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Research ArticleCharacteristic Value Method of Well TestAnalysis for Horizontal Gas Well

Xiao-Ping Li,1 Ning-Ping Yan,1,2 and Xiao-Hua Tan1

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

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

Correspondence should be addressed to Xiao-Hua Tan; [email protected]

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

Academic Editor: Kim M. Liew

Copyright © 2014 Xiao-Ping Li et al. This is an open access article distributed under the Creative Commons Attribution License,which 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 [1–4]. 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 Newman’s product method,Gringarten and Ramey [5] found an access to solve the

unsteady-flow problems in reservoirs by means of the useof source and Green’s 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 pageshttp://dx.doi.org/10.1155/2014/472728

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. Therefore,this 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 𝑝

𝑖and equal every-

where, the gas reservoir is anisotropic, the horizontal per-meability is 𝐾

ℎ, the vertical permeability is 𝐾V, horizontal

section length is 2𝐿, and the position of horizontal sec-tion in the gas reservoir which is parallel to the closedtop and bottom boundary is 𝑧

𝑤. The surface flow rate

of horizontal gas well is 𝑞𝑠𝑐

and assumed to be constant.Single-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 model’s solving and calculation more simple,the 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

𝜃

r

z

M(r, 𝜃, z)

Figure 1: Physical model of horizontal gas well seepage.

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

1

𝑟𝐷

𝜕

𝜕𝑟𝐷

(𝑟𝐷

𝜕𝑚𝐷

𝜕𝑟𝐷

) + 𝐿2

𝐷

𝜕2𝑚𝐷

𝜕𝑧2𝐷

= (ℎ𝐷𝐿𝐷)2 𝜕𝑚𝐷

𝜕𝑡𝐷

. (1)

Initial condition is

𝑚𝐷(𝑟𝐷, 0) = 0. (2)

Inner boundary condition is

lim𝜀→0

[ lim𝑟𝐷→0

∫

𝑧𝑤𝐷+𝜀/2

𝑧𝑤𝐷−𝜀/2

𝑟𝐷

𝜕𝑚𝐷

𝜕𝑟𝐷

𝑑𝑧𝑤𝐷

]

=

{{{{{{{{

{{{{{{{{

{

0, 𝑧𝐷

> (𝑧𝑤𝐷

+𝜀

2)

−1

2, (𝑧

𝑤𝐷+

𝜀

2) ≥ 𝑧𝐷

≥ (𝑧𝑤𝐷

−𝜀

2)

0, 𝑧𝐷

< (𝑧𝑤𝐷

−𝜀

2) ,

(3)

where 𝜀 is a tiny variable.Infinite outer boundary condition in horizontal direction

is

lim𝑟𝐷→∞

𝑚𝐷(𝑟𝐷, 𝑡𝐷) = 0. (4)

Closed outer boundary condition in horizontal directionis

𝜕𝑚𝐷

𝜕𝑟𝐷

𝑟𝐷=𝑟𝑒𝐷

= 0. (5)

Constant pressure outer boundary condition in horizon-tal direction is

𝑚𝐷

𝑟𝐷=𝑟𝑒𝐷= 0. (6)

Closed outer boundary conditions in vertical directionare

𝜕𝑚𝐷

𝜕𝑧𝐷

𝑧𝐷=1

= 0,𝜕𝑚𝐷

𝜕𝑧𝐷

𝑧𝐷=0

= 0. (7)

Mathematical Problems in Engineering 3

The dimensionless variables are defined as follows:

𝑚𝐷

=78.489𝐾

ℎℎ

𝑞𝑠𝑐𝑇

(𝑚𝑖− 𝑚) , 𝑡

𝐷=

3.6𝐾ℎ𝑡

𝜙𝜇𝑐𝑡𝑟2𝑤

,

𝐿𝐷

=𝐿

ℎ√

𝐾V

𝐾ℎ

, ℎ𝐷

=ℎ

𝑟𝑤

√𝐾ℎ

𝐾V,

𝑧𝐷

=𝑧

ℎ, 𝑧

𝑟𝐷= 𝑧𝑤𝐷

+ 𝑟𝑤𝐷

𝐿𝐷,

𝑧𝑤𝐷

=𝑧𝑤

ℎ, 𝑟

𝐷=

𝑟

𝐿,

𝑟𝑒𝐷

=𝑟𝑒

𝐿, 𝑟

𝑤𝐷=

𝑟𝑤

𝐿.

