(mPasdots) 00222

Table 2 Well test analysis results of Longping 1 well

Parameter Parameter valuesWellbore storage coefficient 119862 (m3MPa) 1229Horizontal permeability 119870

ℎ(mD) 7742

Vertical permeability 119870119907(mD) 0039

Flow capacity 119870ℎℎ (mDsdotm) 48857

Skin factor 119878 minus249

Effective horizontal section length 119871 (m) 19837

Reservoir pressure 119901119877(MPa) 28385

parameters of fluid and reservoir are shown in Table 1 Thepressure buildup log-log plot of Longping 1 well is shown inFigure 10

Δt (h)

m(p

ws)

50

0 300 900 1200 1500

46

42

38

34

30

Figure 12 The pressure history matching plot of Longping 1 well

As seen from the contrast between Figure 10 and well testanalysis type curves of horizontal well the pseudopressurebehavior of Longping 1 well manifests four characteristicperiods during the pressure buildup test pure wellborestorage period (I) early vertical radial flow period (II) earlylinear flow period (III) and late horizontal pseudoradial flowperiod (IV)

Using the above characteristic value method of well testanalysis for horizontal gas well well test analysis results ofLongping 1 well are shown in Table 2 The pressure buildupsemilog plot and pressure history matching plot of Longping1 well are shown in Figures 11 and 12 respectively

5 Summary and Conclusions

The four main conclusions and summary of this study are asfollows

(1) On the basis of establishing the mathematical modelsof well test analysis for horizontal gas well and obtain-ing the solutions of the mathematical models severaltype curves which can be used to identify flow regimehave been plotted and the seepage characteristic ofhorizontal gas well has been analyzed

(2) The expressions of dimensionless bottomhole pseu-dopressure and pseudopressure derivative duringeach characteristic period of horizontal gas well havebeen obtained formulas have been developed tocalculate gas reservoir fluid flow parameters

(3) The example study verifies that the characteristicvalue method of well test analysis for horizontal gaswell is reliable and practical

(4) The characteristic value method of well test analysiswhich has been included in the well test analysissoftware at present has been widely used in verticalwell As long as the characteristic straight line seg-ments which are manifested by pressure derivativecurve appear the reservoir fluid flow parameterscan be calculated by the characteristic value methodof well test analysis for vertical well The proposedcharacteristic value method of well test analysis for

Mathematical Problems in Engineering 9

horizontal gas well enriches and develops the well testanalysis theory and method

Nomenclature

119862 Wellbore storage coefficient m3MPa119862119863 Dimensionless wellbore storage coefficient

119862119905 Total compressibility MPaminus1

ℎ Reservoir thickness mℎ119863 Dimensionless reservoir thickness

119868119899 Modified Bessel function of first kind of order 119899

119870ℎ Horizontal permeability mD

119870119899 Modified Bessel function of second kind of order 119899

119870119881 Vertical permeability mD

119871 Horizontal section length m119871119863 Dimensionless horizontal section length

119898(119901) Pseudopressure MPa2mPasdots119898119863 Dimensionless pseudopressure

119898119894 Initial formation pseudopressure MPa2mPasdots

119898119908119863

Dimensionless bottomhole pseudopressure119898119908119891 Flowing wellbore pseudopressure MPa2mPasdots

1198981015840

119908119863 Derivative of119898

119908119863

119898119863 Laplace transform of 119898

119863

119898119908119863

Laplace transform of 119898119908119863

Δ119898 Pseudopressure difference MPaΔ1198981015840 Derivative of Δ119898

119901119888ℎ Casing head pressure MPa

119901119894 Initial formation pressure MPa

119901119877 Reservoir pressure MPa

119901119908ℎ Tubing head pressure MPa

119901119908119904 Flowing wellbore pressure at shut-in MPa

119902119904119888 Gas flow rate 104m3d

119902119908 Water production rate m3d

119903 Radial distance m119903119863 Dimensionless radial distance

119903119890 Outer boundary distance m

119903119890119863 Dimensionless outer boundary distance

119903119908 Wellbore radius m

119903119908119863

Dimensionless wellbore radius119878 Skin factor119904 Laplace transform variable119878119908119894 Initial water saturation fraction

119905 Time hours119905119863 Dimensionless time

119905119901 Production time hours

119905119901119863

Dimensionless production timeΔ119905 Shut-in time hours119879 Formation temperature ∘C119911 Vertical distance m119911119863 Dimensionless vertical distance

119911119908 Horizontal section position m

119911119908119863

Dimensionless horizontal section position119885 Gas deviation factor120576 Tiny variable120601 Porosity fraction120583 Gas viscosity mPasdots120574119892 Gas gravity

Subscripts

119863 Dimensionlesser Earlyerb Early of buildupℎ Horizontal119894 Initial119897 Linearlr Late horizontal pseudo-radiallrb Late horizontal pseudo-radial of builduppp Pseudosteady flow period119905 TotalV Vertical119908119891 Flowing wellbore119908119904 Shut-in wellbore

Conflict of Interests

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

References

[1] R S Nie J C Guo Y L Jia S Q Zhu Z Rao and C G ZhangldquoNew modelling of transient well test and rate decline analysisfor a horizontal well in a multiple-zone reservoirrdquo Journal ofGeophysics and Engineering vol 8 no 3 pp 464ndash476 2011

[2] D R Yuan H Sun Y C Li and M Zhang ldquoMulti-stagefractured tight gas horizontal well test data interpretationstudyrdquo in Proceedings of the International Petroleum TechnologyConference March 2013

[3] M Mavaddat A Soleimani M R Rasaei Y Mavaddat and AMomeni ldquoWell test analysis ofMultipleHydraulically FracturedHorizontal Wells (MHFHW) in gas condensate reservoirsrdquo inProceedings of the SPEIADC Middle East Drilling TechnologyConference and Exhibition (MEDT rsquo11) pp 39ndash48 October 2011

[4] MMirza P AditamaM AL Raqmi and Z Anwar ldquoHorizontalwell stimulation a pilot test in southern Omanrdquo in Proceedingsof the SPE EOR Conference at Oil and GasWest Asia April 2014

[5] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquo Society of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973

[6] M D Clonts and H J Ramey ldquoPressure transient analysis forwells with horizontal drainholesrdquo in Proceedings of the SPECalifornia Regional Meeting Oakland Calif USA April 1986paper SPE 15116

[7] P A Goode and R K M Thambynayagam ldquoPressure draw-down and buildup analysis of horizontal wells in anisotropicmediardquo SPE Formation Evaluation vol 2 no 4 pp 683ndash6971987

[8] E Ozkan and R Rajagopal ldquoHorizontal well pressure analysisrdquoSPE Formation Evaluation pp 567ndash575 1989

[9] A S Odeh and D K Babu ldquoTransient flow behavior ofhorizontal wells Pressure drawdown and buildup analysisrdquo SPEFormation Evaluation vol 5 no 1 pp 7ndash15 1990

[10] L G Thompson and K O Temeng ldquoAutomatic type-curvematching for horizontal wellsrdquo in Proceedings of the ProductionOperation Symposium Oklahoma City Okla USA 1993 paperSPE 25507

10 Mathematical Problems in Engineering

[11] R S Rahavan C C Chen and B Agarwal ldquoAn analysis ofhorizontal wells intercepted by multiple fracturesrdquo SPE27652(Sep pp 235ndash245 1997

[12] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal wells in closed systemsrdquo in Proceed-ings of the SPE International Improved Oil Recovery Conferencein Asia Pacific (IIORC rsquo03) pp 295ndash307 Alberta CanadaOctober 2003

[13] T Whittle H Jiang S Young and A C Gringarten ldquoWellproduction forecasting by extrapolation of the deconvolutionof the well test pressure transientsrdquo in Proceedings of the SPEEUROPECEAGE Conference Amsterdam The NetherlandJune 2009 SPE 122299 paper

[14] C C Miller A B Dyes and C A Hutchinson ldquoThe estimationof permeability and reservoir pressure from bottom-hole pres-sure buildup Characteristicrdquo Journal of Petroleum Technologypp 91ndash104 1950

