Research Article Productivity for Horizontal Wells in Low-Permeability Reservoir with Oil/Water Two-Phase Flow Yu-Long Zhao, 1 Lie-Hui Zhang, 1 Zhi-Xiong He, 2 and Bo-Ning Zhang 1 1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China 2 School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China Correspondence should be addressed to Yu-Long Zhao; [email protected] Received 30 October 2013; Revised 1 February 2014; Accepted 30 April 2014; Published 22 May 2014 Academic Editor: Paulo Batista Gonc ¸alves Copyright © 2014 Yu-Long Zhao et al. is 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. is paper presents a semianalytical steady-state productivity of oil/water two-phase flow in low-permeability reservoirs with both top and bottom boundaries closed which takes the permeability stress-sensitive and threshold pressure gradient into account. Using the similar approach as Joshi’s (1988), the three-dimensional (3D) horizontal well problem is divided into two two-dimensional problems (2D), and then the corresponding nonlinear steady seepage mathematical models in vertical and horizontal planes are established. rough the separation of variables method and equivalent flow resistance principle, the productivity equation of horizontal well is obtained. e liquid and oil productivity with different influential factors are plotted, and the related effects are also analyzed. is paper expanded the conventional productivity equations of single phase into multiphase flow which have both theoretical and practical significance in predicting production behaviors in such reservoirs. 1. Introduction With the development of drilling technology and the reduc- tion of its cost, more and more horizontal wells have been used in low-permeability reservoirs, fractured reservoirs, multilayered reservoirs, and bottom water drive reservoirs. And the steady-state productivity of horizontal well is always a hot topic for the petroleum engineers. Aſter several decades of development and research, many methods, including ana- lytical method, conformal transformation method, potential superposition method, equivalent flow resistance method, and the point source function method, were proposed to calculate it [1–9]. Merkulov [1] and Borisov [2] derived analytical producti- vity equation of the horizontal well with single oil phase flow. Giger et al. [3, 4] and Karcher and Giger [5] developed a con- cept of replacement ratio, FR, which indicates the required number of vertical wells to produce at the same rate as that of a single phase from formation well to horizontal well. Reiss [6] proposed an equation to calculate the productivity index for horizontal well. ereaſter, through subdividing the 3D flow of horizontal well into two 2D problems (flow on horizontal plane and vertical plane, resp.), Joshi [7] derived an equation to calculate the productivity of steady-state horizontal well, which is the most popular method nowadays. Babu and Odeh [8] proposed an equation to calculate the productivity of horizontal well under the assumption that the shape of the drainage volume is box and all the boundaries are closed. Renard and Dupuy [9] derived the flow efficiency equation for horizontal well in anisotropic reservoir with the consideration of skin factor. en, Helmy and Wattenbarger [10] and Billiter et al. [11] also proposed their corresponding equations to calculate the productivity. Anklam and Wiggins [12] obtained the steady-state productivity with the consid- eration of the mechanical properties of fluid flow into the wellbore. Using the steady-state point source function theory, Lu [13] achieved the productivity equations for horizontal wells under different boundary conditions. Most of the productivity equations of horizontal well mentioned above are mainly concentrated on single oil phase flow, but the ones related to multiphase are rare. In this paper, we employ the same method described by Joshi [7] and derived a steady-state productivity formula for a horizontal well with oil/water two-phase flow problem Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 364678, 9 pages http://dx.doi.org/10.1155/2014/364678
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Research ArticleProductivity for Horizontal Wells in Low-Permeability Reservoirwith OilWater Two-Phase Flow
Yu-Long Zhao1 Lie-Hui Zhang1 Zhi-Xiong He2 and Bo-Ning Zhang1
1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum UniversityChengdu Sichuan 610500 China
2 School of Sciences Southwest Petroleum University Chengdu Sichuan 610500 China
Correspondence should be addressed to Yu-Long Zhao 373104686qqcom
Received 30 October 2013 Revised 1 February 2014 Accepted 30 April 2014 Published 22 May 2014
Academic Editor Paulo Batista Goncalves
Copyright copy 2014 Yu-Long Zhao 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 semianalytical steady-state productivity of oilwater two-phase flow in low-permeability reservoirs with bothtop and bottom boundaries closed which takes the permeability stress-sensitive and threshold pressure gradient into account Usingthe similar approach as Joshirsquos (1988) the three-dimensional (3D) horizontal well problem is divided into two two-dimensionalproblems (2D) and then the corresponding nonlinear steady seepage mathematical models in vertical and horizontal planes areestablished Through the separation of variables method and equivalent flow resistance principle the productivity equation ofhorizontal well is obtained The liquid and oil productivity with different influential factors are plotted and the related effects arealso analyzed This paper expanded the conventional productivity equations of single phase into multiphase flow which have boththeoretical and practical significance in predicting production behaviors in such reservoirs
1 Introduction
With the development of drilling technology and the reduc-tion of its cost more and more horizontal wells have beenused in low-permeability reservoirs fractured reservoirsmultilayered reservoirs and bottom water drive reservoirsAnd the steady-state productivity of horizontal well is alwaysa hot topic for the petroleum engineers After several decadesof development and research many methods including ana-lytical method conformal transformation method potentialsuperposition method equivalent flow resistance methodand the point source function method were proposed tocalculate it [1ndash9]
Merkulov [1] and Borisov [2] derived analytical producti-vity equation of the horizontal well with single oil phase flowGiger et al [3 4] and Karcher and Giger [5] developed a con-cept of replacement ratio FR which indicates the requirednumber of vertical wells to produce at the same rate as thatof a single phase from formation well to horizontal wellReiss [6] proposed an equation to calculate the productivityindex for horizontal wellThereafter through subdividing the3D flow of horizontal well into two 2D problems (flow on
horizontal plane and vertical plane resp) Joshi [7] derivedan equation to calculate the productivity of steady-statehorizontal well which is themost popularmethod nowadays
Babu and Odeh [8] proposed an equation to calculate theproductivity of horizontal well under the assumption that theshape of the drainage volume is box and all the boundariesare closed Renard and Dupuy [9] derived the flow efficiencyequation for horizontal well in anisotropic reservoir with theconsideration of skin factor Then Helmy and Wattenbarger[10] and Billiter et al [11] also proposed their correspondingequations to calculate the productivity Anklam andWiggins[12] obtained the steady-state productivity with the consid-eration of the mechanical properties of fluid flow into thewellbore Using the steady-state point source function theoryLu [13] achieved the productivity equations for horizontalwells under different boundary conditions
Most of the productivity equations of horizontal wellmentioned above are mainly concentrated on single oilphase flow but the ones related to multiphase are rareIn this paper we employ the same method described byJoshi [7] and derived a steady-state productivity formulafor a horizontal well with oilwater two-phase flow problem
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 364678 9 pageshttpdxdoiorg1011552014364678
2 Mathematical Problems in Engineering
L
B
AA
B
h2
h
re
Figure 1 Schematic of a horizontal well in a top and bottomboundaries closed reservoir
in low-permeability reservoir which takes the permeabilitystress-sensitive and threshold pressure gradient into account
2 Theoretical Analysis
21 Physical Model Figure 1 is a schematic of a horizontalwell drilled in low-permeability oil reservoirs To make theproblemmore tractable the following assumptions aremade(1) the reservoir is horizontal homogeneous with uniformthickness of ℎ and both the top and the bottom boundariesare closed (2) the well length and radius are 119871 and 119903
119908
respectively the well is located at the center of the formationand the pressure at the drainage boundary is 119901
119894with the
radius of 119903119890 (3) two-phase fluid flows from the reservoir to
the well at a constant bottomhole pressure 119901119908119891
and ignore thegravity and capillary pressure effects
To simplify the mathematical solution the 3D problemis subdivided into two 2D problems [7] Figure 2 shows thefollowing subdivision of the ellipsoidal drainage problem (1)fluid flow into the horizontal well in the horizontal plane (2)fluid flow into the horizontal well in the vertical plane
22 PermeabilityModel in Low-Permeability Reservoir In thispaper we use the exponential permeability model to describethe relationship between permeability and pore pressure [14ndash20] This model is based on a permeability modulus 120572
119896
which is defined as the following
120572119896=
1
119896
120597119896
120597119901 (1)
Integrating of (1) with 119901 from 119901119894to 119901 yields
119896
119896119894
= exp [minus120572119896(119901119894minus 119901)] (2)
where 120572119896is the permeability modulus MPaminus1 119901 is the
formation pressure MPa 119901119894is the initial pressure MPa 119896
119894is
the permeability at initial condition D 119896 is the permeabilityat current condition D
23Threshold PressureGradient inOilWater Two-Phase FlowAccording to the experimental investigations (the schematic
of core sample experiments is shown in Figure 3) an appro-priate form of Darcyrsquos law with a threshold gradient shouldbe used as [21ndash28]
V =
minus119896
120583(120597119901
120597119903minus 119866)
120597119901
120597119903gt 119866
0120597119901
120597119903le 119866
(3)
where V is velocity of the fluid flowms120583 is the fluid viscositycp 119903 is the radius m 119866 is the threshold pressure gradientMPam
24 Flow in a Horizontal Plane Figure 2(a) shows aschematic of fluid flow to a horizontal well in a horizontalplane The drainage area is an ellipse and introduces
119911
1198712=1
2(120585 +
1
120585) (4)
by defining 119911 = 119909+119894 sdot119910 and 120585 = 119906+119894 sdotV = 119903119890119867
sdot 119890119894120579 Substitutingthem into (4) and then equating the real and imaginary partsyield
119909
1198712=1
2[119903119890119867
+1
119903119890119867
] cos (120579)
119910
1198712=1
2[119903119890119867
minus1
119903119890119867
] sin (120579) (5)
The real equation of the elliptical drainage area can bedescribed as
1199092
1198862+1199102
1198872= 1 (6)
where 119886 and 119887 are the major and minor axes of the ellipse mCombing (5) with (6) we have