(8)

The defined gas pseudopressure is

𝑚(𝑝) = 2∫

𝑝

𝑝ref

𝑝

𝜇 (𝑝)𝑍 (𝑝)𝑑𝑝. (9)

2.2. The Mathematical Model with Considering the Effect ofWellbore Storage and Skin. According to Duhamel’s 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:

𝑚𝑤𝐷

= 𝑚𝐷

+ ∫

𝑡𝐷

0

𝐶𝐷

𝑑𝑚𝑤𝐷

𝑑𝜏𝐷

𝑑𝑚𝑤𝐷

(𝑡𝐷

− 𝜏𝐷)

𝑑𝜏𝐷

𝑑𝜏𝐷

+ (1 − 𝐶𝐷

𝑑𝑚𝑤𝐷

𝑑𝜏𝐷

)ℎ𝐷𝑆,

(10)

where

𝐶𝐷

=𝐶

2𝜋𝜙𝐶𝑡ℎ𝐿2

. (11)

2.3. 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 𝑠with respect to 𝑡

𝐷.

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

𝑚𝐷

=1

2𝑠{∫

1

−1

𝐾0(√(𝑥

𝐷− 𝛼)2𝜀0)𝑑𝛼

+ 2

∞

∑

𝑛=1

∫

1

−1

𝐾0(√(𝑥

𝐷− 𝛼)2𝜀𝑛) cos (𝛽

𝑛𝑧𝑟𝐷

)

× cos (𝛽𝑛𝑧𝑤𝐷

) 𝑑𝛼} ,

(12)

where

𝛽𝑛= 𝑛𝜋,

𝜀𝑛= √𝑠(ℎ

𝐷𝐿𝐷)2+ 𝛽𝑛𝐿2𝐷.

(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 obtained.Thisresults in

𝑚𝐷

=1

2𝑠{∫

1

−1

𝐾0(√(𝑥


+𝐾1(𝑟𝑒𝐷

𝜀0)

𝐼1(𝑟𝑒𝐷

𝜀0)

∫

1

−1

𝐼0(√(𝑥


+ 2

∞

∑

𝑛=1

[∫

1

−1

𝐾0(√(𝑥

𝐷− 𝛼)2𝜀𝑛)𝑑𝛼

+𝐾1(𝑟𝑒𝐷

𝜀𝑛)

𝐼1(𝑟𝑒𝐷

𝜀𝑛)

∫

1

−1

𝐼0(√(𝑥


⋅ cos (𝛽𝑛𝑧𝑟𝐷

) cos (𝛽𝑛𝑧𝑤𝐷

) ]} .

(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

𝑚𝐷

=1

2𝑠{∫

1

−1

𝐾0(√(𝑥


−𝐾0(𝑟𝑒𝐷

𝜀0)

𝐼0(𝑟𝑒𝐷

𝜀0)

∫

1

−1

𝐼0(√(𝑥


+ 2

∞

∑

𝑛=1

[∫

1

−1

𝐾0(√(𝑥


−𝐾0(𝑟𝑒𝐷

𝜀𝑛)

𝐼0(𝑟𝑒𝐷

𝜀𝑛)

∫

1

−1

𝐼0(√(𝑥


⋅ cos (𝛽𝑛𝑧𝑟𝐷

) cos (𝛽𝑛𝑧𝑤𝐷

) ]} .