[15] D R Horner ldquoPressure buildup in wellsrdquo in Proceedings of the3rd World Petroleum Congress Hague The Netherlands May-June 1951 paper SPE 4135

[16] R Raghavan ldquoThe effect of producing time on type curveanalysisrdquo Journal of Petroleum Technology vol 32 no 6 pp1053ndash1064 1980

[17] D Bourdet ldquoA new set of type curves simplifies well testanalysisrdquoWorld Oil pp 95ndash106 1983

[18] T M Hegre ldquoHydraulically fractured horizontal well simu-lationrdquo in Proceedings of the NPFSPE European ProductionOperations Conference (EPOC rsquo96) pp 59ndash62 April 1996

[19] R G Agarwal R Al-Hussaing and H J Ramey ldquoAn investiga-tion of wellbore storage and skin effect in unsteady liquid flowI Analytical treatmentrdquo Society of Petroleum Engineers Journalvol 10 no 3 pp 291ndash297 1970

[20] H J J Ramey and R G Agarwal ldquoAnnulus unloading ratesas influenced by wellbore storage and skin effectrdquo Society ofPetroleum Engineers Journal vol 12 no 5 pp 453ndash462 1972

[21] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems part 1-analytical considerationsrdquo SPEForma-tion Evaluation 1991

[22] L G Thompson Analysis of variable rate pressure data usingduhamels principle [PhD thesis] University of Tulsa TulsaOkla USA 1985

[23] H Stehfest ldquoNumerical inversion of laplace transformsrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

[24] N Al-Ajmi M Ahmadi E Ozkan and H Kazemi ldquoNumericalinversion of laplace transforms in the solution of transient flowproblemswith discontinuitiesrdquo inProceedings of the SPEAnnualTechnical Conference and Exhibition Paper SPE 116255 pp 21ndash24 Denver Colorado 2008

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

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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

6 Mathematical Problems in Engineering

in horizontal direction The smaller the distance of the outerboundary the sooner the appearance of the steady state flowperiod The greater the distance of the outer boundary thelater the appearance of the steady state flow period

34 Characteristic Value Method of Well Test Analysis forHorizontal GasWell The characteristic value method of welltest analysis for horizontal gas well can determine the gasreservoir fluid flowparameters according to the characteristiclines which are manifested by pseudopressure derivativecurve of each flow period on the log-log plot

341 PureWellbore Storage Period The characteristic of purewellbore storage period of horizontal well is manifested asa straight line segment with a slope of 1 on the log-log plotof 119898119908119863

1198981015840119908119863

versus 119905119863119862119863 The expression of dimensionless

bottomhole pseudopressure during this period is

119898119908119863

=119905119863

119862119863

(22)

Equation (22) can be converted to dimensional formand then according to the time and pressure data duringpure wellbore storage period the method to determine thewellbore storage coefficient can be obtained By plottingthe log-log plot of Δ119898 Δ119898

1015840 versus 119905 the wellbore storagecoefficient can be determined by the straight line segmentwith a slope of 1 on the log-log plot

The following can be obtained from the definitions ofdimensionless variables

119905119863

119862119863

=36119870ℎ1199051206011205831198881199051199032

119908

1198622120587120601119888119905ℎ1198712

=72120587119870

ℎℎ1198712

1205831199032119908

119905

119862 (23)

According to (22) (23) and the definition of dimension-less pseudopressure the wellbore storage coefficient can beobtained This results in

119862 =0288119902

119904119888119879

120583

1198712

1199032119908

119905

Δ119898 (24)

where 119905Δ119898 represents the actual value on the log-log plot ofΔ119898 Δ119898

1015840 versus 119905 during the pure wellbore storage period

342 Early Vertical Radial Flow Period

The Determination of Geometric Mean Permeability Thedimensionless bottomhole pseudopressure derivative curveis manifested as a horizontal straight line segment with thevalue of 1(4119871

119863) during the early vertical radial flow period

The expression of dimensionless bottomhole pseudopressurederivative during this period is

1198981015840

119908119863=

119889119898119908119863

119889 ln (119905119863119862119863)=

1

4119871119863

(25)

According to the definitions of dimensionless variablesthe dimensional form of (25) can be obtained This results in

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)er

=1

4 (119871ℎ)radic119870V119870ℎ (26)

The geometric mean permeability of gas reservoir can bedetermined by (26) This results in

radic119870ℎ119870V =

3185 times 10minus3

119902119904119888119879

119871(119905Δ1198981015840)er (27)

where (119905Δ1198981015840)er represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early vertical radial flow period

The Determination of Skin Factor and Initial Reservoir Pres-sure The expression of dimensionless bottomhole pseudo-pressure during the early vertical radial flow period is

119898119908119863

=1

4119871119863

[ln 225 (119905119863

119862119863

) + ln1198621198631198902119878] (28)

According to the definitions of dimensionless variablesand (28) the skin factor can be obtained This results in

119878 = 05 [Δ119898er

(119905Δ1198981015840)erminus ln

119870ℎ119905er

1206011205831198621199051199032119908

minus 080907] (29)

where Δ119898er and 119905er represent the pseudopressure differenceand time corresponding to the (119905Δ119898

1015840)er respectively

For pressure buildup analysis when Δ119905 rarr infin thelnΔ119905(Δ119905 + 119905

119901) rarr 0 (30) can be obtained through

the use of the definitions of dimensionless variables andpseudopressure difference during the early vertical radial flowperiod

119898119894minus 119898119908119891

119905Δ1198981015840= ln 119905119901119863

+ 080907 + 2119878 (30)

The initial reservoir pseudopressure can be determinedby (30) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)erb

(ln 119905119901119863

+ 080907 + 2119878) (31)

where (119905Δ1198981015840)erb represents the actual value on the pressure

buildup log-log plot of Δ1198981015840 versus 119905 during the early vertical

radial flow period

343 Early Linear Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 05 during theearly linear flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables thefollowing can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)119897=

119903119908

119871radic

36120587119870ℎ119905

1206011205831198621199051199032119908

(32)

The horizontal permeability of gas reservoir can bedetermined by (32) This results in

radic119870ℎ=

428 times 10minus2

119902119904119888119879

119871ℎradic120601120583119862119905

[radic119905

(119905Δ1198981015840)]

119897

(33)

Mathematical Problems in Engineering 7

where (119905Δ1198981015840)119897represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early linear flow period

Combining (27) with (33) the vertical permeability canbe obtained This results in

radic119870V = 744 times 10minus2

ℎradic120601120583119862119905

[(119905Δ1198981015840) radic119905]

119897

(119905Δ1198981015840)er (34)

344 Late Horizontal Pseudoradial Flow Period The dimen-sionless bottomhole pseudopressure derivative curve is man-ifested as a horizontal straight line segment with the valueof 05 during the late horizontal pseudoradial flow periodAccording to the expression of dimensionless bottomholepseudopressure derivative during this period and the defini-tions of dimensionless variables (35) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)lr= 05 (35)

The horizontal permeability of gas reservoir can bedetermined by (35) This results in

119870ℎ=

637 times 10minus3

119902119904119888119879

ℎ(119905Δ1198981015840)lr (36)

where (119905Δ1198981015840)lr represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the late horizontal pseudoradial flow

periodFor pressure buildup analysis when Δ119905 rarr infin the

lnΔ119905(Δ119905 + 119905119901) rarr 0 (37) can be obtained through the use

of the definitions of dimensionless variables and pseudopres-sure difference during the late horizontal pseudoradial flowperiod

119898119894minus 119898119908119891

(119905Δ1198981015840)lrb= ln 119903119908119863

119905119901119863

+ 080907 + 2119878 (37)

The initial reservoir pseudopressure can be determinedby (37) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)lrb

(ln 119903119908119863

119905119901119863

+ 080907 + 2119878) (38)

where (119905Δ1198981015840)lrb represents the actual value on the pressure

buildup log-log plot ofΔ1198981015840 versus 119905 during the late horizontal

pseudoradial flow period

345 Pseudosteady Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 1 during thepseudosteady flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables(39) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)pp