119903119890119867
=119886 + 119887
05119871 (7)
Moreover +1198712 and minus1198712 represent foci of the ellipsewhich have the following relationship with 119886 and 119887
1198862
minus 1198872
= (119871
2)2
(8)
The drainage radius of horizontal well 119903119890 can be obtained
by equaling the areas of a circle and ellipse which is
119903119890= radic119886119887 (9)
Combing (9) with (8) we have
119886 = 05119871[
[
05 + radic025 + (2119903119890
119871)4
]
]
05
119887 = radic1198862 minus (119871
2)2
(10)
Mathematical Problems in Engineering 3
+
z
x
h2
h2x
minusL2
y
L2
(a) (b)
Figure 2 Schematic of the fluid flow to a horizontal well (a) horizontal plane and (b) vertical plane
Pressure gradient (MPam)
Velo
city
(ms
)
G (TPG)
Figure 3 A schematic of core sample laboratory experiments inlow-permeability reservoir
So as shown in Figure 4 the fluid flow in the 119906-V planecan be viewed as a unit radius vertical well produced at a circledrainage area with the radius of (119886 + 119887)(05119871) Combiningthe modified Darcy flow equation with the steady-stategoverning equation the fluid flow in the horizontal plane canbe described as
1
119903
dd119903
[119903119890minus120572119896(119901119894minus119901)120582
119905(d119901d119903
minus 1198661015840
)] = 0 (11)
1199011003816100381610038161003816119903=1 = 119901
119888 (12)
1199011003816100381610038161003816119903=119903119890119867
= 119901119894 (13)
where 120582119905is the total mobility of the oilwater phase 120582
119905=
119896119903119900120583119900+ 119896119903119908120583119908 which is a function of the radius and 1198661015840 is
the equivalent threshold pressure gradient in the horizontalplane which is
1198661015840
=119866 (119903119890minus 1198712)
((119886 + 119887) 05119871) minus 1 (14)
The solution of mathematical models of (11)ndash(13) is (thedetailed derivations are showed in the Appendix)
119901 (119903) = 119901119894
+1
120572119896
ln1198901205721198961198661015840119903
[119890minus120572119896(119866
1015840+119901119894minus119901119888)
+119890minus120572119896119866
1015840119903119890119867 minus 119890minus120572119896(119866
1015840+119901119894minus119901119888)
int119903119890119867
1
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
sdot int119903
1
119890minus120572119896119866
1015840119903
1
119903120582119905(119903)
d119903]
(15)
The liquid flow rate in horizontal plane can be calculatedby
119902119871119867
= 119862ℎ119896120582119905(1) sdot (119903
d119901d119903
)119903=1
(16)
where 119862 is the unit conversion factor 119862 = 542 119902119871119867
is theliquid flow rate at reservoir condition m3d
Combining (15) with (16) the liquid flow rate in the hori-zontal plane is
119902119871119867
= 119862119896119894119890minus120572119896(119901119894minus119901119888)ℎ
sdot
120582119905(1) sdot 119866
1015840
+[119890minus120572119896[119866
1015840119903119890119867minus(119901119894minus119901119888)] minus 119890minus120572119896119866
1015840
]
(120572119896int119903119890119867
1
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903)
(17)
The corresponding oil production rate in the horizontalplane is
119902119900119867
= 119862119896119894119890minus120572119896(119901119894minus119901119888)ℎ120582
119900(1)
sdot [1198661015840
+1
120582119905(1)
sdot119890minus120572119896[119866
1015840119903119890119867minus(119901119894minus119901119888)] minus 119890minus120572119896119866
1015840
120572119896int119903119890119867
1
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
]
(18)
4 Mathematical Problems in Engineering
y
O
(minusL2 0) (L2 0)x
1u
reH
z
L2=
1
2(120585 + 1
120585)
Figure 4 The schematic of fluid flow in horizontal plane after conformal transformation
h2
h2
z
x120585 = (1 minus eminus120587zh) times (1 + e120587zh)
u1
120585
2120587rwh
Figure 5 The schematic of fluid flow in vertical plane after conformal transformation
where 119902119900119867
is the oil flow rate at reservoir condition m3d120582119905(119903119908) is the total mobility of the oilwater phase in the well
bottomhole DcpIf we do not take into account the effects of permeability
stress-sensitive and threshold pressure gradient on the pro-ductivity equation in horizontal plane (120572
119896= 119866 = 0) and
assume only oil flow in the formation (120582119905(119903) = 1120583
119900) then
(17) becomes
119902SOH =0542 times 10minus3119896
119894ℎ (119901119894minus 119901119888)
119861119900120583119900ln (2 (119886 + 119887) 119871)
(19)
where 119902SOH is the oil flow rate at surface condition m3d 119861119900
is the oil volume factor sm3m3Equation (19) is the same as the productivity equation in
horizontal plane of (1) derived by Joshi [7]
25 Calculation of Flow in a Vertical Plane The fluid flowin the vertical plane with top and bottom boundaries closedreservoir can be viewed as a vertical well with the radiusof 120585119908
= 2120587119903119908ℎ produced in a unit circle area after the
conformal transformation (as shown in Figure 5)The mathematical models to describe the steady-state
fluid flow of oilwater flow in the vertical plane are
1
119903
dd119903
[119903119890minus120572119896(119901119894minus119901)120582
119905(d119901d119903
minus 11986610158401015840
)] = 0
1199011003816100381610038161003816119903=120585119908
= 119901119908119891
1199011003816100381610038161003816119903=1 = 119901
119888
(20)
where 11986610158401015840 is the equivalent threshold pressure gradient in thevertical plane which is
11986610158401015840
=(ℎ2) minus 119903
119908
1 minus (2120587119903119908ℎ)
119866 (21)
The solution of the pressure distribution along the radiuscan be solvedwith the samemethod showed in the AppendixThen the productivity equation in the vertical plane can becalculated by the following equation
119902119871119881
= 119862119871119896120582119905(120585119908) sdot (119903
d119901d119903
)119903=120585119908
(22)
The liquid and oil productivity in the vertical plane are
119902119871119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)119871
sdot
120582119905(120585119908) sdot 11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120572119896int1
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowast)
119902119900119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)119871120582
119900(120585119908)
times
11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120582119905(120585119908) 120572119896int1
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowastlowast)
Assuming the reservoir with single phase flow andneglecting the permeability stress-sensitive and thresholdpressure gradient (120572
119896= 119866 = 0 120582
119905(119903) = 1120583
119900) yield
119902119900119881
=119862119896119894119871 (119901119888minus 119901119908119891)
119861119900120583119900ln (ℎ2119903
119908) (23)
Mathematical Problems in Engineering 5
Equation (23) is similar to the productivity equation invertical plane of (D-3) derived by Joshi [7] which proves thecorrectness of the productivity equation in this paper
26 Horizontal Well Eccentricity Figure 5 and (lowast)-(lowastlowast) areobtained under the assumption that the horizontal well islocated at the center of the reservoir in the vertical planeAccording to Muskatrsquos [29] formulation for off-centeredwells the liquid production rate of a well placed at a distance120575 from the mid-height of the reservoir in a vertical plane is
119902119871119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)ℎ
sdot
120582119905(120585119908) sdot 1198661015840
sdot 120585119908
+[119890minus120572119896[119866
1015840120573minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
1015840120585119908]
(120572119896int120573
120585119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowast1015840
)
The oil production rate is
119902119900119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)ℎ120582
119900(120585119908)
times
11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840120573minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120582119905(120585119908) 120572119896int120573
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowastlowast1015840)
where 120575 is the vertical distance between the reservoir centerand horizontal well location m 120573 = 1 minus (2120575ℎ)
2
3 The Solving Method of 120582119905(119903)
In order to correctly calculate the productivity of horizontalwell with oilwater phase flow we must determinate theexpression of total mobility 120582
119905(119903) along the radius 119903 The
following steps are the procedure to calculate it
Step 1 According to the relative permeability curves and theviscosity of oil and water the relationship between 120582
119905(119903) and
119878119908can be obtained
Step 2 When the water displacement front breaks throughthe oil well the distribution of water saturation along the wellradius satisfies the following Buckley-Leverett equation
120587ℎ120601 (1199032
119890minus 1199032
) =d119891119908
d119904119908
sdot sum119876119897 (24)
where ℎ is the formation thickness m 120601 is the porosityfractionsum119876
119897is the cumulative fluid production m3d 119891
119908is
the water ratio fraction 119891119908= (119896119903119908120583119908)(119896119903119900120583119900) + (119896119903119908120583119908)
120582t(r) sim r
p1 larrminus pe p2 larrminus pwf
pc = (p1 + p2)2
qL eq (lowast)qHL eq (17)
If |qL minus qHL| gt eps
No
Yes If qL lt qHL p2 larrminus pc
If qL gt qHL p1 larrminus pc
0 re h Q1 120583o 120583w
kro sim Sw krw sim Sw
re h L rw G120572k pe pwf ki
Output the liquid and oil production rate
Figure 6 The productivity calculation diagram
In (24) the expressions of the d119891119908d119878119908versus 119904
119908can be
calculated by the relative permeability curves
Step 3 Combining the relationship of 120582119905(119903) versus 119878
119908and 119878119908
versus 119903 derived in Steps 1 and 2 the relationship of 120582119905(119903)with
119903 can be obtained
Statistical results show that the relationships of 119878119908versus
119903 120582119905(119903) versus 119878
119908 and 120582
119905(119903) versus 119903 can be approximated by
the following analytical functions
119878119908(119903) = 119886
1+ 11988711199032
120582119905(119878119908) = 1198862+ 1198872119878119908+ 11988821198782
119908
120582119905(119903) = 119886
3+ 11988731199032
+ 11988831199034
(25)
4 The Productivity Calculation Process
Using the electrical analog concept the well production ratein the horizontal plane must equal the production rate in thevertical plane Because of the nonlinearity of (17)-(18) and(lowast)-(lowastlowast) we cannot obtain productivity expressions similarto Joshi [7] With the aid of computer programs the resultscan be obtained and the calculation diagram is showed inFigure 6
5 Results and Their Sensitive Analysis
In this section the liquid and oil production rate are calcu-lated and the essential parameters of well reservoir and fluidproperties are listed in Table 1 and the relative permeabilitycurves are showed in Figure 7
According to the relative permeability curves and theparameters in Table 1 the relationship of 120582
119905(119878119908) versus 119878
119908
6 Mathematical Problems in Engineering
00
02
04
06
08
10
00 01 02 03 04 05 06 07 08 09 10
krok
rw
krwkro
Sw
Figure 7 The relative permeability curves
00
02
04
06
08
10
00 01 02 03 04 05 06 07 08 09 10Sw
120582t
120582t = 32547S2w minus 23096Sw + 04917
R2 = 1
Figure 8 The relationship between 120582119905(119878119908) and 119878
119908
