-
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
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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)
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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
𝐷− 𝛼)2𝜀0)𝑑𝛼
+𝐾1(𝑟𝑒𝐷
𝜀0)
𝐼1(𝑟𝑒𝐷
𝜀0)
∫
1
−1
𝐼0(√(𝑥
𝐷− 𝛼)2𝜀0)𝑑𝛼
+ 2
∞
∑
𝑛=1
[∫
1
−1
𝐾0(√(𝑥
𝐷− 𝛼)2𝜀𝑛)𝑑𝛼
+𝐾1(𝑟𝑒𝐷
𝜀𝑛)
𝐼1(𝑟𝑒𝐷
𝜀𝑛)
∫
1
−1
𝐼0(√(𝑥
𝐷− 𝛼)2𝜀𝑛)𝑑𝛼
⋅ 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
𝐷− 𝛼)2𝜀0)𝑑𝛼
−𝐾0(𝑟𝑒𝐷
𝜀0)
𝐼0(𝑟𝑒𝐷
𝜀0)
∫
1
−1
𝐼0(√(𝑥
𝐷− 𝛼)2𝜀0)𝑑𝛼
+ 2
∞
∑
𝑛=1
[∫
1
−1
𝐾0(√(𝑥
𝐷− 𝛼)2𝜀𝑛)𝑑𝛼
−𝐾0(𝑟𝑒𝐷
𝜀𝑛)
𝐼0(𝑟𝑒𝐷
𝜀𝑛)
∫
1
−1
𝐼0(√(𝑥
𝐷− 𝛼)2𝜀𝑛)𝑑𝛼
⋅ 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)
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4 Mathematical Problems in Engineering
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.
Expressions of dimensionless bottomhole pseudopres-sure and
pseudopressure derivative during this period can beobtained. This
results in
𝑚𝑤𝐷
=1
4𝐿𝐷
[ln 2.25 ( 𝑡𝐷𝐶𝐷
) + ln𝐶𝐷𝑒2𝑆] ,
𝑚
𝑤𝐷=
𝑑𝑚𝑤𝐷
𝑑 ln (𝑡𝐷/𝐶𝐷)=
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.
Expressions of dimensionless bottomhole pseudopres-sure and
pseudopressure derivative during this period can beobtained. This
results in
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
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Mathematical Problems in Engineering 5
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.
Expressions of dimensionless bottomhole pseudopres-sure and
pseudopressure derivative during this period can beobtained. This
results in
𝑚𝑤𝐷
=1
2[ln(
𝑟𝑤𝐷
𝑡𝐷
𝐶𝐷
) + ln𝐶𝐷𝑒2𝑆] ,
𝑚
𝑤𝐷=
𝑑𝑚𝑤𝐷
𝑑 ln (𝑡𝐷/𝐶𝐷)=
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
-
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.
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
𝑑 ln (𝑡𝐷/𝐶𝐷)=
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)
-
Mathematical Problems in Engineering 7
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
-
Mathematical Problems in Engineering 9
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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