=72120587119870

ℎ119905pp

1199032119890ℎ120601120583119862

119905

(39)

The pore volume of gas reservoir can be determined by(39) This results in

1205871199032

119890ℎ120601 =

0905119902119904119888119879

ℎ120583119862119905

119905pp

(119905Δ1198981015840)pp (40)

01020304050

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

0246810

Gas flow rateWater flow rate

qsc(104m

3d)

qw(m

3d)

Figure 8 The gas flow rate and water flow rate curve of Longping 1well

5

10

15

20

25

5

10

15

20

25

Tubing head pressureCasing head pressure

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

Pwh

(MPa

)

Pch

(MPa

)

Figure 9 The tubing head pressure and casing head pressure curveof Longping 1 well

where (119905Δ1198981015840)pp represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the pseudosteady flow period

4 Example Analysis

The Longping 1 well is a horizontal development well inJingBian gas field the well total depth is 4672m the drilledformation name is Majiagou group the mid-depth of reser-voir is 342563m and the well completion system is screencompletion According to the deliverability test during 26ndash29December 2006 the calculated absolute open flow was 9426times 104m3d The commissioning data of Longping 1 well wasin 12May 2007 the initial formation pressure was 2939MPabefore production and the surface tubing pressure and casingpressure were both 2390MPa The production performancecurves of Longping 1 well are shown in Figures 8 and 9respectively

Longping 1 well has been conducted pressure builduptest during 14 August 2007 and 23 October 2007 The gasflow rate of Longping 1 well was 40 times 104m3d before theshut-in The bottomhole pressure recovered from 2238MPato 2783MPa during the pressure buildup test Physical

8 Mathematical Problems in EngineeringΔmΔ

m998400

Δt (h)10minus3

10minus4

10minus5

10minus6

10minus7

10minus2 10minus1 100 101 102 103 104

III

IIIIV

Figure 10 The pressure buildup log-log plot of Longping 1 well

100 101 102 103 104 105 106 107

(tp + Δt)Δt

m(p

wt)

50

46

42

38

34

30

Figure 11 The pressure buildup semilog plot of Longping 1 well

Table 1 Physical parameters of fluid and reservoir

Parameter ValueInitial formation pressure 119901

119894(MPa) 2939

Formation temperature 119879 (∘C) 9580Formation thickness ℎ (m) 631Porosity 120601 () 777Initial water saturation 119878wi () 1360Well radius 119903

119908(m) 00797

Gas gravity 120574119892

0608Gas deviation factor 119885 09738Gas viscosity 120583

119892(mPasdots) 00222

Table 2 Well test analysis results of Longping 1 well

Parameter Parameter valuesWellbore storage coefficient 119862 (m3MPa) 1229Horizontal permeability 119870

ℎ(mD) 7742

Vertical permeability 119870119907(mD) 0039

Flow capacity 119870ℎℎ (mDsdotm) 48857

Skin factor 119878 minus249

Effective horizontal section length 119871 (m) 19837

Reservoir pressure 119901119877(MPa) 28385

parameters of fluid and reservoir are shown in Table 1 Thepressure buildup log-log plot of Longping 1 well is shown inFigure 10

Δt (h)

m(p

ws)

50

0 300 900 1200 1500

46

42

38

34

30

Figure 12 The pressure history matching plot of Longping 1 well

As seen from the contrast between Figure 10 and well testanalysis type curves of horizontal well the pseudopressurebehavior of Longping 1 well manifests four characteristicperiods during the pressure buildup test pure wellborestorage period (I) early vertical radial flow period (II) earlylinear flow period (III) and late horizontal pseudoradial flowperiod (IV)

Using the above characteristic value method of well testanalysis for horizontal gas well well test analysis results ofLongping 1 well are shown in Table 2 The pressure buildupsemilog plot and pressure history matching plot of Longping1 well are shown in Figures 11 and 12 respectively

5 Summary and Conclusions

The four main conclusions and summary of this study are asfollows

(1) On the basis of establishing the mathematical modelsof well test analysis for horizontal gas well and obtain-ing the solutions of the mathematical models severaltype curves which can be used to identify flow regimehave been plotted and the seepage characteristic ofhorizontal gas well has been analyzed

(2) The expressions of dimensionless bottomhole pseu-dopressure and pseudopressure derivative duringeach characteristic period of horizontal gas well havebeen obtained formulas have been developed tocalculate gas reservoir fluid flow parameters

(3) The example study verifies that the characteristicvalue method of well test analysis for horizontal gaswell is reliable and practical

(4) The characteristic value method of well test analysiswhich has been included in the well test analysissoftware at present has been widely used in verticalwell As long as the characteristic straight line seg-ments which are manifested by pressure derivativecurve appear the reservoir fluid flow parameterscan be calculated by the characteristic value methodof well test analysis for vertical well The proposedcharacteristic value method of well test analysis for

Mathematical Problems in Engineering 9

horizontal gas well enriches and develops the well testanalysis theory and method

Nomenclature

119862 Wellbore storage coefficient m3MPa119862119863 Dimensionless wellbore storage coefficient

119862119905 Total compressibility MPaminus1

ℎ Reservoir thickness mℎ119863 Dimensionless reservoir thickness

119868119899 Modified Bessel function of first kind of order 119899

119870ℎ Horizontal permeability mD

119870119899 Modified Bessel function of second kind of order 119899

119870119881 Vertical permeability mD

119871 Horizontal section length m119871119863 Dimensionless horizontal section length

119898(119901) Pseudopressure MPa2mPasdots119898119863 Dimensionless pseudopressure

119898119894 Initial formation pseudopressure MPa2mPasdots

119898119908119863

Dimensionless bottomhole pseudopressure119898119908119891 Flowing wellbore pseudopressure MPa2mPasdots

1198981015840

119908119863 Derivative of119898

119908119863

119898119863 Laplace transform of 119898

119863

119898119908119863

Laplace transform of 119898119908119863

Δ119898 Pseudopressure difference MPaΔ1198981015840 Derivative of Δ119898

119901119888ℎ Casing head pressure MPa

119901119894 Initial formation pressure MPa

119901119877 Reservoir pressure MPa

119901119908ℎ Tubing head pressure MPa

119901119908119904 Flowing wellbore pressure at shut-in MPa

119902119904119888 Gas flow rate 104m3d

119902119908 Water production rate m3d

119903 Radial distance m119903119863 Dimensionless radial distance

119903119890 Outer boundary distance m

119903119890119863 Dimensionless outer boundary distance

119903119908 Wellbore radius m

119903119908119863

Dimensionless wellbore radius119878 Skin factor119904 Laplace transform variable119878119908119894 Initial water saturation fraction

119905 Time hours119905119863 Dimensionless time

119905119901 Production time hours

119905119901119863

Dimensionless production timeΔ119905 Shut-in time hours119879 Formation temperature ∘C119911 Vertical distance m119911119863 Dimensionless vertical distance

119911119908 Horizontal section position m

119911119908119863

Dimensionless horizontal section position119885 Gas deviation factor120576 Tiny variable120601 Porosity fraction120583 Gas viscosity mPasdots120574119892 Gas gravity

Subscripts

119863 Dimensionlesser Earlyerb Early of buildupℎ Horizontal119894 Initial119897 Linearlr Late horizontal pseudo-radiallrb Late horizontal pseudo-radial of builduppp Pseudosteady flow period119905 TotalV Vertical119908119891 Flowing wellbore119908119904 Shut-in wellbore

Conflict of Interests

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

References

[1] R S Nie J C Guo Y L Jia S Q Zhu Z Rao and C G ZhangldquoNew modelling of transient well test and rate decline analysisfor a horizontal well in a multiple-zone reservoirrdquo Journal ofGeophysics and Engineering vol 8 no 3 pp 464ndash476 2011

[2] D R Yuan H Sun Y C Li and M Zhang ldquoMulti-stagefractured tight gas horizontal well test data interpretationstudyrdquo in Proceedings of the International Petroleum TechnologyConference March 2013