Table 1 The parameters of reservoir and fluid properties
Variables valuePorosity 0 (fraction) 005Drainage radius 119903
119890(m) 600
Horizontal well length 119871 (m) 300Initial permeability of the reservoir 119896
119894(D) 2 times 10minus3
Cumulative fluid production sum119876119897(m3) 114 times 106
Oil viscosity in the reservoir 120583119900(mPasdots) 3
Formation thickness ℎ (m) 20Initial reservoir pressure 119901
119890(MPa) 30
Permeability modulus 120572119896(MPaminus1) 01
Water saturation in the bottomhole 119878119908119861
(fraction) 06Threshold pressure gradient 119866 (MPam) 0001Water viscosity in the reservoir 120583
119908(mPasdots) 1
and d119891119908d119878119908versus 119878
119908can be plotted when water cut ratio
reaches 06 (as shown in Figures 8 and 9)The correspondingregression curve equations can be obtained as follows
120582119905= 04917 minus 23096119878
119908+ 32547119878
2
119908
d119891119908
d119878119908
= minus60226119878119908+ 43193 119878
119908gt 06
(26)
0
1
2
3
4
5
6
02 03 04 05 06 07 08 09 10
000204060810
060 062 064 066 068 070dfwdS w df
wdS w
dfwdSw = minus60226Sw + 43193
R2 = 09716
Sw
Sw
Figure 9 The relationship between d119891119908d119878119908and 119878
119908
0
5
10
15
20
25
0 005 01 015 02 025 03
Liquid production rateOil production rate
Rate
(m3d
)
120572k (MPaminus1)
Figure 10 The effect of permeability modulus (120572119896) on liquid
production rate
Taking (26) as well as the cumulative fluid productioninto (24) the expressions between the 120582
119905(119903) and 119903 can be
obtained which is
120582119905= 02 + 153 times 10
minus6
1199032
+ 68 times 10minus13
1199034
(27)
Combining (27) and other parameters in Table 1 with(17)-(18) and (lowast)-(lowastlowast) the steady-state fluid productivity canbe calculated
Figure 10 shows the effect of permeability modulus 120572119896
on liquid and oil productivity of horizontal well in lowpermeability reservoir It can be seen from the figure thatthe permeability stress-sensitive has a significant effect on theproductivity the bigger the 120572
119896is the smaller the liquid and
oil productivity are which is mainly because with the samepressure drop of the reservoir big 120572
119896will lead to a serious
permeability decreasing When we do not take into accountthe permeability stress sensitive (120572
119896= 0) the liquid and oil
productivity can be calculated with the limit of 120572119896tending to
zero for (17)-(18) and (lowast)-(lowastlowast)Figures 11 and 12 show the effect of threshold pressure
gradient (119866) and well length (119871) on liquid and oil pro-ductivity It can be seen from the chart that the thresholdpressure gradient has small effect on the productivity ofhorizontal well for big drainage volume In general the bigger
Mathematical Problems in Engineering 7
Table 2 The productivity of liquid and oil in different bottomhole pressure
119901119908119891
(MPa)120572119896= 01 119866 = 0001 120572
119896= 0 119866 = 0
Liquid production rate(m3d)
Oil production rate(m3d)
Liquid production rate(m3d)
Oil production rate(m3d)
3000 000 000 000 0002980 000 000 039 0192960 000 000 079 0372950 033 016 098 0472700 458 218 589 2802400 861 409 1177 5602100 1171 556 1766 84018 1407 669 235 111915 1585 754 2944 13992 1720 818 3532 1679
0
2
4
6
8
10
12
14
0 0003 0006 0009 0012 0015
Liquid production rateOil production rate
G (MPaminus1m)
Rate
(m3d
)
Figure 11 The effect of threshold pressure gradient (119866) on liquidproduction rate
0
5
10
15
20
100 150 200 250 300 350 400 450
Liquid production rateOil production rate
Rate
(m3d
)
L (m)
Figure 12 The effect of well length (119871) on liquid production rate
the 119866 is the smaller the liquid and oil productivity areWhen the reservoir has both threshold pressure gradient andpermeability stress sensitive the longer the well length is thebigger the productivity is
048
12162024283236
12 14 16 18 20 22 24 26 28 30
Liquid production rate Oil production rate
Prod
uctio
n ra
te (m
3d
)
Liquid production rate Oil production rate
pwf (MPa)
(120572k = G = 0) (120572k = G = 0)
Figure 13 The productivity of horizontal well in different bottom-hole pressure
Figure 13 shows the liquid productivity with differentbottomhole pressure when 120572
119896= 01 119866 = 0001 and 120572
119896=
0 119866 = 0 the corresponding values are listed in Table 2It can be clearly seen that the permeability stress-sensitiveand threshold pressure gradient have significant effects onthe well productivity and the bigger 120572
119896and 119866 are the more
obvious the effect is And when the pressure drop is smallthe fluid cannot flow for the existing of threshold pressuregradient which is mainly because only the fluid can flowwhen the pressure drop overcomes the threshold pressure formultiphase flow
6 Conclusions
In this paper a semianalytical productivity equation of hor-izontal well in low-permeability oil reservoir with oilwatertwo-phase flow is established with the consideration ofpermeability stress-sensitive and threshold pressure gradientBased on the above study the following conclusions can besummarized
8 Mathematical Problems in Engineering
(1) The steady-state percolation mathematical modelsof horizontal well with oilwater two-phase floware established and the corresponding solutions aresolved by the method of separation of variables
(2) For low-permeability reservoir there always existsthe phenomenon of permeability stress-sensitive (120572
119896)
which has a significant influence on the well produc-tivity the bigger the 120572
119896is the smaller the productivity
is(3) Due to the existence of capillary pressure of two-
phase flow there always is threshold pressure gradient(119866) in the fluid seepage process Although the 119866 hasa smaller effect on the productivity of the horizontalwell for a big drainage volume we cannot neglect itseffect on the productivity
Appendix
From the expressions of (11) we have
119903120582119905119890minus120572119896(119901119894minus119901) (
d119901d119903
minus 1198661015840
) = 119888 (A1)
where 119888 is a constantWe define the following expression
119910 = 119890minus120572119896(119901119894minus119901) (A2)
Substituting (A2) into (A1) and (12)-(13) yields
d119910d119903
= 1205721198961198661015840
119910 +119888120572119896
119903120582119905
(A3)
1199101003816100381610038161003816119903=1 = 119890
minus120572119896(119901119894minus119901119888) (A4)
1199101003816100381610038161003816119903=119903119890119867
= 1 (A5)
According to the general solution of the Bernoulli differ-ential equation [30] the solution of (A3) can be obtained as
119910 (119903) = 1198901205721198961198661015840119903
sdot [119890minus120572119896119866
1015840
119890minus120572119896(119901119894minus119901119888) + int
119903
119903119908
119890minus120572119896119866
1015840119903
119888120572119896
119903120582119905(119903)
d119903] (A6)
Substituting (A6) into (A5) the value of constant 119888 canbe solved and then combining it with (A6) we have
119901 (119903)
= 119901119894+
1
120572119896
ln
1198901205721198961198661015840119903
sdot [
[
119890minus120572119896(119866
1015840+119901119894minus119901119888) +
119890minus1205721198961198661015840119903119890119867 minus 119890minus120572119896(119866
1015840+119901119894minus119901119888)
int119903119890119867
119903119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
sdot int119903
119903119908
119890minus120572119896119866
1015840119903
1
119903120582119905(119903)
d119903]
]
(A7)
Equation (A7) is the pressure distribution relation alongthe radius 119903 with the oilwater two-phase flows in thehorizontal plane of the horizontal well
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Natural Science Foundationof China (Grant no 51374181) and the project of NationalScience Fund for Distinguished Young Scholars of China(Grant no 51125019)The authors would also like to thank thereviewers and editors for their patience to read this paper andvaluable comments
References
[1] V P Merkulov ldquoLe debit des puits devies et horizontauxrdquo NeftKhoz vol 6 no 1 pp 51ndash56 1958
[2] J P Borisov Oil Production Using Horizontal and MultipleDeviation Wells The RampD Translation Company BartlesvilleOkla USA 1984
[3] F M Giger L H Reiss and A P Jourdan ldquoThe reservoir engi-neering aspects of horizontal drillingrdquo in Proceedings of the59th Annual Technical Conference and Exhibition Houston TexUSA 1984
[4] F M Giger ldquoHorizontal wells production techniques in het-erogeneous reservoirsrdquo in Proceedings of the Middle East OilTechnical Conference and Exhibition Bahrain 1985
[5] B J Karcher and F M Giger ldquoSome practical formulas to pre-dict horizontal well behaviorrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition New Orleans La USA1986
[6] L H Reiss ldquoProduction from horizontal wells after five yearsrdquoJournal of Petroleum Technology vol 39 no 11 pp 1411ndash14161987
[7] S D Joshi ldquoAugmentation of well productivity using slant andhorizontal wellsrdquo Journal of Petroleum Technology vol 40 no6 pp 729ndash739 1988
[8] D K Babu and A S Odeh ldquoProductivity of a horizontal wellrdquoSPE Reservoir Engineering vol 4 no 4 pp 417ndash421 1989
[9] G Renard and JM Dupuy ldquoFormation damage effects on hori-zontal-well flow efficiencyrdquo Journal of Petroleum Technologyvol 43 no 7 pp 786ndash789 1991
[10] M W Helmy and R A Wattenbarger ldquoSimplified productivityequations for horizontal wells producing at constant rate andconstant pressurerdquo in Proceedings of the SPE Technical Con-ference and Exhibition pp 379ndash388 New Orleans La USASeptember 1998
[11] T Billiter J Lee and R Chase ldquoDimensionless inflow-perform-ance-relationship curve for unfractured horizontal gas wellsrdquo inProceedings of the SPE Eastern Regional Meeting Canton OhioUSA 2001
[12] E G Anklam and M L Wiggins ldquoHorizontal well produc-tivity and wellbore pressure behavior incorporating wellborehydraulicsrdquo in Proceedings of the SPE Production andOperations
Mathematical Problems in Engineering 9
Symposium pp 565ndash584 Oklahoma City Okla USA April2005
[13] J Lu ldquoNewproductivity formulae of horizontal wellsrdquo Journal ofCanadian Petroleum Technology vol 40 no 10 pp 55ndash67 2001
[14] F Samaniego W E Brigham and F G Miller ldquoPerformance-prediction procedure for transient flow of fluids throughpressure-sensitive formationsrdquo Journal of PetroleumTechnologyvol 31 no 6 pp 779ndash786 1979
[15] G K Falade ldquoTransient flow of fluids in reservoirs with stresssensitive rock and fluid propertiesrdquo International Journal ofNon-Linear Mechanics vol 17 no 4 pp 277ndash283 1982
[16] R W Ostensen ldquoMicrocrack Permeability in tight gas sand-stonerdquo Society of Petroleum Engineers Journal vol 23 no 6 pp919ndash927 1983
[17] J Pedrosa and O A Petrobras ldquoPressure transient response instress-sensitive formationsrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA 1986
[18] D A Barry D A Lockington D-S Jeng J-Y Parlange L Liand F Stagnitti ldquoAnalytical approximations for flow in com-pressible saturated one-dimensional porous mediardquo Advancesin Water Resources vol 30 no 4 pp 927ndash936 2007
[19] T Friedel and H D Voigt ldquoAnalytical solutions for the radialflow equation with constant-rate and constant-pressure bound-ary conditions in reservoirs with pressure-sensitive perme-abilityrdquo in Proceedings of the SPE Rocky Mountain PetroleumTechnology Conference Denver Colo USA 2009