(15)

Making the Laplace transform to 𝑠 with respect to𝑡𝐷/𝐶𝐷, (10) can be solved for the dimensionless bottomhole

pseudopressure of horizontal gas well considering the effectof wellbore storage and skin in the Laplace space.This resultsin

𝑚𝑤𝐷

=𝑠𝑚𝐷

+ ℎ𝐷𝑆

𝑠 + 𝑠2(𝑠𝑚𝐷

+ ℎ𝐷𝑆)

=1

𝑠 (𝑠 + 1/ (𝑠𝑚𝐷

+ ℎ𝐷𝑆))

.

(16)


0.01

0.1

1

10

hD = 50

hD = 100

hD = 200

tD/CD

1E−2

1E−1

1E0

1E1

1E2

1E3

1E4

1E5

1E6

1E7

1E8

mwD,m

wD

I II

III

IV

CDe2S = 104, LD = 10, ZwD = 0.5

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

3.1. 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).

3.1.1. 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 𝑚

𝑤𝐷, 𝑚𝑤𝐷

versus 𝑡𝐷/𝐶𝐷, 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

𝑚𝑤𝐷

=𝑡𝐷

𝐶𝐷

,

𝑚

𝑤𝐷=

𝑑𝑚𝑤𝐷

𝑑 ln (𝑡𝐷/𝐶𝐷)=

𝑡𝐷

𝐶𝐷

.

(17)

3.1.2. Early Vertical Radial Flow Period. The early verticalradial flow period appears after the effect of wellbore storage;the characteristic of this period is manifested as a horizontalstraight line segment on the log-log plot of 𝑚

𝑤𝐷versus

𝑡𝐷/𝐶𝐷.The 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.


𝑚𝑤𝐷

=1

4𝐿𝐷

[ln 2.25 ( 𝑡𝐷𝐶𝐷

) + ln𝐶𝐷𝑒2𝑆] ,

𝑚

𝑤𝐷=

𝑑𝑚𝑤𝐷


1

4𝐿𝐷

.

(18)

3.1.3. 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 0.5 on the log-log plot of𝑚

𝑤𝐷versus

𝑡𝐷/𝐶𝐷. This characteristic describes the linear flow of fluid

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

𝐷, dimensionless horizontal section length 𝐿

𝐷,

and the position of horizontal section in the gas reservoir 𝑧𝑤𝐷

.The flow regime of this period is shown in Figure 4.


𝑚𝑤𝐷

= 2𝑟𝑤𝐷

√𝜋𝑡𝐷

+ 𝑆,

𝑚

𝑤𝐷=

𝑑𝑚𝑤𝐷

𝑑 ln (𝑡𝐷/𝐶𝐷)= 𝑟𝑤𝐷

√𝜋𝑡𝐷.

(19)

3.1.4. 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


Figure 5: The schematic diagram of late horizontal pseudoradialflow.

0.01

0.1

1

10

100

tD/CD

1E−2

1E−1

1E0

1E1

1E2

1E3

1E4

1E5

1E6

1E7

1E8

mwD,m

wD

CD = 1, S = 5, hD = 30, LD = 5, ZwD = 0.5

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 0.5on the log-log plot of 𝑚

𝑤𝐷versus 𝑡

𝐷/𝐶𝐷. 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.


𝑚𝑤𝐷

=1

2[ln(

𝑟𝑤𝐷

𝑡𝐷

𝐶𝐷

) + ln𝐶𝐷𝑒2𝑆] ,

𝑚

𝑤𝐷=

𝑑𝑚𝑤𝐷


1

2.

(20)

3.2. 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

0.001

0.01

0.1

1

10

1E−2

1E0

1E2

1E4

1E6

1E8

1E10

tD/CD

reD = 10

reD = 20

reD = 40

CD = 1, S = 10 , hD = 30, LD = 10 , ZwD = 0.5

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 response.The characteristic of the pseudosteady state flow period ismanifested as a straight line segment with a slope of 1 onthe log-log plot of 𝑚

𝑤𝐷, 𝑚𝑤𝐷

versus 𝑡𝐷/𝐶𝐷. 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

𝑚𝑤𝐷

= 2𝜋(𝑟𝑤𝐷

𝑟𝑒𝐷

)

2

𝑡𝐷

+ 𝑆,

𝑚

𝑤𝐷=

𝑑𝑚𝑤𝐷

𝑑 ln (𝑡𝐷/𝐶𝐷)= 2𝜋(

𝑟𝑤𝐷

𝑟𝑒𝐷

)

2

𝑡𝐷.