[3] M Mavaddat A Soleimani M R Rasaei Y Mavaddat and AMomeni ldquoWell test analysis ofMultipleHydraulically FracturedHorizontal Wells (MHFHW) in gas condensate reservoirsrdquo inProceedings of the SPEIADC Middle East Drilling TechnologyConference and Exhibition (MEDT rsquo11) pp 39ndash48 October 2011

[4] MMirza P AditamaM AL Raqmi and Z Anwar ldquoHorizontalwell stimulation a pilot test in southern Omanrdquo in Proceedingsof the SPE EOR Conference at Oil and GasWest Asia April 2014

[5] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquo Society of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973

[6] M D Clonts and H J Ramey ldquoPressure transient analysis forwells with horizontal drainholesrdquo in Proceedings of the SPECalifornia Regional Meeting Oakland Calif USA April 1986paper SPE 15116

[7] P A Goode and R K M Thambynayagam ldquoPressure draw-down and buildup analysis of horizontal wells in anisotropicmediardquo SPE Formation Evaluation vol 2 no 4 pp 683ndash6971987

[8] E Ozkan and R Rajagopal ldquoHorizontal well pressure analysisrdquoSPE Formation Evaluation pp 567ndash575 1989

[9] A S Odeh and D K Babu ldquoTransient flow behavior ofhorizontal wells Pressure drawdown and buildup analysisrdquo SPEFormation Evaluation vol 5 no 1 pp 7ndash15 1990

[10] L G Thompson and K O Temeng ldquoAutomatic type-curvematching for horizontal wellsrdquo in Proceedings of the ProductionOperation Symposium Oklahoma City Okla USA 1993 paperSPE 25507

10 Mathematical Problems in Engineering

[11] R S Rahavan C C Chen and B Agarwal ldquoAn analysis ofhorizontal wells intercepted by multiple fracturesrdquo SPE27652(Sep pp 235ndash245 1997

[12] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal wells in closed systemsrdquo in Proceed-ings of the SPE International Improved Oil Recovery Conferencein Asia Pacific (IIORC rsquo03) pp 295ndash307 Alberta CanadaOctober 2003

[13] T Whittle H Jiang S Young and A C Gringarten ldquoWellproduction forecasting by extrapolation of the deconvolutionof the well test pressure transientsrdquo in Proceedings of the SPEEUROPECEAGE Conference Amsterdam The NetherlandJune 2009 SPE 122299 paper

[14] C C Miller A B Dyes and C A Hutchinson ldquoThe estimationof permeability and reservoir pressure from bottom-hole pres-sure buildup Characteristicrdquo Journal of Petroleum Technologypp 91ndash104 1950

[15] D R Horner ldquoPressure buildup in wellsrdquo in Proceedings of the3rd World Petroleum Congress Hague The Netherlands May-June 1951 paper SPE 4135

[16] R Raghavan ldquoThe effect of producing time on type curveanalysisrdquo Journal of Petroleum Technology vol 32 no 6 pp1053ndash1064 1980

[17] D Bourdet ldquoA new set of type curves simplifies well testanalysisrdquoWorld Oil pp 95ndash106 1983

[18] T M Hegre ldquoHydraulically fractured horizontal well simu-lationrdquo in Proceedings of the NPFSPE European ProductionOperations Conference (EPOC rsquo96) pp 59ndash62 April 1996

[19] R G Agarwal R Al-Hussaing and H J Ramey ldquoAn investiga-tion of wellbore storage and skin effect in unsteady liquid flowI Analytical treatmentrdquo Society of Petroleum Engineers Journalvol 10 no 3 pp 291ndash297 1970

[20] H J J Ramey and R G Agarwal ldquoAnnulus unloading ratesas influenced by wellbore storage and skin effectrdquo Society ofPetroleum Engineers Journal vol 12 no 5 pp 453ndash462 1972

[21] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems part 1-analytical considerationsrdquo SPEForma-tion Evaluation 1991

[22] L G Thompson Analysis of variable rate pressure data usingduhamels principle [PhD thesis] University of Tulsa TulsaOkla USA 1985

[23] H Stehfest ldquoNumerical inversion of laplace transformsrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

[24] N Al-Ajmi M Ahmadi E Ozkan and H Kazemi ldquoNumericalinversion of laplace transforms in the solution of transient flowproblemswith discontinuitiesrdquo inProceedings of the SPEAnnualTechnical Conference and Exhibition Paper SPE 116255 pp 21ndash24 Denver Colorado 2008

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

Mathematical Problems in Engineering 7

where (119905Δ1198981015840)119897represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the early linear flow period

Combining (27) with (33) the vertical permeability canbe obtained This results in

radic119870V = 744 times 10minus2

ℎradic120601120583119862119905

[(119905Δ1198981015840) radic119905]

119897

(119905Δ1198981015840)er (34)

344 Late Horizontal Pseudoradial Flow Period The dimen-sionless bottomhole pseudopressure derivative curve is man-ifested as a horizontal straight line segment with the valueof 05 during the late horizontal pseudoradial flow periodAccording to the expression of dimensionless bottomholepseudopressure derivative during this period and the defini-tions of dimensionless variables (35) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)lr= 05 (35)

The horizontal permeability of gas reservoir can bedetermined by (35) This results in

119870ℎ=

637 times 10minus3

119902119904119888119879

ℎ(119905Δ1198981015840)lr (36)

where (119905Δ1198981015840)lr represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the late horizontal pseudoradial flow

periodFor pressure buildup analysis when Δ119905 rarr infin the

lnΔ119905(Δ119905 + 119905119901) rarr 0 (37) can be obtained through the use

of the definitions of dimensionless variables and pseudopres-sure difference during the late horizontal pseudoradial flowperiod

119898119894minus 119898119908119891

(119905Δ1198981015840)lrb= ln 119903119908119863

119905119901119863

+ 080907 + 2119878 (37)

The initial reservoir pseudopressure can be determinedby (37) This results in

119898119894= 119898119908119891

+ (119905Δ1198981015840)lrb

(ln 119903119908119863

119905119901119863

+ 080907 + 2119878) (38)

where (119905Δ1198981015840)lrb represents the actual value on the pressure

buildup log-log plot ofΔ1198981015840 versus 119905 during the late horizontal

pseudoradial flow period

345 Pseudosteady Flow Period The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 1 during thepseudosteady flow period According to the expression ofdimensionless bottomhole pseudopressure derivative duringthis period and the definitions of dimensionless variables(39) can be obtained

78489119870ℎℎ

119902119904119888119879

(119905Δ1198981015840)pp

=72120587119870

ℎ119905pp

1199032119890ℎ120601120583119862

119905

(39)

The pore volume of gas reservoir can be determined by(39) This results in

1205871199032

119890ℎ120601 =

0905119902119904119888119879

ℎ120583119862119905

119905pp

(119905Δ1198981015840)pp (40)

01020304050

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

0246810

Gas flow rateWater flow rate

qsc(104m

3d)

qw(m

3d)

Figure 8 The gas flow rate and water flow rate curve of Longping 1well

5

10

15

20

25

5

10

15

20

25

Tubing head pressureCasing head pressure

2007

05

12

2007

08

24

2007

12

06

2008

03

19

2008

07

01

2008

10

13

Date

Pwh

(MPa

)

Pch

(MPa

)

Figure 9 The tubing head pressure and casing head pressure curveof Longping 1 well

where (119905Δ1198981015840)pp represents the actual value on the log-log plot

of Δ1198981015840 versus 119905 during the pseudosteady flow period

4 Example Analysis

The Longping 1 well is a horizontal development well inJingBian gas field the well total depth is 4672m the drilledformation name is Majiagou group the mid-depth of reser-voir is 342563m and the well completion system is screencompletion According to the deliverability test during 26ndash29December 2006 the calculated absolute open flow was 9426times 104m3d The commissioning data of Longping 1 well wasin 12May 2007 the initial formation pressure was 2939MPabefore production and the surface tubing pressure and casingpressure were both 2390MPa The production performancecurves of Longping 1 well are shown in Figures 8 and 9respectively