[20] B Ju YWu and T Fan ldquoStudy on fluid flow in nonlinear elasticporousmedia experimental andmodeling approachesrdquo Journalof Petroleum Science and Engineering vol 76 no 3-4 pp 205ndash211 2011
[21] D Swartzendruber ldquoNon-Darcy flow behavior in liquid satu-rated porousmediardquo Journal of Geophysical Research vol 67 no13 pp 5205ndash5213 1962
[22] R J Miller and F L Philip ldquoThreshold Gradient for water flowin clay systemrdquo Soil Science Society of America Journal vol 27no 6 pp 605ndash609 1963
[23] H W Olsen ldquoDeviations from Darcyrsquos law in saturated claysrdquoSoil Science Society of America Journal vol 29 no 2 pp 135ndash1401965
[24] H Pascal ldquoNonsteady flow through porous media in the pre-sence of a threshold gradientrdquo Acta Mechanica vol 39 no 3-4pp 207ndash224 1981
[25] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999
[26] J Lu and S Ghedan ldquoPressure behavior of vertical wells inlow-permeability reservoirs with threshold pressure gradientrdquoSpecial Topics and Reviews in Porous Media vol 2 no 3 pp157ndash169 2011
[27] Y L Zhao L H Zhang F Wu B N Zhang and Q G LiuldquoAnalysis of horizontal well pressure behaviour in fracturedlow permeability reservoirs with consideration of the thresholdpressure gradientrdquo Journal of Geophysics and Engineering vol10 no 3 pp 1ndash10 2013
[28] Y L Zhao L H Zhang J Z Zhao S Y Hu and B N ZhangldquoTransient pressure analysis of horizontal well in low per-meability oil reservoirrdquo International Journal of Oil Gas andCoal Technology 2014
[29] M Muskat The Flow of Homogeneous Fluids through a PorousMedia Intl Human Resources Development Corp BostonMass USA 1937
[30] I S Gradshteyn and I M Ryzhik Table of Integrals Seriesand Products Academic Press San Diego Calif USA Seventhedition 2007
Submit your manuscripts athttpwwwhindawicom
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OptimizationJournal of
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International 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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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
L
B
AA
B
h2
h
re
Figure 1 Schematic of a horizontal well in a top and bottomboundaries closed reservoir
in low-permeability reservoir which takes the permeabilitystress-sensitive and threshold pressure gradient into account
2 Theoretical Analysis
21 Physical Model Figure 1 is a schematic of a horizontalwell drilled in low-permeability oil reservoirs To make theproblemmore tractable the following assumptions aremade(1) the reservoir is horizontal homogeneous with uniformthickness of ℎ and both the top and the bottom boundariesare closed (2) the well length and radius are 119871 and 119903
119908
respectively the well is located at the center of the formationand the pressure at the drainage boundary is 119901
119894with the
radius of 119903119890 (3) two-phase fluid flows from the reservoir to
the well at a constant bottomhole pressure 119901119908119891
and ignore thegravity and capillary pressure effects
To simplify the mathematical solution the 3D problemis subdivided into two 2D problems [7] Figure 2 shows thefollowing subdivision of the ellipsoidal drainage problem (1)fluid flow into the horizontal well in the horizontal plane (2)fluid flow into the horizontal well in the vertical plane
22 PermeabilityModel in Low-Permeability Reservoir In thispaper we use the exponential permeability model to describethe relationship between permeability and pore pressure [14ndash20] This model is based on a permeability modulus 120572
119896
which is defined as the following
120572119896=
1
119896
120597119896
120597119901 (1)
Integrating of (1) with 119901 from 119901119894to 119901 yields
119896
119896119894
= exp [minus120572119896(119901119894minus 119901)] (2)
where 120572119896is the permeability modulus MPaminus1 119901 is the
formation pressure MPa 119901119894is the initial pressure MPa 119896
119894is
the permeability at initial condition D 119896 is the permeabilityat current condition D
23Threshold PressureGradient inOilWater Two-Phase FlowAccording to the experimental investigations (the schematic
of core sample experiments is shown in Figure 3) an appro-priate form of Darcyrsquos law with a threshold gradient shouldbe used as [21ndash28]
V =
minus119896
120583(120597119901
120597119903minus 119866)
120597119901
120597119903gt 119866
0120597119901
120597119903le 119866
(3)
where V is velocity of the fluid flowms120583 is the fluid viscositycp 119903 is the radius m 119866 is the threshold pressure gradientMPam
24 Flow in a Horizontal Plane Figure 2(a) shows aschematic of fluid flow to a horizontal well in a horizontalplane The drainage area is an ellipse and introduces
119911
1198712=1
2(120585 +
1
120585) (4)
by defining 119911 = 119909+119894 sdot119910 and 120585 = 119906+119894 sdotV = 119903119890119867
sdot 119890119894120579 Substitutingthem into (4) and then equating the real and imaginary partsyield
119909
1198712=1
2[119903119890119867
+1
119903119890119867
] cos (120579)
119910
1198712=1
2[119903119890119867
minus1
119903119890119867
] sin (120579) (5)
The real equation of the elliptical drainage area can bedescribed as
1199092
1198862+1199102
1198872= 1 (6)
where 119886 and 119887 are the major and minor axes of the ellipse mCombing (5) with (6) we have
119903119890119867
=119886 + 119887
05119871 (7)
Moreover +1198712 and minus1198712 represent foci of the ellipsewhich have the following relationship with 119886 and 119887
1198862
minus 1198872
= (119871
2)2
(8)
The drainage radius of horizontal well 119903119890 can be obtained
by equaling the areas of a circle and ellipse which is
119903119890= radic119886119887 (9)
Combing (9) with (8) we have
119886 = 05119871[
[
05 + radic025 + (2119903119890
119871)4
]
]
05
119887 = radic1198862 minus (119871
2)2
(10)
Mathematical Problems in Engineering 3
+
z
x
h2
h2x
minusL2
y
L2
(a) (b)
Figure 2 Schematic of the fluid flow to a horizontal well (a) horizontal plane and (b) vertical plane
Pressure gradient (MPam)
Velo
city
(ms
)
G (TPG)
Figure 3 A schematic of core sample laboratory experiments inlow-permeability reservoir
So as shown in Figure 4 the fluid flow in the 119906-V planecan be viewed as a unit radius vertical well produced at a circledrainage area with the radius of (119886 + 119887)(05119871) Combiningthe modified Darcy flow equation with the steady-stategoverning equation the fluid flow in the horizontal plane canbe described as
1
119903
dd119903
[119903119890minus120572119896(119901119894minus119901)120582
119905(d119901d119903
minus 1198661015840
)] = 0 (11)
1199011003816100381610038161003816119903=1 = 119901
119888 (12)
1199011003816100381610038161003816119903=119903119890119867
= 119901119894 (13)
where 120582119905is the total mobility of the oilwater phase 120582
119905=
119896119903119900120583119900+ 119896119903119908120583119908 which is a function of the radius and 1198661015840 is
the equivalent threshold pressure gradient in the horizontalplane which is
1198661015840
=119866 (119903119890minus 1198712)
((119886 + 119887) 05119871) minus 1 (14)
The solution of mathematical models of (11)ndash(13) is (thedetailed derivations are showed in the Appendix)
119901 (119903) = 119901119894
+1
120572119896
ln1198901205721198961198661015840119903
[119890minus120572119896(119866
1015840+119901119894minus119901119888)
+119890minus120572119896119866
1015840119903119890119867 minus 119890minus120572119896(119866
1015840+119901119894minus119901119888)
int119903119890119867
1
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
sdot int119903
1
119890minus120572119896119866
1015840119903
1
119903120582119905(119903)
d119903]
(15)
The liquid flow rate in horizontal plane can be calculatedby
119902119871119867
= 119862ℎ119896120582119905(1) sdot (119903
d119901d119903
)119903=1
(16)
where 119862 is the unit conversion factor 119862 = 542 119902119871119867
is theliquid flow rate at reservoir condition m3d
Combining (15) with (16) the liquid flow rate in the hori-zontal plane is
119902119871119867
= 119862119896119894119890minus120572119896(119901119894minus119901119888)ℎ
sdot
120582119905(1) sdot 119866
1015840
+[119890minus120572119896[119866
1015840119903119890119867minus(119901119894minus119901119888)] minus 119890minus120572119896119866
1015840
]
(120572119896int119903119890119867
1
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903)
(17)
The corresponding oil production rate in the horizontalplane is
119902119900119867
= 119862119896119894119890minus120572119896(119901119894minus119901119888)ℎ120582
119900(1)
sdot [1198661015840
+1
120582119905(1)
sdot119890minus120572119896[119866
1015840119903119890119867minus(119901119894minus119901119888)] minus 119890minus120572119896119866
1015840
120572119896int119903119890119867
1
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
]
(18)
4 Mathematical Problems in Engineering
y
O
(minusL2 0) (L2 0)x
1u
reH
z
L2=
1
2(120585 + 1
120585)
Figure 4 The schematic of fluid flow in horizontal plane after conformal transformation
h2
h2
z
x120585 = (1 minus eminus120587zh) times (1 + e120587zh)
u1
120585
2120587rwh
Figure 5 The schematic of fluid flow in vertical plane after conformal transformation
where 119902119900119867
is the oil flow rate at reservoir condition m3d120582119905(119903119908) is the total mobility of the oilwater phase in the well
bottomhole DcpIf we do not take into account the effects of permeability
stress-sensitive and threshold pressure gradient on the pro-ductivity equation in horizontal plane (120572
119896= 119866 = 0) and
assume only oil flow in the formation (120582119905(119903) = 1120583
119900) then
(17) becomes
119902SOH =0542 times 10minus3119896
119894ℎ (119901119894minus 119901119888)
119861119900120583119900ln (2 (119886 + 119887) 119871)
(19)
where 119902SOH is the oil flow rate at surface condition m3d 119861119900
is the oil volume factor sm3m3Equation (19) is the same as the productivity equation in
horizontal plane of (1) derived by Joshi [7]
25 Calculation of Flow in a Vertical Plane The fluid flowin the vertical plane with top and bottom boundaries closedreservoir can be viewed as a vertical well with the radiusof 120585119908
= 2120587119903119908ℎ produced in a unit circle area after the
conformal transformation (as shown in Figure 5)The mathematical models to describe the steady-state
fluid flow of oilwater flow in the vertical plane are
1
119903
dd119903
[119903119890minus120572119896(119901119894minus119901)120582
119905(d119901d119903
minus 11986610158401015840
)] = 0
1199011003816100381610038161003816119903=120585119908
= 119901119908119891
1199011003816100381610038161003816119903=1 = 119901
119888
(20)
where 11986610158401015840 is the equivalent threshold pressure gradient in thevertical plane which is
11986610158401015840
=(ℎ2) minus 119903
119908
1 minus (2120587119903119908ℎ)
119866 (21)
The solution of the pressure distribution along the radiuscan be solvedwith the samemethod showed in the AppendixThen the productivity equation in the vertical plane can becalculated by the following equation
119902119871119881
= 119862119871119896120582119905(120585119908) sdot (119903
d119901d119903
)119903=120585119908
(22)
The liquid and oil productivity in the vertical plane are
119902119871119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)119871
sdot
120582119905(120585119908) sdot 11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120572119896int1
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowast)