(21)

3.3. 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 boundary,but 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


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.

3.4. 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.

3.4.1. 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 𝑚𝑤𝐷

, 𝑚𝑤𝐷

versus 𝑡𝐷/𝐶𝐷. The expression of dimensionless

bottomhole pseudopressure during this period is

𝑚𝑤𝐷

=𝑡𝐷

𝐶𝐷

. (22)

Equation (22) can be converted to dimensional form,and 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 Δ𝑚, Δ𝑚 versus 𝑡, 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:

𝑡𝐷

𝐶𝐷

=3.6𝐾ℎ𝑡/𝜙𝜇𝑐𝑡𝑟2

𝑤

𝐶/2𝜋𝜙𝑐𝑡ℎ𝐿2

=7.2𝜋𝐾

ℎℎ𝐿2

𝜇𝑟2𝑤

𝑡

𝐶. (23)

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

𝐶 =0.288𝑞

𝑠𝑐𝑇

𝜇

𝐿2

𝑟2𝑤

𝑡

Δ𝑚, (24)

where 𝑡/Δ𝑚 represents the actual value on the log-log plot ofΔ𝑚, Δ𝑚 versus 𝑡 during the pure wellbore storage period.

3.4.2. 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/(4𝐿

𝐷) during the early vertical radial flow period.

The expression of dimensionless bottomhole pseudopressurederivative during this period is

𝑚

𝑤𝐷=

𝑑𝑚𝑤𝐷


1

4𝐿𝐷

. (25)

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

78.489𝐾ℎℎ

𝑞𝑠𝑐𝑇

(𝑡Δ𝑚)er

=1

4 (𝐿/ℎ)√𝐾V/𝐾ℎ. (26)

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

√𝐾ℎ𝐾V =

3.185 × 10−3

𝑞𝑠𝑐𝑇

𝐿(𝑡Δ𝑚)er, (27)

where (𝑡Δ𝑚)er represents the actual value on the log-log plotof Δ𝑚 versus 𝑡 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

𝑚𝑤𝐷

=1

4𝐿𝐷

[ln 2.25 ( 𝑡𝐷𝐶𝐷

) + ln𝐶𝐷𝑒2𝑆] . (28)

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

𝑆 = 0.5 [Δ𝑚er

(𝑡Δ𝑚)er− ln

𝐾ℎ𝑡er

𝜙𝜇𝐶𝑡𝑟2𝑤

− 0.80907] , (29)

where Δ𝑚er and 𝑡er represent the pseudopressure differenceand time corresponding to the (𝑡Δ𝑚)er, respectively.

For pressure buildup analysis, when Δ𝑡 → ∞, thelnΔ𝑡/(Δ𝑡 + 𝑡

𝑝) → 0, (30) can be obtained through

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

𝑚𝑖− 𝑚𝑤𝑓

𝑡Δ𝑚= ln 𝑡𝑝𝐷

+ 0.80907 + 2𝑆. (30)

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

𝑚𝑖= 𝑚𝑤𝑓

+ (𝑡Δ𝑚)erb

(ln 𝑡𝑝𝐷

+ 0.80907 + 2𝑆) , (31)

where (𝑡Δ𝑚)erb represents the actual value on the pressurebuildup log-log plot of Δ𝑚 versus 𝑡 during the early verticalradial flow period.