Longping 1 well has been conducted pressure builduptest during 14 August 2007 and 23 October 2007 The gasflow rate of Longping 1 well was 40 times 104m3d before theshut-in The bottomhole pressure recovered from 2238MPato 2783MPa during the pressure buildup test Physical

8 Mathematical Problems in EngineeringΔmΔ

m998400

Δt (h)10minus3

10minus4

10minus5

10minus6

10minus7

10minus2 10minus1 100 101 102 103 104

III

IIIIV

Figure 10 The pressure buildup log-log plot of Longping 1 well

100 101 102 103 104 105 106 107

(tp + Δt)Δt

m(p

wt)

50

46

42

38

34

30

Figure 11 The pressure buildup semilog plot of Longping 1 well

Table 1 Physical parameters of fluid and reservoir

Parameter ValueInitial formation pressure 119901

119894(MPa) 2939

Formation temperature 119879 (∘C) 9580Formation thickness ℎ (m) 631Porosity 120601 () 777Initial water saturation 119878wi () 1360Well radius 119903

119908(m) 00797

Gas gravity 120574119892

0608Gas deviation factor 119885 09738Gas viscosity 120583

119892(mPasdots) 00222

Table 2 Well test analysis results of Longping 1 well

Parameter Parameter valuesWellbore storage coefficient 119862 (m3MPa) 1229Horizontal permeability 119870

ℎ(mD) 7742

Vertical permeability 119870119907(mD) 0039

Flow capacity 119870ℎℎ (mDsdotm) 48857

Skin factor 119878 minus249

Effective horizontal section length 119871 (m) 19837

Reservoir pressure 119901119877(MPa) 28385

parameters of fluid and reservoir are shown in Table 1 Thepressure buildup log-log plot of Longping 1 well is shown inFigure 10

Δt (h)

m(p

ws)

50

0 300 900 1200 1500

46

42

38

34

30

Figure 12 The pressure history matching plot of Longping 1 well

As seen from the contrast between Figure 10 and well testanalysis type curves of horizontal well the pseudopressurebehavior of Longping 1 well manifests four characteristicperiods during the pressure buildup test pure wellborestorage period (I) early vertical radial flow period (II) earlylinear flow period (III) and late horizontal pseudoradial flowperiod (IV)

Using the above characteristic value method of well testanalysis for horizontal gas well well test analysis results ofLongping 1 well are shown in Table 2 The pressure buildupsemilog plot and pressure history matching plot of Longping1 well are shown in Figures 11 and 12 respectively

5 Summary and Conclusions

The four main conclusions and summary of this study are asfollows

(1) On the basis of establishing the mathematical modelsof well test analysis for horizontal gas well and obtain-ing the solutions of the mathematical models severaltype curves which can be used to identify flow regimehave been plotted and the seepage characteristic ofhorizontal gas well has been analyzed

(2) The expressions of dimensionless bottomhole pseu-dopressure and pseudopressure derivative duringeach characteristic period of horizontal gas well havebeen obtained formulas have been developed tocalculate gas reservoir fluid flow parameters

(3) The example study verifies that the characteristicvalue method of well test analysis for horizontal gaswell is reliable and practical

(4) The characteristic value method of well test analysiswhich has been included in the well test analysissoftware at present has been widely used in verticalwell As long as the characteristic straight line seg-ments which are manifested by pressure derivativecurve appear the reservoir fluid flow parameterscan be calculated by the characteristic value methodof well test analysis for vertical well The proposedcharacteristic value method of well test analysis for

Mathematical Problems in Engineering 9

horizontal gas well enriches and develops the well testanalysis theory and method

Nomenclature

119862 Wellbore storage coefficient m3MPa119862119863 Dimensionless wellbore storage coefficient

119862119905 Total compressibility MPaminus1

ℎ Reservoir thickness mℎ119863 Dimensionless reservoir thickness

119868119899 Modified Bessel function of first kind of order 119899

119870ℎ Horizontal permeability mD

119870119899 Modified Bessel function of second kind of order 119899

119870119881 Vertical permeability mD

119871 Horizontal section length m119871119863 Dimensionless horizontal section length

119898(119901) Pseudopressure MPa2mPasdots119898119863 Dimensionless pseudopressure

119898119894 Initial formation pseudopressure MPa2mPasdots

119898119908119863

Dimensionless bottomhole pseudopressure119898119908119891 Flowing wellbore pseudopressure MPa2mPasdots

1198981015840

119908119863 Derivative of119898

119908119863

119898119863 Laplace transform of 119898

119863

119898119908119863

Laplace transform of 119898119908119863

Δ119898 Pseudopressure difference MPaΔ1198981015840 Derivative of Δ119898

119901119888ℎ Casing head pressure MPa

119901119894 Initial formation pressure MPa

119901119877 Reservoir pressure MPa

119901119908ℎ Tubing head pressure MPa

119901119908119904 Flowing wellbore pressure at shut-in MPa

119902119904119888 Gas flow rate 104m3d

119902119908 Water production rate m3d

119903 Radial distance m119903119863 Dimensionless radial distance

119903119890 Outer boundary distance m

119903119890119863 Dimensionless outer boundary distance

119903119908 Wellbore radius m

119903119908119863

Dimensionless wellbore radius119878 Skin factor119904 Laplace transform variable119878119908119894 Initial water saturation fraction

119905 Time hours119905119863 Dimensionless time

119905119901 Production time hours

119905119901119863

Dimensionless production timeΔ119905 Shut-in time hours119879 Formation temperature ∘C119911 Vertical distance m119911119863 Dimensionless vertical distance

119911119908 Horizontal section position m

119911119908119863

Dimensionless horizontal section position119885 Gas deviation factor120576 Tiny variable120601 Porosity fraction120583 Gas viscosity mPasdots120574119892 Gas gravity

Subscripts

119863 Dimensionlesser Earlyerb Early of buildupℎ Horizontal119894 Initial119897 Linearlr Late horizontal pseudo-radiallrb Late horizontal pseudo-radial of builduppp Pseudosteady flow period119905 TotalV Vertical119908119891 Flowing wellbore119908119904 Shut-in wellbore

Conflict of Interests

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

References

[1] R S Nie J C Guo Y L Jia S Q Zhu Z Rao and C G ZhangldquoNew modelling of transient well test and rate decline analysisfor a horizontal well in a multiple-zone reservoirrdquo Journal ofGeophysics and Engineering vol 8 no 3 pp 464ndash476 2011

[2] D R Yuan H Sun Y C Li and M Zhang ldquoMulti-stagefractured tight gas horizontal well test data interpretationstudyrdquo in Proceedings of the International Petroleum TechnologyConference March 2013

[3] M Mavaddat A Soleimani M R Rasaei Y Mavaddat and AMomeni ldquoWell test analysis ofMultipleHydraulically FracturedHorizontal Wells (MHFHW) in gas condensate reservoirsrdquo inProceedings of the SPEIADC Middle East Drilling TechnologyConference and Exhibition (MEDT rsquo11) pp 39ndash48 October 2011

[4] MMirza P AditamaM AL Raqmi and Z Anwar ldquoHorizontalwell stimulation a pilot test in southern Omanrdquo in Proceedingsof the SPE EOR Conference at Oil and GasWest Asia April 2014

[5] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquo Society of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973

[6] M D Clonts and H J Ramey ldquoPressure transient analysis forwells with horizontal drainholesrdquo in Proceedings of the SPECalifornia Regional Meeting Oakland Calif USA April 1986paper SPE 15116

[7] P A Goode and R K M Thambynayagam ldquoPressure draw-down and buildup analysis of horizontal wells in anisotropicmediardquo SPE Formation Evaluation vol 2 no 4 pp 683ndash6971987

[8] E Ozkan and R Rajagopal ldquoHorizontal well pressure analysisrdquoSPE Formation Evaluation pp 567ndash575 1989

[9] A S Odeh and D K Babu ldquoTransient flow behavior ofhorizontal wells Pressure drawdown and buildup analysisrdquo SPEFormation Evaluation vol 5 no 1 pp 7ndash15 1990

[10] L G Thompson and K O Temeng ldquoAutomatic type-curvematching for horizontal wellsrdquo in Proceedings of the ProductionOperation Symposium Oklahoma City Okla USA 1993 paperSPE 25507