119902119900119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)119871120582
119900(120585119908)
times
11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120582119905(120585119908) 120572119896int1
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowastlowast)
Assuming the reservoir with single phase flow andneglecting the permeability stress-sensitive and thresholdpressure gradient (120572
119896= 119866 = 0 120582
119905(119903) = 1120583
119900) yield
119902119900119881
=119862119896119894119871 (119901119888minus 119901119908119891)
119861119900120583119900ln (ℎ2119903
119908) (23)
Mathematical Problems in Engineering 5
Equation (23) is similar to the productivity equation invertical plane of (D-3) derived by Joshi [7] which proves thecorrectness of the productivity equation in this paper
26 Horizontal Well Eccentricity Figure 5 and (lowast)-(lowastlowast) areobtained under the assumption that the horizontal well islocated at the center of the reservoir in the vertical planeAccording to Muskatrsquos [29] formulation for off-centeredwells the liquid production rate of a well placed at a distance120575 from the mid-height of the reservoir in a vertical plane is
119902119871119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)ℎ
sdot
120582119905(120585119908) sdot 1198661015840
sdot 120585119908
+[119890minus120572119896[119866
1015840120573minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
1015840120585119908]
(120572119896int120573
120585119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowast1015840
)
The oil production rate is
119902119900119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)ℎ120582
119900(120585119908)
times
11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840120573minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120582119905(120585119908) 120572119896int120573
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowastlowast1015840)
where 120575 is the vertical distance between the reservoir centerand horizontal well location m 120573 = 1 minus (2120575ℎ)
2
3 The Solving Method of 120582119905(119903)
In order to correctly calculate the productivity of horizontalwell with oilwater phase flow we must determinate theexpression of total mobility 120582
119905(119903) along the radius 119903 The
following steps are the procedure to calculate it
Step 1 According to the relative permeability curves and theviscosity of oil and water the relationship between 120582
119905(119903) and
119878119908can be obtained
Step 2 When the water displacement front breaks throughthe oil well the distribution of water saturation along the wellradius satisfies the following Buckley-Leverett equation
120587ℎ120601 (1199032
119890minus 1199032
) =d119891119908
d119904119908
sdot sum119876119897 (24)
where ℎ is the formation thickness m 120601 is the porosityfractionsum119876
119897is the cumulative fluid production m3d 119891
119908is
the water ratio fraction 119891119908= (119896119903119908120583119908)(119896119903119900120583119900) + (119896119903119908120583119908)
120582t(r) sim r
p1 larrminus pe p2 larrminus pwf
pc = (p1 + p2)2
qL eq (lowast)qHL eq (17)
If |qL minus qHL| gt eps
No
Yes If qL lt qHL p2 larrminus pc
If qL gt qHL p1 larrminus pc
0 re h Q1 120583o 120583w
kro sim Sw krw sim Sw
re h L rw G120572k pe pwf ki
Output the liquid and oil production rate
Figure 6 The productivity calculation diagram
In (24) the expressions of the d119891119908d119878119908versus 119904
119908can be
calculated by the relative permeability curves
Step 3 Combining the relationship of 120582119905(119903) versus 119878
119908and 119878119908
versus 119903 derived in Steps 1 and 2 the relationship of 120582119905(119903)with
119903 can be obtained
Statistical results show that the relationships of 119878119908versus
119903 120582119905(119903) versus 119878
119908 and 120582
119905(119903) versus 119903 can be approximated by
the following analytical functions
119878119908(119903) = 119886
1+ 11988711199032
120582119905(119878119908) = 1198862+ 1198872119878119908+ 11988821198782
119908
120582119905(119903) = 119886
3+ 11988731199032
+ 11988831199034
(25)
4 The Productivity Calculation Process
Using the electrical analog concept the well production ratein the horizontal plane must equal the production rate in thevertical plane Because of the nonlinearity of (17)-(18) and(lowast)-(lowastlowast) we cannot obtain productivity expressions similarto Joshi [7] With the aid of computer programs the resultscan be obtained and the calculation diagram is showed inFigure 6
5 Results and Their Sensitive Analysis
In this section the liquid and oil production rate are calcu-lated and the essential parameters of well reservoir and fluidproperties are listed in Table 1 and the relative permeabilitycurves are showed in Figure 7
According to the relative permeability curves and theparameters in Table 1 the relationship of 120582
119905(119878119908) versus 119878
119908
6 Mathematical Problems in Engineering
00
02
04
06
08
10
00 01 02 03 04 05 06 07 08 09 10
krok
rw
krwkro
Sw
Figure 7 The relative permeability curves
00
02
04
06
08
10
00 01 02 03 04 05 06 07 08 09 10Sw
120582t
120582t = 32547S2w minus 23096Sw + 04917
R2 = 1
Figure 8 The relationship between 120582119905(119878119908) and 119878
119908
Table 1 The parameters of reservoir and fluid properties
Variables valuePorosity 0 (fraction) 005Drainage radius 119903
119890(m) 600
Horizontal well length 119871 (m) 300Initial permeability of the reservoir 119896
119894(D) 2 times 10minus3
Cumulative fluid production sum119876119897(m3) 114 times 106
Oil viscosity in the reservoir 120583119900(mPasdots) 3
Formation thickness ℎ (m) 20Initial reservoir pressure 119901
119890(MPa) 30
Permeability modulus 120572119896(MPaminus1) 01
Water saturation in the bottomhole 119878119908119861
(fraction) 06Threshold pressure gradient 119866 (MPam) 0001Water viscosity in the reservoir 120583
119908(mPasdots) 1
and d119891119908d119878119908versus 119878
119908can be plotted when water cut ratio
reaches 06 (as shown in Figures 8 and 9)The correspondingregression curve equations can be obtained as follows
120582119905= 04917 minus 23096119878
119908+ 32547119878
2
119908
d119891119908
d119878119908
= minus60226119878119908+ 43193 119878
119908gt 06
(26)
0
1
2
3
4
5
6
02 03 04 05 06 07 08 09 10
000204060810
060 062 064 066 068 070dfwdS w df
wdS w
dfwdSw = minus60226Sw + 43193
R2 = 09716
Sw
Sw
Figure 9 The relationship between d119891119908d119878119908and 119878
119908
0
5
10
15
20
25
0 005 01 015 02 025 03
Liquid production rateOil production rate
Rate
(m3d
)
120572k (MPaminus1)
Figure 10 The effect of permeability modulus (120572119896) on liquid
production rate
Taking (26) as well as the cumulative fluid productioninto (24) the expressions between the 120582
119905(119903) and 119903 can be
obtained which is
120582119905= 02 + 153 times 10
minus6
1199032
+ 68 times 10minus13
1199034
(27)
Combining (27) and other parameters in Table 1 with(17)-(18) and (lowast)-(lowastlowast) the steady-state fluid productivity canbe calculated
Figure 10 shows the effect of permeability modulus 120572119896
on liquid and oil productivity of horizontal well in lowpermeability reservoir It can be seen from the figure thatthe permeability stress-sensitive has a significant effect on theproductivity the bigger the 120572
119896is the smaller the liquid and
oil productivity are which is mainly because with the samepressure drop of the reservoir big 120572
119896will lead to a serious
permeability decreasing When we do not take into accountthe permeability stress sensitive (120572
119896= 0) the liquid and oil
productivity can be calculated with the limit of 120572119896tending to
zero for (17)-(18) and (lowast)-(lowastlowast)Figures 11 and 12 show the effect of threshold pressure
gradient (119866) and well length (119871) on liquid and oil pro-ductivity It can be seen from the chart that the thresholdpressure gradient has small effect on the productivity ofhorizontal well for big drainage volume In general the bigger
Mathematical Problems in Engineering 7
Table 2 The productivity of liquid and oil in different bottomhole pressure
119901119908119891
(MPa)120572119896= 01 119866 = 0001 120572
119896= 0 119866 = 0
Liquid production rate(m3d)
Oil production rate(m3d)
Liquid production rate(m3d)
Oil production rate(m3d)
3000 000 000 000 0002980 000 000 039 0192960 000 000 079 0372950 033 016 098 0472700 458 218 589 2802400 861 409 1177 5602100 1171 556 1766 84018 1407 669 235 111915 1585 754 2944 13992 1720 818 3532 1679
0
2
4
6
8
10
12
14
0 0003 0006 0009 0012 0015
Liquid production rateOil production rate
G (MPaminus1m)
Rate
(m3d
)
Figure 11 The effect of threshold pressure gradient (119866) on liquidproduction rate
0
5
10
15
20
100 150 200 250 300 350 400 450
Liquid production rateOil production rate
Rate
(m3d
)
L (m)
Figure 12 The effect of well length (119871) on liquid production rate
the 119866 is the smaller the liquid and oil productivity areWhen the reservoir has both threshold pressure gradient andpermeability stress sensitive the longer the well length is thebigger the productivity is
048
12162024283236
12 14 16 18 20 22 24 26 28 30
Liquid production rate Oil production rate
Prod
uctio
n ra
te (m
3d
)
Liquid production rate Oil production rate
pwf (MPa)
(120572k = G = 0) (120572k = G = 0)
Figure 13 The productivity of horizontal well in different bottom-hole pressure
Figure 13 shows the liquid productivity with differentbottomhole pressure when 120572
119896= 01 119866 = 0001 and 120572
119896=
0 119866 = 0 the corresponding values are listed in Table 2It can be clearly seen that the permeability stress-sensitiveand threshold pressure gradient have significant effects onthe well productivity and the bigger 120572
119896and 119866 are the more
obvious the effect is And when the pressure drop is smallthe fluid cannot flow for the existing of threshold pressuregradient which is mainly because only the fluid can flowwhen the pressure drop overcomes the threshold pressure formultiphase flow
6 Conclusions
In this paper a semianalytical productivity equation of hor-izontal well in low-permeability oil reservoir with oilwatertwo-phase flow is established with the consideration ofpermeability stress-sensitive and threshold pressure gradientBased on the above study the following conclusions can besummarized
8 Mathematical Problems in Engineering
(1) The steady-state percolation mathematical modelsof horizontal well with oilwater two-phase floware established and the corresponding solutions aresolved by the method of separation of variables
(2) For low-permeability reservoir there always existsthe phenomenon of permeability stress-sensitive (120572
119896)
which has a significant influence on the well produc-tivity the bigger the 120572
119896is the smaller the productivity
is(3) Due to the existence of capillary pressure of two-
phase flow there always is threshold pressure gradient(119866) in the fluid seepage process Although the 119866 hasa smaller effect on the productivity of the horizontalwell for a big drainage volume we cannot neglect itseffect on the productivity
Appendix
From the expressions of (11) we have
119903120582119905119890minus120572119896(119901119894minus119901) (
d119901d119903
minus 1198661015840
) = 119888 (A1)
where 119888 is a constantWe define the following expression
119910 = 119890minus120572119896(119901119894minus119901) (A2)
Substituting (A2) into (A1) and (12)-(13) yields
d119910d119903