3.4.3. Early Linear Flow Period. The dimensionless bot-tomhole pseudopressure derivative curve is manifested asa straight line segment with the slope of 0.5 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:

78.489𝐾ℎℎ

𝑞𝑠𝑐𝑇

(𝑡Δ𝑚)𝑙=

𝑟𝑤

𝐿√

3.6𝜋𝐾ℎ𝑡

𝜙𝜇𝐶𝑡𝑟2𝑤

. (32)

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

√𝐾ℎ=

4.28 × 10−2

𝑞𝑠𝑐𝑇

𝐿ℎ√𝜙𝜇𝐶𝑡

[√𝑡

(𝑡Δ𝑚)]

𝑙

, (33)


where (𝑡Δ𝑚)𝑙represents the actual value on the log-log plot

of Δ𝑚 versus 𝑡 during the early linear flow period.Combining (27) with (33), the vertical permeability can

be obtained. This results in

√𝐾V = 7.44 × 10−2

ℎ√𝜙𝜇𝐶𝑡

[(𝑡Δ𝑚) /√𝑡]

𝑙

(𝑡Δ𝑚)er. (34)

3.4.4. Late Horizontal Pseudoradial Flow Period. The dimen-sionless bottomhole pseudopressure derivative curve is man-ifested as a horizontal straight line segment with the valueof 0.5 during the late horizontal pseudoradial flow period.According to the expression of dimensionless bottomholepseudopressure derivative during this period and the defini-tions of dimensionless variables, (35) can be obtained:

78.489𝐾ℎℎ

𝑞𝑠𝑐𝑇

(𝑡Δ𝑚)lr= 0.5. (35)

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

𝐾ℎ=

6.37 × 10−3

𝑞𝑠𝑐𝑇

ℎ(𝑡Δ𝑚)lr, (36)

where (𝑡Δ𝑚)lr represents the actual value on the log-log plotof Δ𝑚 versus 𝑡 during the late horizontal pseudoradial flowperiod.

For pressure buildup analysis, when Δ𝑡 → ∞, thelnΔ𝑡/(Δ𝑡 + 𝑡

𝑝) → 0, (37) can be obtained through the use

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

𝑚𝑖− 𝑚𝑤𝑓

(𝑡Δ𝑚)lrb= ln 𝑟𝑤𝐷

𝑡𝑝𝐷

+ 0.80907 + 2𝑆. (37)

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

𝑚𝑖= 𝑚𝑤𝑓

+ (𝑡Δ𝑚)lrb

(ln 𝑟𝑤𝐷

𝑡𝑝𝐷

+ 0.80907 + 2𝑆) , (38)

where (𝑡Δ𝑚)lrb represents the actual value on the pressurebuildup log-log plot ofΔ𝑚 versus 𝑡 during the late horizontalpseudoradial flow period.

3.4.5. 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:

78.489𝐾ℎℎ

𝑞𝑠𝑐𝑇

(𝑡Δ𝑚)pp

=7.2𝜋𝐾

ℎ𝑡pp

𝑟2𝑒ℎ𝜙𝜇𝐶

𝑡

. (39)

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

𝜋𝑟2

𝑒ℎ𝜙 =

0.905𝑞𝑠𝑐𝑇

ℎ𝜇𝐶𝑡

𝑡pp

(𝑡Δ𝑚)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

3/d)

qw(m

3/d)

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 (𝑡Δ𝑚)pp represents the actual value on the log-log plotof Δ𝑚 versus 𝑡 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 3425.63m, and the well completion system is screencompletion. According to the deliverability test during 26–29December, 2006, the calculated absolute open flow was 94.26× 104m3/d. The commissioning data of Longping 1 well wasin 12May, 2007, the initial formation pressure was 29.39MPa,before production, and the surface tubing pressure and casingpressure were both 23.90MPa. The production performancecurves of Longping 1 well are shown in Figures 8 and 9,respectively.

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 × 104m3/d before theshut-in. The bottomhole pressure recovered from 22.38MPato 27.83MPa during the pressure buildup test. Physical

8 Mathematical Problems in EngineeringΔm,Δ

m

Δt (h)10−3

10−4

10−5

10−6

10−7

10−2 10−1 100 101 102 103 104

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 𝑝

𝑖(MPa) 29.39

Formation temperature 𝑇 (∘C) 95.80Formation thickness ℎ (m) 6.31Porosity 𝜙 (%) 7.77Initial water saturation 𝑆wi (%) 13.60Well radius 𝑟

𝑤(m) 0.0797

Gas gravity 𝛾𝑔

0.608Gas deviation factor 𝑍 0.9738Gas viscosity 𝜇

𝑔(mPa⋅s) 0.0222

Table 2: Well test analysis results of Longping 1 well.