10 Mathematical Problems in Engineering

[11] R S Rahavan C C Chen and B Agarwal ldquoAn analysis ofhorizontal wells intercepted by multiple fracturesrdquo SPE27652(Sep pp 235ndash245 1997

[12] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal wells in closed systemsrdquo in Proceed-ings of the SPE International Improved Oil Recovery Conferencein Asia Pacific (IIORC rsquo03) pp 295ndash307 Alberta CanadaOctober 2003

[13] T Whittle H Jiang S Young and A C Gringarten ldquoWellproduction forecasting by extrapolation of the deconvolutionof the well test pressure transientsrdquo in Proceedings of the SPEEUROPECEAGE Conference Amsterdam The NetherlandJune 2009 SPE 122299 paper

[14] C C Miller A B Dyes and C A Hutchinson ldquoThe estimationof permeability and reservoir pressure from bottom-hole pres-sure buildup Characteristicrdquo Journal of Petroleum Technologypp 91ndash104 1950

[15] D R Horner ldquoPressure buildup in wellsrdquo in Proceedings of the3rd World Petroleum Congress Hague The Netherlands May-June 1951 paper SPE 4135

[16] R Raghavan ldquoThe effect of producing time on type curveanalysisrdquo Journal of Petroleum Technology vol 32 no 6 pp1053ndash1064 1980

[17] D Bourdet ldquoA new set of type curves simplifies well testanalysisrdquoWorld Oil pp 95ndash106 1983

[18] T M Hegre ldquoHydraulically fractured horizontal well simu-lationrdquo in Proceedings of the NPFSPE European ProductionOperations Conference (EPOC rsquo96) pp 59ndash62 April 1996

[19] R G Agarwal R Al-Hussaing and H J Ramey ldquoAn investiga-tion of wellbore storage and skin effect in unsteady liquid flowI Analytical treatmentrdquo Society of Petroleum Engineers Journalvol 10 no 3 pp 291ndash297 1970

[20] H J J Ramey and R G Agarwal ldquoAnnulus unloading ratesas influenced by wellbore storage and skin effectrdquo Society ofPetroleum Engineers Journal vol 12 no 5 pp 453ndash462 1972

[21] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems part 1-analytical considerationsrdquo SPEForma-tion Evaluation 1991

[22] L G Thompson Analysis of variable rate pressure data usingduhamels principle [PhD thesis] University of Tulsa TulsaOkla USA 1985

[23] H Stehfest ldquoNumerical inversion of laplace transformsrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

[24] N Al-Ajmi M Ahmadi E Ozkan and H Kazemi ldquoNumericalinversion of laplace transforms in the solution of transient flowproblemswith discontinuitiesrdquo inProceedings of the SPEAnnualTechnical Conference and Exhibition Paper SPE 116255 pp 21ndash24 Denver Colorado 2008

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 Mathematical Problems in EngineeringΔmΔ

m998400

Δt (h)10minus3

10minus4

10minus5

10minus6

10minus7

10minus2 10minus1 100 101 102 103 104

III

IIIIV

Figure 10 The pressure buildup log-log plot of Longping 1 well

100 101 102 103 104 105 106 107

(tp + Δt)Δt

m(p

wt)

50

46

42

38

34

30

Figure 11 The pressure buildup semilog plot of Longping 1 well

Table 1 Physical parameters of fluid and reservoir

Parameter ValueInitial formation pressure 119901

119894(MPa) 2939

Formation temperature 119879 (∘C) 9580Formation thickness ℎ (m) 631Porosity 120601 () 777Initial water saturation 119878wi () 1360Well radius 119903

119908(m) 00797

Gas gravity 120574119892

0608Gas deviation factor 119885 09738Gas viscosity 120583

119892(mPasdots) 00222

Table 2 Well test analysis results of Longping 1 well

Parameter Parameter valuesWellbore storage coefficient 119862 (m3MPa) 1229Horizontal permeability 119870

ℎ(mD) 7742

Vertical permeability 119870119907(mD) 0039

Flow capacity 119870ℎℎ (mDsdotm) 48857

Skin factor 119878 minus249

Effective horizontal section length 119871 (m) 19837

Reservoir pressure 119901119877(MPa) 28385

parameters of fluid and reservoir are shown in Table 1 Thepressure buildup log-log plot of Longping 1 well is shown inFigure 10

Δt (h)

m(p

ws)

50

0 300 900 1200 1500

46

42

38

34

30

Figure 12 The pressure history matching plot of Longping 1 well

As seen from the contrast between Figure 10 and well testanalysis type curves of horizontal well the pseudopressurebehavior of Longping 1 well manifests four characteristicperiods during the pressure buildup test pure wellborestorage period (I) early vertical radial flow period (II) earlylinear flow period (III) and late horizontal pseudoradial flowperiod (IV)

Using the above characteristic value method of well testanalysis for horizontal gas well well test analysis results ofLongping 1 well are shown in Table 2 The pressure buildupsemilog plot and pressure history matching plot of Longping1 well are shown in Figures 11 and 12 respectively

5 Summary and Conclusions

The four main conclusions and summary of this study are asfollows

(1) On the basis of establishing the mathematical modelsof well test analysis for horizontal gas well and obtain-ing the solutions of the mathematical models severaltype curves which can be used to identify flow regimehave been plotted and the seepage characteristic ofhorizontal gas well has been analyzed

(2) The expressions of dimensionless bottomhole pseu-dopressure and pseudopressure derivative duringeach characteristic period of horizontal gas well havebeen obtained formulas have been developed tocalculate gas reservoir fluid flow parameters

(3) The example study verifies that the characteristicvalue method of well test analysis for horizontal gaswell is reliable and practical

(4) The characteristic value method of well test analysiswhich has been included in the well test analysissoftware at present has been widely used in verticalwell As long as the characteristic straight line seg-ments which are manifested by pressure derivativecurve appear the reservoir fluid flow parameterscan be calculated by the characteristic value methodof well test analysis for vertical well The proposedcharacteristic value method of well test analysis for

Mathematical Problems in Engineering 9

horizontal gas well enriches and develops the well testanalysis theory and method

Nomenclature

119862 Wellbore storage coefficient m3MPa119862119863 Dimensionless wellbore storage coefficient

119862119905 Total compressibility MPaminus1

ℎ Reservoir thickness mℎ119863 Dimensionless reservoir thickness

119868119899 Modified Bessel function of first kind of order 119899

119870ℎ Horizontal permeability mD

119870119899 Modified Bessel function of second kind of order 119899

119870119881 Vertical permeability mD

119871 Horizontal section length m119871119863 Dimensionless horizontal section length

119898(119901) Pseudopressure MPa2mPasdots119898119863 Dimensionless pseudopressure

119898119894 Initial formation pseudopressure MPa2mPasdots

119898119908119863

Dimensionless bottomhole pseudopressure119898119908119891 Flowing wellbore pseudopressure MPa2mPasdots

1198981015840

119908119863 Derivative of119898

119908119863

119898119863 Laplace transform of 119898

119863

119898119908119863

Laplace transform of 119898119908119863

Δ119898 Pseudopressure difference MPaΔ1198981015840 Derivative of Δ119898

119901119888ℎ Casing head pressure MPa

119901119894 Initial formation pressure MPa

119901119877 Reservoir pressure MPa

119901119908ℎ Tubing head pressure MPa

119901119908119904 Flowing wellbore pressure at shut-in MPa

119902119904119888 Gas flow rate 104m3d

119902119908 Water production rate m3d

119903 Radial distance m119903119863 Dimensionless radial distance

119903119890 Outer boundary distance m

119903119890119863 Dimensionless outer boundary distance

119903119908 Wellbore radius m

119903119908119863

Dimensionless wellbore radius119878 Skin factor119904 Laplace transform variable119878119908119894 Initial water saturation fraction

119905 Time hours119905119863 Dimensionless time

119905119901 Production time hours

119905119901119863

Dimensionless production timeΔ119905 Shut-in time hours119879 Formation temperature ∘C119911 Vertical distance m119911119863 Dimensionless vertical distance