= 1205721198961198661015840
119910 +119888120572119896
119903120582119905
(A3)
1199101003816100381610038161003816119903=1 = 119890
minus120572119896(119901119894minus119901119888) (A4)
1199101003816100381610038161003816119903=119903119890119867
= 1 (A5)
According to the general solution of the Bernoulli differ-ential equation [30] the solution of (A3) can be obtained as
119910 (119903) = 1198901205721198961198661015840119903
sdot [119890minus120572119896119866
1015840
119890minus120572119896(119901119894minus119901119888) + int
119903
119903119908
119890minus120572119896119866
1015840119903
119888120572119896
119903120582119905(119903)
d119903] (A6)
Substituting (A6) into (A5) the value of constant 119888 canbe solved and then combining it with (A6) we have
119901 (119903)
= 119901119894+
1
120572119896
ln
1198901205721198961198661015840119903
sdot [
[
119890minus120572119896(119866
1015840+119901119894minus119901119888) +
119890minus1205721198961198661015840119903119890119867 minus 119890minus120572119896(119866
1015840+119901119894minus119901119888)
int119903119890119867
119903119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
sdot int119903
119903119908
119890minus120572119896119866
1015840119903
1
119903120582119905(119903)
d119903]
]
(A7)
Equation (A7) is the pressure distribution relation alongthe radius 119903 with the oilwater two-phase flows in thehorizontal plane of the horizontal well
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Natural Science Foundationof China (Grant no 51374181) and the project of NationalScience Fund for Distinguished Young Scholars of China(Grant no 51125019)The authors would also like to thank thereviewers and editors for their patience to read this paper andvaluable comments
References
[1] V P Merkulov ldquoLe debit des puits devies et horizontauxrdquo NeftKhoz vol 6 no 1 pp 51ndash56 1958
[2] J P Borisov Oil Production Using Horizontal and MultipleDeviation Wells The RampD Translation Company BartlesvilleOkla USA 1984
[3] F M Giger L H Reiss and A P Jourdan ldquoThe reservoir engi-neering aspects of horizontal drillingrdquo in Proceedings of the59th Annual Technical Conference and Exhibition Houston TexUSA 1984
[4] F M Giger ldquoHorizontal wells production techniques in het-erogeneous reservoirsrdquo in Proceedings of the Middle East OilTechnical Conference and Exhibition Bahrain 1985
[5] B J Karcher and F M Giger ldquoSome practical formulas to pre-dict horizontal well behaviorrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition New Orleans La USA1986
[6] L H Reiss ldquoProduction from horizontal wells after five yearsrdquoJournal of Petroleum Technology vol 39 no 11 pp 1411ndash14161987
[7] S D Joshi ldquoAugmentation of well productivity using slant andhorizontal wellsrdquo Journal of Petroleum Technology vol 40 no6 pp 729ndash739 1988
[8] D K Babu and A S Odeh ldquoProductivity of a horizontal wellrdquoSPE Reservoir Engineering vol 4 no 4 pp 417ndash421 1989
[9] G Renard and JM Dupuy ldquoFormation damage effects on hori-zontal-well flow efficiencyrdquo Journal of Petroleum Technologyvol 43 no 7 pp 786ndash789 1991
[10] M W Helmy and R A Wattenbarger ldquoSimplified productivityequations for horizontal wells producing at constant rate andconstant pressurerdquo in Proceedings of the SPE Technical Con-ference and Exhibition pp 379ndash388 New Orleans La USASeptember 1998
[11] T Billiter J Lee and R Chase ldquoDimensionless inflow-perform-ance-relationship curve for unfractured horizontal gas wellsrdquo inProceedings of the SPE Eastern Regional Meeting Canton OhioUSA 2001
[12] E G Anklam and M L Wiggins ldquoHorizontal well produc-tivity and wellbore pressure behavior incorporating wellborehydraulicsrdquo in Proceedings of the SPE Production andOperations
Mathematical Problems in Engineering 9
Symposium pp 565ndash584 Oklahoma City Okla USA April2005
[13] J Lu ldquoNewproductivity formulae of horizontal wellsrdquo Journal ofCanadian Petroleum Technology vol 40 no 10 pp 55ndash67 2001
[14] F Samaniego W E Brigham and F G Miller ldquoPerformance-prediction procedure for transient flow of fluids throughpressure-sensitive formationsrdquo Journal of PetroleumTechnologyvol 31 no 6 pp 779ndash786 1979
[15] G K Falade ldquoTransient flow of fluids in reservoirs with stresssensitive rock and fluid propertiesrdquo International Journal ofNon-Linear Mechanics vol 17 no 4 pp 277ndash283 1982
[16] R W Ostensen ldquoMicrocrack Permeability in tight gas sand-stonerdquo Society of Petroleum Engineers Journal vol 23 no 6 pp919ndash927 1983
[17] J Pedrosa and O A Petrobras ldquoPressure transient response instress-sensitive formationsrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA 1986
[18] D A Barry D A Lockington D-S Jeng J-Y Parlange L Liand F Stagnitti ldquoAnalytical approximations for flow in com-pressible saturated one-dimensional porous mediardquo Advancesin Water Resources vol 30 no 4 pp 927ndash936 2007
[19] T Friedel and H D Voigt ldquoAnalytical solutions for the radialflow equation with constant-rate and constant-pressure bound-ary conditions in reservoirs with pressure-sensitive perme-abilityrdquo in Proceedings of the SPE Rocky Mountain PetroleumTechnology Conference Denver Colo USA 2009
[20] B Ju YWu and T Fan ldquoStudy on fluid flow in nonlinear elasticporousmedia experimental andmodeling approachesrdquo Journalof Petroleum Science and Engineering vol 76 no 3-4 pp 205ndash211 2011
[21] D Swartzendruber ldquoNon-Darcy flow behavior in liquid satu-rated porousmediardquo Journal of Geophysical Research vol 67 no13 pp 5205ndash5213 1962
[22] R J Miller and F L Philip ldquoThreshold Gradient for water flowin clay systemrdquo Soil Science Society of America Journal vol 27no 6 pp 605ndash609 1963
[23] H W Olsen ldquoDeviations from Darcyrsquos law in saturated claysrdquoSoil Science Society of America Journal vol 29 no 2 pp 135ndash1401965
[24] H Pascal ldquoNonsteady flow through porous media in the pre-sence of a threshold gradientrdquo Acta Mechanica vol 39 no 3-4pp 207ndash224 1981
[25] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999
[26] J Lu and S Ghedan ldquoPressure behavior of vertical wells inlow-permeability reservoirs with threshold pressure gradientrdquoSpecial Topics and Reviews in Porous Media vol 2 no 3 pp157ndash169 2011
[27] Y L Zhao L H Zhang F Wu B N Zhang and Q G LiuldquoAnalysis of horizontal well pressure behaviour in fracturedlow permeability reservoirs with consideration of the thresholdpressure gradientrdquo Journal of Geophysics and Engineering vol10 no 3 pp 1ndash10 2013
[28] Y L Zhao L H Zhang J Z Zhao S Y Hu and B N ZhangldquoTransient pressure analysis of horizontal well in low per-meability oil reservoirrdquo International Journal of Oil Gas andCoal Technology 2014
[29] M Muskat The Flow of Homogeneous Fluids through a PorousMedia Intl Human Resources Development Corp BostonMass USA 1937
[30] I S Gradshteyn and I M Ryzhik Table of Integrals Seriesand Products Academic Press San Diego Calif USA Seventhedition 2007
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
+
z
x
h2
h2x
minusL2
y
L2
(a) (b)
Figure 2 Schematic of the fluid flow to a horizontal well (a) horizontal plane and (b) vertical plane
Pressure gradient (MPam)
Velo
city
(ms
)
G (TPG)
Figure 3 A schematic of core sample laboratory experiments inlow-permeability reservoir
So as shown in Figure 4 the fluid flow in the 119906-V planecan be viewed as a unit radius vertical well produced at a circledrainage area with the radius of (119886 + 119887)(05119871) Combiningthe modified Darcy flow equation with the steady-stategoverning equation the fluid flow in the horizontal plane canbe described as
1
119903
dd119903
[119903119890minus120572119896(119901119894minus119901)120582
119905(d119901d119903
minus 1198661015840
)] = 0 (11)
1199011003816100381610038161003816119903=1 = 119901
119888 (12)
1199011003816100381610038161003816119903=119903119890119867
= 119901119894 (13)
where 120582119905is the total mobility of the oilwater phase 120582
119905=
119896119903119900120583119900+ 119896119903119908120583119908 which is a function of the radius and 1198661015840 is
the equivalent threshold pressure gradient in the horizontalplane which is
1198661015840
=119866 (119903119890minus 1198712)
((119886 + 119887) 05119871) minus 1 (14)
The solution of mathematical models of (11)ndash(13) is (thedetailed derivations are showed in the Appendix)
119901 (119903) = 119901119894
+1
120572119896
ln1198901205721198961198661015840119903
[119890minus120572119896(119866
1015840+119901119894minus119901119888)
+119890minus120572119896119866
1015840119903119890119867 minus 119890minus120572119896(119866
1015840+119901119894minus119901119888)
int119903119890119867
1
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
sdot int119903
1
119890minus120572119896119866
1015840119903
1
119903120582119905(119903)
d119903]
(15)
The liquid flow rate in horizontal plane can be calculatedby
119902119871119867
= 119862ℎ119896120582119905(1) sdot (119903
d119901d119903
)119903=1
(16)
where 119862 is the unit conversion factor 119862 = 542 119902119871119867
is theliquid flow rate at reservoir condition m3d
Combining (15) with (16) the liquid flow rate in the hori-zontal plane is
119902119871119867
= 119862119896119894119890minus120572119896(119901119894minus119901119888)ℎ
sdot
120582119905(1) sdot 119866
1015840
+[119890minus120572119896[119866
1015840119903119890119867minus(119901119894minus119901119888)] minus 119890minus120572119896119866
1015840
]
(120572119896int119903119890119867
1
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903)
(17)
The corresponding oil production rate in the horizontalplane is
119902119900119867
= 119862119896119894119890minus120572119896(119901119894minus119901119888)ℎ120582
119900(1)
sdot [1198661015840
+1
120582119905(1)
sdot119890minus120572119896[119866
1015840119903119890119867minus(119901119894minus119901119888)] minus 119890minus120572119896119866
1015840
120572119896int119903119890119867
1
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
]
(18)
4 Mathematical Problems in Engineering
y
O
(minusL2 0) (L2 0)x
1u
reH
z
L2=
1
2(120585 + 1
120585)
Figure 4 The schematic of fluid flow in horizontal plane after conformal transformation
h2
h2
z
x120585 = (1 minus eminus120587zh) times (1 + e120587zh)
u1
120585
2120587rwh
Figure 5 The schematic of fluid flow in vertical plane after conformal transformation
where 119902119900119867
is the oil flow rate at reservoir condition m3d120582119905(119903119908) is the total mobility of the oilwater phase in the well
bottomhole DcpIf we do not take into account the effects of permeability
stress-sensitive and threshold pressure gradient on the pro-ductivity equation in horizontal plane (120572
119896= 119866 = 0) and
assume only oil flow in the formation (120582119905(119903) = 1120583
119900) then
(17) becomes
119902SOH =0542 times 10minus3119896
119894ℎ (119901119894minus 119901119888)
119861119900120583119900ln (2 (119886 + 119887) 119871)
(19)
where 119902SOH is the oil flow rate at surface condition m3d 119861119900
is the oil volume factor sm3m3Equation (19) is the same as the productivity equation in
horizontal plane of (1) derived by Joshi [7]
25 Calculation of Flow in a Vertical Plane The fluid flowin the vertical plane with top and bottom boundaries closedreservoir can be viewed as a vertical well with the radiusof 120585119908
= 2120587119903119908ℎ produced in a unit circle area after the