Parameter Parameter valuesWellbore storage coefficient 𝐶 (m3/MPa) 1.229Horizontal permeability 𝐾

ℎ(mD) 7.742

Vertical permeability 𝐾𝑣(mD) 0.039

Flow capacity 𝐾ℎℎ (mD⋅m) 48.857

Skin factor 𝑆 −2.49Effective horizontal section length 𝐿 (m) 198.37Reservoir pressure 𝑝

𝑅(MPa) 28.385

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 analysis,which 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


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

Nomenclature

𝐶: Wellbore storage coefficient, m3/MPa𝐶𝐷: Dimensionless wellbore storage coefficient

𝐶𝑡: Total compressibility, MPa−1

ℎ: Reservoir thickness, mℎ𝐷: Dimensionless reservoir thickness

𝐼𝑛: Modified Bessel function of first kind of order 𝑛

𝐾ℎ: Horizontal permeability, mD

𝐾𝑛: Modified Bessel function of second kind of order 𝑛

𝐾𝑉: Vertical permeability, mD

𝐿: Horizontal section length, m𝐿𝐷: Dimensionless horizontal section length

𝑚(𝑝): Pseudopressure, MPa2/mPa⋅s𝑚𝐷: Dimensionless pseudopressure

𝑚𝑖: Initial formation pseudopressure, MPa2/mPa⋅s

𝑚𝑤𝐷

: Dimensionless bottomhole pseudopressure𝑚𝑤𝑓: Flowing wellbore pseudopressure, MPa2/mPa⋅s

𝑚

𝑤𝐷: Derivative of𝑚

𝑤𝐷

𝑚𝐷: Laplace transform of 𝑚

𝐷

𝑚𝑤𝐷

: Laplace transform of 𝑚𝑤𝐷

Δ𝑚: Pseudopressure difference, MPaΔ𝑚: Derivative of Δ𝑚

𝑝𝑐ℎ: Casing head pressure, MPa

𝑝𝑖: Initial formation pressure, MPa

𝑝𝑅: Reservoir pressure, MPa

𝑝𝑤ℎ: Tubing head pressure, MPa

𝑝𝑤𝑠: Flowing wellbore pressure at shut-in, MPa

𝑞𝑠𝑐: Gas flow rate, 104m3/d

𝑞𝑤: Water production rate, m3/d

𝑟: Radial distance, m𝑟𝐷: Dimensionless radial distance

𝑟𝑒: Outer boundary distance, m

𝑟𝑒𝐷: Dimensionless outer boundary distance

𝑟𝑤: Wellbore radius, m

𝑟𝑤𝐷

: Dimensionless wellbore radius𝑆: Skin factor𝑠: Laplace transform variable𝑆𝑤𝑖: Initial water saturation, fraction

𝑡: Time, hours𝑡𝐷: Dimensionless time

𝑡𝑝: Production time, hours

𝑡𝑝𝐷

: Dimensionless production timeΔ𝑡: Shut-in time, hours𝑇: Formation temperature, ∘C𝑧: Vertical distance, m𝑧𝐷: Dimensionless vertical distance

𝑧𝑤: Horizontal section position, m

𝑧𝑤𝐷

: Dimensionless horizontal section position𝑍: Gas deviation factor𝜀: Tiny variable𝜙: Porosity, fraction𝜇: Gas viscosity, mPa⋅s𝛾𝑔: Gas gravity.

Subscripts

𝐷: Dimensionlesser: Earlyerb: Early of buildupℎ: Horizontal𝑖: Initial𝑙: Linearlr: Late horizontal pseudo-radiallrb: Late horizontal pseudo-radial of builduppp: Pseudosteady flow period𝑡: TotalV: Vertical𝑤𝑓: Flowing wellbore𝑤𝑠: Shut-in wellbore.

Conflict of Interests

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

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