119911119908 Horizontal section position m

119911119908119863

Dimensionless horizontal section position119885 Gas deviation factor120576 Tiny variable120601 Porosity fraction120583 Gas viscosity mPasdots120574119892 Gas gravity

Subscripts

119863 Dimensionlesser Earlyerb Early of buildupℎ Horizontal119894 Initial119897 Linearlr Late horizontal pseudo-radiallrb Late horizontal pseudo-radial of builduppp Pseudosteady flow period119905 TotalV Vertical119908119891 Flowing wellbore119908119904 Shut-in wellbore

Conflict of Interests

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

References

[1] R S Nie J C Guo Y L Jia S Q Zhu Z Rao and C G ZhangldquoNew modelling of transient well test and rate decline analysisfor a horizontal well in a multiple-zone reservoirrdquo Journal ofGeophysics and Engineering vol 8 no 3 pp 464ndash476 2011

[2] D R Yuan H Sun Y C Li and M Zhang ldquoMulti-stagefractured tight gas horizontal well test data interpretationstudyrdquo in Proceedings of the International Petroleum TechnologyConference March 2013

[3] M Mavaddat A Soleimani M R Rasaei Y Mavaddat and AMomeni ldquoWell test analysis ofMultipleHydraulically FracturedHorizontal Wells (MHFHW) in gas condensate reservoirsrdquo inProceedings of the SPEIADC Middle East Drilling TechnologyConference and Exhibition (MEDT rsquo11) pp 39ndash48 October 2011

[4] MMirza P AditamaM AL Raqmi and Z Anwar ldquoHorizontalwell stimulation a pilot test in southern Omanrdquo in Proceedingsof the SPE EOR Conference at Oil and GasWest Asia April 2014

[5] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquo Society of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973

[6] M D Clonts and H J Ramey ldquoPressure transient analysis forwells with horizontal drainholesrdquo in Proceedings of the SPECalifornia Regional Meeting Oakland Calif USA April 1986paper SPE 15116

[7] P A Goode and R K M Thambynayagam ldquoPressure draw-down and buildup analysis of horizontal wells in anisotropicmediardquo SPE Formation Evaluation vol 2 no 4 pp 683ndash6971987

[8] E Ozkan and R Rajagopal ldquoHorizontal well pressure analysisrdquoSPE Formation Evaluation pp 567ndash575 1989

[9] A S Odeh and D K Babu ldquoTransient flow behavior ofhorizontal wells Pressure drawdown and buildup analysisrdquo SPEFormation Evaluation vol 5 no 1 pp 7ndash15 1990

[10] L G Thompson and K O Temeng ldquoAutomatic type-curvematching for horizontal wellsrdquo in Proceedings of the ProductionOperation Symposium Oklahoma City Okla USA 1993 paperSPE 25507

10 Mathematical Problems in Engineering

[11] R S Rahavan C C Chen and B Agarwal ldquoAn analysis ofhorizontal wells intercepted by multiple fracturesrdquo SPE27652(Sep pp 235ndash245 1997

[12] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal wells in closed systemsrdquo in Proceed-ings of the SPE International Improved Oil Recovery Conferencein Asia Pacific (IIORC rsquo03) pp 295ndash307 Alberta CanadaOctober 2003

[13] T Whittle H Jiang S Young and A C Gringarten ldquoWellproduction forecasting by extrapolation of the deconvolutionof the well test pressure transientsrdquo in Proceedings of the SPEEUROPECEAGE Conference Amsterdam The NetherlandJune 2009 SPE 122299 paper

[14] C C Miller A B Dyes and C A Hutchinson ldquoThe estimationof permeability and reservoir pressure from bottom-hole pres-sure buildup Characteristicrdquo Journal of Petroleum Technologypp 91ndash104 1950

[15] D R Horner ldquoPressure buildup in wellsrdquo in Proceedings of the3rd World Petroleum Congress Hague The Netherlands May-June 1951 paper SPE 4135

[16] R Raghavan ldquoThe effect of producing time on type curveanalysisrdquo Journal of Petroleum Technology vol 32 no 6 pp1053ndash1064 1980

[17] D Bourdet ldquoA new set of type curves simplifies well testanalysisrdquoWorld Oil pp 95ndash106 1983

[18] T M Hegre ldquoHydraulically fractured horizontal well simu-lationrdquo in Proceedings of the NPFSPE European ProductionOperations Conference (EPOC rsquo96) pp 59ndash62 April 1996

[19] R G Agarwal R Al-Hussaing and H J Ramey ldquoAn investiga-tion of wellbore storage and skin effect in unsteady liquid flowI Analytical treatmentrdquo Society of Petroleum Engineers Journalvol 10 no 3 pp 291ndash297 1970

[20] H J J Ramey and R G Agarwal ldquoAnnulus unloading ratesas influenced by wellbore storage and skin effectrdquo Society ofPetroleum Engineers Journal vol 12 no 5 pp 453ndash462 1972

[21] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems part 1-analytical considerationsrdquo SPEForma-tion Evaluation 1991

[22] L G Thompson Analysis of variable rate pressure data usingduhamels principle [PhD thesis] University of Tulsa TulsaOkla USA 1985

[23] H Stehfest ldquoNumerical inversion of laplace transformsrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

[24] N Al-Ajmi M Ahmadi E Ozkan and H Kazemi ldquoNumericalinversion of laplace transforms in the solution of transient flowproblemswith discontinuitiesrdquo inProceedings of the SPEAnnualTechnical Conference and Exhibition Paper SPE 116255 pp 21ndash24 Denver Colorado 2008

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

Mathematical Problems in Engineering 9

horizontal gas well enriches and develops the well testanalysis theory and method

Nomenclature

119862 Wellbore storage coefficient m3MPa119862119863 Dimensionless wellbore storage coefficient

119862119905 Total compressibility MPaminus1

ℎ Reservoir thickness mℎ119863 Dimensionless reservoir thickness

119868119899 Modified Bessel function of first kind of order 119899

119870ℎ Horizontal permeability mD

119870119899 Modified Bessel function of second kind of order 119899

119870119881 Vertical permeability mD

119871 Horizontal section length m119871119863 Dimensionless horizontal section length

119898(119901) Pseudopressure MPa2mPasdots119898119863 Dimensionless pseudopressure

119898119894 Initial formation pseudopressure MPa2mPasdots

119898119908119863

Dimensionless bottomhole pseudopressure119898119908119891 Flowing wellbore pseudopressure MPa2mPasdots

1198981015840

119908119863 Derivative of119898

119908119863

119898119863 Laplace transform of 119898

119863

119898119908119863

Laplace transform of 119898119908119863

Δ119898 Pseudopressure difference MPaΔ1198981015840 Derivative of Δ119898

119901119888ℎ Casing head pressure MPa

119901119894 Initial formation pressure MPa

119901119877 Reservoir pressure MPa

119901119908ℎ Tubing head pressure MPa

119901119908119904 Flowing wellbore pressure at shut-in MPa

119902119904119888 Gas flow rate 104m3d

119902119908 Water production rate m3d

119903 Radial distance m119903119863 Dimensionless radial distance

119903119890 Outer boundary distance m

119903119890119863 Dimensionless outer boundary distance

119903119908 Wellbore radius m

119903119908119863

Dimensionless wellbore radius119878 Skin factor119904 Laplace transform variable119878119908119894 Initial water saturation fraction

119905 Time hours119905119863 Dimensionless time

119905119901 Production time hours

119905119901119863

Dimensionless production timeΔ119905 Shut-in time hours119879 Formation temperature ∘C119911 Vertical distance m119911119863 Dimensionless vertical distance

119911119908 Horizontal section position m

119911119908119863

Dimensionless horizontal section position119885 Gas deviation factor120576 Tiny variable120601 Porosity fraction120583 Gas viscosity mPasdots120574119892 Gas gravity

Subscripts

119863 Dimensionlesser Earlyerb Early of buildupℎ Horizontal119894 Initial119897 Linearlr Late horizontal pseudo-radiallrb Late horizontal pseudo-radial of builduppp Pseudosteady flow period119905 TotalV Vertical119908119891 Flowing wellbore119908119904 Shut-in wellbore