conformal transformation (as shown in Figure 5)The mathematical models to describe the steady-state
fluid flow of oilwater flow in the vertical plane are
1
119903
dd119903
[119903119890minus120572119896(119901119894minus119901)120582
119905(d119901d119903
minus 11986610158401015840
)] = 0
1199011003816100381610038161003816119903=120585119908
= 119901119908119891
1199011003816100381610038161003816119903=1 = 119901
119888
(20)
where 11986610158401015840 is the equivalent threshold pressure gradient in thevertical plane which is
11986610158401015840
=(ℎ2) minus 119903
119908
1 minus (2120587119903119908ℎ)
119866 (21)
The solution of the pressure distribution along the radiuscan be solvedwith the samemethod showed in the AppendixThen the productivity equation in the vertical plane can becalculated by the following equation
119902119871119881
= 119862119871119896120582119905(120585119908) sdot (119903
d119901d119903
)119903=120585119908
(22)
The liquid and oil productivity in the vertical plane are
119902119871119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)119871
sdot
120582119905(120585119908) sdot 11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120572119896int1
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowast)
119902119900119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)119871120582
119900(120585119908)
times
11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120582119905(120585119908) 120572119896int1
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowastlowast)
Assuming the reservoir with single phase flow andneglecting the permeability stress-sensitive and thresholdpressure gradient (120572
119896= 119866 = 0 120582
119905(119903) = 1120583
119900) yield
119902119900119881
=119862119896119894119871 (119901119888minus 119901119908119891)
119861119900120583119900ln (ℎ2119903
119908) (23)
Mathematical Problems in Engineering 5
Equation (23) is similar to the productivity equation invertical plane of (D-3) derived by Joshi [7] which proves thecorrectness of the productivity equation in this paper
26 Horizontal Well Eccentricity Figure 5 and (lowast)-(lowastlowast) areobtained under the assumption that the horizontal well islocated at the center of the reservoir in the vertical planeAccording to Muskatrsquos [29] formulation for off-centeredwells the liquid production rate of a well placed at a distance120575 from the mid-height of the reservoir in a vertical plane is
119902119871119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)ℎ
sdot
120582119905(120585119908) sdot 1198661015840
sdot 120585119908
+[119890minus120572119896[119866
1015840120573minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
1015840120585119908]
(120572119896int120573
120585119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowast1015840
)
The oil production rate is
119902119900119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)ℎ120582
119900(120585119908)
times
11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840120573minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120582119905(120585119908) 120572119896int120573
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowastlowast1015840)
where 120575 is the vertical distance between the reservoir centerand horizontal well location m 120573 = 1 minus (2120575ℎ)
2
3 The Solving Method of 120582119905(119903)
In order to correctly calculate the productivity of horizontalwell with oilwater phase flow we must determinate theexpression of total mobility 120582
119905(119903) along the radius 119903 The
following steps are the procedure to calculate it
Step 1 According to the relative permeability curves and theviscosity of oil and water the relationship between 120582
119905(119903) and
119878119908can be obtained
Step 2 When the water displacement front breaks throughthe oil well the distribution of water saturation along the wellradius satisfies the following Buckley-Leverett equation
120587ℎ120601 (1199032
119890minus 1199032
) =d119891119908
d119904119908
sdot sum119876119897 (24)
where ℎ is the formation thickness m 120601 is the porosityfractionsum119876
119897is the cumulative fluid production m3d 119891
119908is
the water ratio fraction 119891119908= (119896119903119908120583119908)(119896119903119900120583119900) + (119896119903119908120583119908)
120582t(r) sim r
p1 larrminus pe p2 larrminus pwf
pc = (p1 + p2)2
qL eq (lowast)qHL eq (17)
If |qL minus qHL| gt eps
No
Yes If qL lt qHL p2 larrminus pc
If qL gt qHL p1 larrminus pc
0 re h Q1 120583o 120583w
kro sim Sw krw sim Sw
re h L rw G120572k pe pwf ki
Output the liquid and oil production rate
Figure 6 The productivity calculation diagram
In (24) the expressions of the d119891119908d119878119908versus 119904
119908can be
calculated by the relative permeability curves
Step 3 Combining the relationship of 120582119905(119903) versus 119878
119908and 119878119908
versus 119903 derived in Steps 1 and 2 the relationship of 120582119905(119903)with
119903 can be obtained
Statistical results show that the relationships of 119878119908versus
119903 120582119905(119903) versus 119878
119908 and 120582
119905(119903) versus 119903 can be approximated by
the following analytical functions
119878119908(119903) = 119886
1+ 11988711199032
120582119905(119878119908) = 1198862+ 1198872119878119908+ 11988821198782
119908
120582119905(119903) = 119886
3+ 11988731199032
+ 11988831199034
(25)
4 The Productivity Calculation Process
Using the electrical analog concept the well production ratein the horizontal plane must equal the production rate in thevertical plane Because of the nonlinearity of (17)-(18) and(lowast)-(lowastlowast) we cannot obtain productivity expressions similarto Joshi [7] With the aid of computer programs the resultscan be obtained and the calculation diagram is showed inFigure 6
5 Results and Their Sensitive Analysis
In this section the liquid and oil production rate are calcu-lated and the essential parameters of well reservoir and fluidproperties are listed in Table 1 and the relative permeabilitycurves are showed in Figure 7
According to the relative permeability curves and theparameters in Table 1 the relationship of 120582
119905(119878119908) versus 119878
119908
6 Mathematical Problems in Engineering
00
02
04
06
08
10
00 01 02 03 04 05 06 07 08 09 10
krok
rw
krwkro
Sw
Figure 7 The relative permeability curves
00
02
04
06
08
10
00 01 02 03 04 05 06 07 08 09 10Sw
120582t
120582t = 32547S2w minus 23096Sw + 04917
R2 = 1
Figure 8 The relationship between 120582119905(119878119908) and 119878
119908
Table 1 The parameters of reservoir and fluid properties
Variables valuePorosity 0 (fraction) 005Drainage radius 119903
119890(m) 600
Horizontal well length 119871 (m) 300Initial permeability of the reservoir 119896
119894(D) 2 times 10minus3
Cumulative fluid production sum119876119897(m3) 114 times 106
Oil viscosity in the reservoir 120583119900(mPasdots) 3
Formation thickness ℎ (m) 20Initial reservoir pressure 119901
119890(MPa) 30
Permeability modulus 120572119896(MPaminus1) 01
Water saturation in the bottomhole 119878119908119861
(fraction) 06Threshold pressure gradient 119866 (MPam) 0001Water viscosity in the reservoir 120583
119908(mPasdots) 1
and d119891119908d119878119908versus 119878
119908can be plotted when water cut ratio
reaches 06 (as shown in Figures 8 and 9)The correspondingregression curve equations can be obtained as follows
120582119905= 04917 minus 23096119878
119908+ 32547119878
2
119908
d119891119908
d119878119908
= minus60226119878119908+ 43193 119878
119908gt 06
(26)
0
1
2
3
4
5
6
02 03 04 05 06 07 08 09 10
000204060810
060 062 064 066 068 070dfwdS w df
wdS w
dfwdSw = minus60226Sw + 43193
R2 = 09716
Sw
Sw
Figure 9 The relationship between d119891119908d119878119908and 119878
119908
0
5
10
15
20
25
0 005 01 015 02 025 03
Liquid production rateOil production rate
Rate
(m3d
)
120572k (MPaminus1)
Figure 10 The effect of permeability modulus (120572119896) on liquid
production rate
Taking (26) as well as the cumulative fluid productioninto (24) the expressions between the 120582
119905(119903) and 119903 can be
obtained which is
120582119905= 02 + 153 times 10
minus6
1199032
+ 68 times 10minus13
1199034
(27)
Combining (27) and other parameters in Table 1 with(17)-(18) and (lowast)-(lowastlowast) the steady-state fluid productivity canbe calculated
Figure 10 shows the effect of permeability modulus 120572119896
on liquid and oil productivity of horizontal well in lowpermeability reservoir It can be seen from the figure thatthe permeability stress-sensitive has a significant effect on theproductivity the bigger the 120572
119896is the smaller the liquid and
oil productivity are which is mainly because with the samepressure drop of the reservoir big 120572
119896will lead to a serious
permeability decreasing When we do not take into accountthe permeability stress sensitive (120572
119896= 0) the liquid and oil
productivity can be calculated with the limit of 120572119896tending to
zero for (17)-(18) and (lowast)-(lowastlowast)Figures 11 and 12 show the effect of threshold pressure
gradient (119866) and well length (119871) on liquid and oil pro-ductivity It can be seen from the chart that the thresholdpressure gradient has small effect on the productivity ofhorizontal well for big drainage volume In general the bigger
Mathematical Problems in Engineering 7
Table 2 The productivity of liquid and oil in different bottomhole pressure
119901119908119891
(MPa)120572119896= 01 119866 = 0001 120572
119896= 0 119866 = 0
Liquid production rate(m3d)
Oil production rate(m3d)
Liquid production rate(m3d)
Oil production rate(m3d)
3000 000 000 000 0002980 000 000 039 0192960 000 000 079 0372950 033 016 098 0472700 458 218 589 2802400 861 409 1177 5602100 1171 556 1766 84018 1407 669 235 111915 1585 754 2944 13992 1720 818 3532 1679
0
2
4
6
8
10
12
14
0 0003 0006 0009 0012 0015
Liquid production rateOil production rate
G (MPaminus1m)
Rate
(m3d
)
Figure 11 The effect of threshold pressure gradient (119866) on liquidproduction rate
0
5
10
15
20
100 150 200 250 300 350 400 450
Liquid production rateOil production rate
Rate
(m3d
)
L (m)
Figure 12 The effect of well length (119871) on liquid production rate
the 119866 is the smaller the liquid and oil productivity areWhen the reservoir has both threshold pressure gradient andpermeability stress sensitive the longer the well length is thebigger the productivity is
048
12162024283236
12 14 16 18 20 22 24 26 28 30
Liquid production rate Oil production rate
Prod
uctio
n ra
te (m
3d
)
Liquid production rate Oil production rate
pwf (MPa)
(120572k = G = 0) (120572k = G = 0)
Figure 13 The productivity of horizontal well in different bottom-hole pressure
Figure 13 shows the liquid productivity with differentbottomhole pressure when 120572
119896= 01 119866 = 0001 and 120572
119896=
0 119866 = 0 the corresponding values are listed in Table 2It can be clearly seen that the permeability stress-sensitiveand threshold pressure gradient have significant effects onthe well productivity and the bigger 120572
119896and 119866 are the more
obvious the effect is And when the pressure drop is smallthe fluid cannot flow for the existing of threshold pressuregradient which is mainly because only the fluid can flowwhen the pressure drop overcomes the threshold pressure formultiphase flow
6 Conclusions
In this paper a semianalytical productivity equation of hor-izontal well in low-permeability oil reservoir with oilwatertwo-phase flow is established with the consideration ofpermeability stress-sensitive and threshold pressure gradientBased on the above study the following conclusions can besummarized