Conflict of Interests

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

References

[1] R S Nie J C Guo Y L Jia S Q Zhu Z Rao and C G ZhangldquoNew modelling of transient well test and rate decline analysisfor a horizontal well in a multiple-zone reservoirrdquo Journal ofGeophysics and Engineering vol 8 no 3 pp 464ndash476 2011

[2] D R Yuan H Sun Y C Li and M Zhang ldquoMulti-stagefractured tight gas horizontal well test data interpretationstudyrdquo in Proceedings of the International Petroleum TechnologyConference March 2013

[3] M Mavaddat A Soleimani M R Rasaei Y Mavaddat and AMomeni ldquoWell test analysis ofMultipleHydraulically FracturedHorizontal Wells (MHFHW) in gas condensate reservoirsrdquo inProceedings of the SPEIADC Middle East Drilling TechnologyConference and Exhibition (MEDT rsquo11) pp 39ndash48 October 2011

[4] MMirza P AditamaM AL Raqmi and Z Anwar ldquoHorizontalwell stimulation a pilot test in southern Omanrdquo in Proceedingsof the SPE EOR Conference at Oil and GasWest Asia April 2014

[5] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquo Society of Petroleum Engineers Journal vol 13 no 5 pp 285ndash296 1973

[6] M D Clonts and H J Ramey ldquoPressure transient analysis forwells with horizontal drainholesrdquo in Proceedings of the SPECalifornia Regional Meeting Oakland Calif USA April 1986paper SPE 15116

[7] P A Goode and R K M Thambynayagam ldquoPressure draw-down and buildup analysis of horizontal wells in anisotropicmediardquo SPE Formation Evaluation vol 2 no 4 pp 683ndash6971987

[8] E Ozkan and R Rajagopal ldquoHorizontal well pressure analysisrdquoSPE Formation Evaluation pp 567ndash575 1989

[9] A S Odeh and D K Babu ldquoTransient flow behavior ofhorizontal wells Pressure drawdown and buildup analysisrdquo SPEFormation Evaluation vol 5 no 1 pp 7ndash15 1990

[10] L G Thompson and K O Temeng ldquoAutomatic type-curvematching for horizontal wellsrdquo in Proceedings of the ProductionOperation Symposium Oklahoma City Okla USA 1993 paperSPE 25507

10 Mathematical Problems in Engineering

[11] R S Rahavan C C Chen and B Agarwal ldquoAn analysis ofhorizontal wells intercepted by multiple fracturesrdquo SPE27652(Sep pp 235ndash245 1997

[12] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal wells in closed systemsrdquo in Proceed-ings of the SPE International Improved Oil Recovery Conferencein Asia Pacific (IIORC rsquo03) pp 295ndash307 Alberta CanadaOctober 2003

[13] T Whittle H Jiang S Young and A C Gringarten ldquoWellproduction forecasting by extrapolation of the deconvolutionof the well test pressure transientsrdquo in Proceedings of the SPEEUROPECEAGE Conference Amsterdam The NetherlandJune 2009 SPE 122299 paper

[14] C C Miller A B Dyes and C A Hutchinson ldquoThe estimationof permeability and reservoir pressure from bottom-hole pres-sure buildup Characteristicrdquo Journal of Petroleum Technologypp 91ndash104 1950

[15] D R Horner ldquoPressure buildup in wellsrdquo in Proceedings of the3rd World Petroleum Congress Hague The Netherlands May-June 1951 paper SPE 4135

[16] R Raghavan ldquoThe effect of producing time on type curveanalysisrdquo Journal of Petroleum Technology vol 32 no 6 pp1053ndash1064 1980

[17] D Bourdet ldquoA new set of type curves simplifies well testanalysisrdquoWorld Oil pp 95ndash106 1983

[18] T M Hegre ldquoHydraulically fractured horizontal well simu-lationrdquo in Proceedings of the NPFSPE European ProductionOperations Conference (EPOC rsquo96) pp 59ndash62 April 1996

[19] R G Agarwal R Al-Hussaing and H J Ramey ldquoAn investiga-tion of wellbore storage and skin effect in unsteady liquid flowI Analytical treatmentrdquo Society of Petroleum Engineers Journalvol 10 no 3 pp 291ndash297 1970

[20] H J J Ramey and R G Agarwal ldquoAnnulus unloading ratesas influenced by wellbore storage and skin effectrdquo Society ofPetroleum Engineers Journal vol 12 no 5 pp 453ndash462 1972

[21] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems part 1-analytical considerationsrdquo SPEForma-tion Evaluation 1991

[22] L G Thompson Analysis of variable rate pressure data usingduhamels principle [PhD thesis] University of Tulsa TulsaOkla USA 1985

[23] H Stehfest ldquoNumerical inversion of laplace transformsrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

[24] N Al-Ajmi M Ahmadi E Ozkan and H Kazemi ldquoNumericalinversion of laplace transforms in the solution of transient flowproblemswith discontinuitiesrdquo inProceedings of the SPEAnnualTechnical Conference and Exhibition Paper SPE 116255 pp 21ndash24 Denver Colorado 2008

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 Mathematical Problems in Engineering

[11] R S Rahavan C C Chen and B Agarwal ldquoAn analysis ofhorizontal wells intercepted by multiple fracturesrdquo SPE27652(Sep pp 235ndash245 1997

[12] A Zerzar and Y Bettam ldquoInterpretation of multiple hydrauli-cally fractured horizontal wells in closed systemsrdquo in Proceed-ings of the SPE International Improved Oil Recovery Conferencein Asia Pacific (IIORC rsquo03) pp 295ndash307 Alberta CanadaOctober 2003

[13] T Whittle H Jiang S Young and A C Gringarten ldquoWellproduction forecasting by extrapolation of the deconvolutionof the well test pressure transientsrdquo in Proceedings of the SPEEUROPECEAGE Conference Amsterdam The NetherlandJune 2009 SPE 122299 paper

[14] C C Miller A B Dyes and C A Hutchinson ldquoThe estimationof permeability and reservoir pressure from bottom-hole pres-sure buildup Characteristicrdquo Journal of Petroleum Technologypp 91ndash104 1950

[15] D R Horner ldquoPressure buildup in wellsrdquo in Proceedings of the3rd World Petroleum Congress Hague The Netherlands May-June 1951 paper SPE 4135

[16] R Raghavan ldquoThe effect of producing time on type curveanalysisrdquo Journal of Petroleum Technology vol 32 no 6 pp1053ndash1064 1980

[17] D Bourdet ldquoA new set of type curves simplifies well testanalysisrdquoWorld Oil pp 95ndash106 1983

[18] T M Hegre ldquoHydraulically fractured horizontal well simu-lationrdquo in Proceedings of the NPFSPE European ProductionOperations Conference (EPOC rsquo96) pp 59ndash62 April 1996

[19] R G Agarwal R Al-Hussaing and H J Ramey ldquoAn investiga-tion of wellbore storage and skin effect in unsteady liquid flowI Analytical treatmentrdquo Society of Petroleum Engineers Journalvol 10 no 3 pp 291ndash297 1970

[20] H J J Ramey and R G Agarwal ldquoAnnulus unloading ratesas influenced by wellbore storage and skin effectrdquo Society ofPetroleum Engineers Journal vol 12 no 5 pp 453ndash462 1972

[21] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems part 1-analytical considerationsrdquo SPEForma-tion Evaluation 1991

[22] L G Thompson Analysis of variable rate pressure data usingduhamels principle [PhD thesis] University of Tulsa TulsaOkla USA 1985

[23] H Stehfest ldquoNumerical inversion of laplace transformsrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

[24] N Al-Ajmi M Ahmadi E Ozkan and H Kazemi ldquoNumericalinversion of laplace transforms in the solution of transient flowproblemswith discontinuitiesrdquo inProceedings of the SPEAnnualTechnical Conference and Exhibition Paper SPE 116255 pp 21ndash24 Denver Colorado 2008

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