8 Mathematical Problems in Engineering
(1) The steady-state percolation mathematical modelsof horizontal well with oilwater two-phase floware established and the corresponding solutions aresolved by the method of separation of variables
(2) For low-permeability reservoir there always existsthe phenomenon of permeability stress-sensitive (120572
119896)
which has a significant influence on the well produc-tivity the bigger the 120572
119896is the smaller the productivity
is(3) Due to the existence of capillary pressure of two-
phase flow there always is threshold pressure gradient(119866) in the fluid seepage process Although the 119866 hasa smaller effect on the productivity of the horizontalwell for a big drainage volume we cannot neglect itseffect on the productivity
Appendix
From the expressions of (11) we have
119903120582119905119890minus120572119896(119901119894minus119901) (
d119901d119903
minus 1198661015840
) = 119888 (A1)
where 119888 is a constantWe define the following expression
119910 = 119890minus120572119896(119901119894minus119901) (A2)
Substituting (A2) into (A1) and (12)-(13) yields
d119910d119903
= 1205721198961198661015840
119910 +119888120572119896
119903120582119905
(A3)
1199101003816100381610038161003816119903=1 = 119890
minus120572119896(119901119894minus119901119888) (A4)
1199101003816100381610038161003816119903=119903119890119867
= 1 (A5)
According to the general solution of the Bernoulli differ-ential equation [30] the solution of (A3) can be obtained as
119910 (119903) = 1198901205721198961198661015840119903
sdot [119890minus120572119896119866
1015840
119890minus120572119896(119901119894minus119901119888) + int
119903
119903119908
119890minus120572119896119866
1015840119903
119888120572119896
119903120582119905(119903)
d119903] (A6)
Substituting (A6) into (A5) the value of constant 119888 canbe solved and then combining it with (A6) we have
119901 (119903)
= 119901119894+
1
120572119896
ln
1198901205721198961198661015840119903
sdot [
[
119890minus120572119896(119866
1015840+119901119894minus119901119888) +
119890minus1205721198961198661015840119903119890119867 minus 119890minus120572119896(119866
1015840+119901119894minus119901119888)
int119903119890119867
119903119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
sdot int119903
119903119908
119890minus120572119896119866
1015840119903
1
119903120582119905(119903)
d119903]
]
(A7)
Equation (A7) is the pressure distribution relation alongthe radius 119903 with the oilwater two-phase flows in thehorizontal plane of the horizontal well
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Natural Science Foundationof China (Grant no 51374181) and the project of NationalScience Fund for Distinguished Young Scholars of China(Grant no 51125019)The authors would also like to thank thereviewers and editors for their patience to read this paper andvaluable comments
References
[1] V P Merkulov ldquoLe debit des puits devies et horizontauxrdquo NeftKhoz vol 6 no 1 pp 51ndash56 1958
[2] J P Borisov Oil Production Using Horizontal and MultipleDeviation Wells The RampD Translation Company BartlesvilleOkla USA 1984
[3] F M Giger L H Reiss and A P Jourdan ldquoThe reservoir engi-neering aspects of horizontal drillingrdquo in Proceedings of the59th Annual Technical Conference and Exhibition Houston TexUSA 1984
[4] F M Giger ldquoHorizontal wells production techniques in het-erogeneous reservoirsrdquo in Proceedings of the Middle East OilTechnical Conference and Exhibition Bahrain 1985
[5] B J Karcher and F M Giger ldquoSome practical formulas to pre-dict horizontal well behaviorrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition New Orleans La USA1986
[6] L H Reiss ldquoProduction from horizontal wells after five yearsrdquoJournal of Petroleum Technology vol 39 no 11 pp 1411ndash14161987
[7] S D Joshi ldquoAugmentation of well productivity using slant andhorizontal wellsrdquo Journal of Petroleum Technology vol 40 no6 pp 729ndash739 1988
[8] D K Babu and A S Odeh ldquoProductivity of a horizontal wellrdquoSPE Reservoir Engineering vol 4 no 4 pp 417ndash421 1989
[9] G Renard and JM Dupuy ldquoFormation damage effects on hori-zontal-well flow efficiencyrdquo Journal of Petroleum Technologyvol 43 no 7 pp 786ndash789 1991
[10] M W Helmy and R A Wattenbarger ldquoSimplified productivityequations for horizontal wells producing at constant rate andconstant pressurerdquo in Proceedings of the SPE Technical Con-ference and Exhibition pp 379ndash388 New Orleans La USASeptember 1998
[11] T Billiter J Lee and R Chase ldquoDimensionless inflow-perform-ance-relationship curve for unfractured horizontal gas wellsrdquo inProceedings of the SPE Eastern Regional Meeting Canton OhioUSA 2001
[12] E G Anklam and M L Wiggins ldquoHorizontal well produc-tivity and wellbore pressure behavior incorporating wellborehydraulicsrdquo in Proceedings of the SPE Production andOperations
Mathematical Problems in Engineering 9
Symposium pp 565ndash584 Oklahoma City Okla USA April2005
[13] J Lu ldquoNewproductivity formulae of horizontal wellsrdquo Journal ofCanadian Petroleum Technology vol 40 no 10 pp 55ndash67 2001
[14] F Samaniego W E Brigham and F G Miller ldquoPerformance-prediction procedure for transient flow of fluids throughpressure-sensitive formationsrdquo Journal of PetroleumTechnologyvol 31 no 6 pp 779ndash786 1979
[15] G K Falade ldquoTransient flow of fluids in reservoirs with stresssensitive rock and fluid propertiesrdquo International Journal ofNon-Linear Mechanics vol 17 no 4 pp 277ndash283 1982
[16] R W Ostensen ldquoMicrocrack Permeability in tight gas sand-stonerdquo Society of Petroleum Engineers Journal vol 23 no 6 pp919ndash927 1983
[17] J Pedrosa and O A Petrobras ldquoPressure transient response instress-sensitive formationsrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA 1986
[18] D A Barry D A Lockington D-S Jeng J-Y Parlange L Liand F Stagnitti ldquoAnalytical approximations for flow in com-pressible saturated one-dimensional porous mediardquo Advancesin Water Resources vol 30 no 4 pp 927ndash936 2007
[19] T Friedel and H D Voigt ldquoAnalytical solutions for the radialflow equation with constant-rate and constant-pressure bound-ary conditions in reservoirs with pressure-sensitive perme-abilityrdquo in Proceedings of the SPE Rocky Mountain PetroleumTechnology Conference Denver Colo USA 2009
[20] B Ju YWu and T Fan ldquoStudy on fluid flow in nonlinear elasticporousmedia experimental andmodeling approachesrdquo Journalof Petroleum Science and Engineering vol 76 no 3-4 pp 205ndash211 2011
[21] D Swartzendruber ldquoNon-Darcy flow behavior in liquid satu-rated porousmediardquo Journal of Geophysical Research vol 67 no13 pp 5205ndash5213 1962
[22] R J Miller and F L Philip ldquoThreshold Gradient for water flowin clay systemrdquo Soil Science Society of America Journal vol 27no 6 pp 605ndash609 1963
[23] H W Olsen ldquoDeviations from Darcyrsquos law in saturated claysrdquoSoil Science Society of America Journal vol 29 no 2 pp 135ndash1401965
[24] H Pascal ldquoNonsteady flow through porous media in the pre-sence of a threshold gradientrdquo Acta Mechanica vol 39 no 3-4pp 207ndash224 1981
[25] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999
[26] J Lu and S Ghedan ldquoPressure behavior of vertical wells inlow-permeability reservoirs with threshold pressure gradientrdquoSpecial Topics and Reviews in Porous Media vol 2 no 3 pp157ndash169 2011
[27] Y L Zhao L H Zhang F Wu B N Zhang and Q G LiuldquoAnalysis of horizontal well pressure behaviour in fracturedlow permeability reservoirs with consideration of the thresholdpressure gradientrdquo Journal of Geophysics and Engineering vol10 no 3 pp 1ndash10 2013
[28] Y L Zhao L H Zhang J Z Zhao S Y Hu and B N ZhangldquoTransient pressure analysis of horizontal well in low per-meability oil reservoirrdquo International Journal of Oil Gas andCoal Technology 2014
[29] M Muskat The Flow of Homogeneous Fluids through a PorousMedia Intl Human Resources Development Corp BostonMass USA 1937
[30] I S Gradshteyn and I M Ryzhik Table of Integrals Seriesand Products Academic Press San Diego Calif USA Seventhedition 2007
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
y
O
(minusL2 0) (L2 0)x
1u
reH
z
L2=
1
2(120585 + 1
120585)
Figure 4 The schematic of fluid flow in horizontal plane after conformal transformation
h2
h2
z
x120585 = (1 minus eminus120587zh) times (1 + e120587zh)
u1
120585
2120587rwh
Figure 5 The schematic of fluid flow in vertical plane after conformal transformation
where 119902119900119867
is the oil flow rate at reservoir condition m3d120582119905(119903119908) is the total mobility of the oilwater phase in the well
bottomhole DcpIf we do not take into account the effects of permeability
stress-sensitive and threshold pressure gradient on the pro-ductivity equation in horizontal plane (120572
119896= 119866 = 0) and
assume only oil flow in the formation (120582119905(119903) = 1120583
119900) then
(17) becomes
119902SOH =0542 times 10minus3119896
119894ℎ (119901119894minus 119901119888)
119861119900120583119900ln (2 (119886 + 119887) 119871)
(19)
where 119902SOH is the oil flow rate at surface condition m3d 119861119900
is the oil volume factor sm3m3Equation (19) is the same as the productivity equation in
horizontal plane of (1) derived by Joshi [7]
25 Calculation of Flow in a Vertical Plane The fluid flowin the vertical plane with top and bottom boundaries closedreservoir can be viewed as a vertical well with the radiusof 120585119908
= 2120587119903119908ℎ produced in a unit circle area after the
conformal transformation (as shown in Figure 5)The mathematical models to describe the steady-state
fluid flow of oilwater flow in the vertical plane are
1
119903
dd119903
[119903119890minus120572119896(119901119894minus119901)120582
119905(d119901d119903
minus 11986610158401015840
)] = 0
1199011003816100381610038161003816119903=120585119908
= 119901119908119891
1199011003816100381610038161003816119903=1 = 119901
119888
(20)
where 11986610158401015840 is the equivalent threshold pressure gradient in thevertical plane which is
11986610158401015840
=(ℎ2) minus 119903
119908
1 minus (2120587119903119908ℎ)
119866 (21)
The solution of the pressure distribution along the radiuscan be solvedwith the samemethod showed in the AppendixThen the productivity equation in the vertical plane can becalculated by the following equation
119902119871119881
= 119862119871119896120582119905(120585119908) sdot (119903
d119901d119903
)119903=120585119908
(22)
The liquid and oil productivity in the vertical plane are
119902119871119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)119871
sdot
120582119905(120585119908) sdot 11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120572119896int1
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowast)
119902119900119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)119871120582
119900(120585119908)
times
11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120582119905(120585119908) 120572119896int1
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowastlowast)
Assuming the reservoir with single phase flow andneglecting the permeability stress-sensitive and thresholdpressure gradient (120572
119896= 119866 = 0 120582
119905(119903) = 1120583
119900) yield
119902119900119881
=119862119896119894119871 (119901119888minus 119901119908119891)
119861119900120583119900ln (ℎ2119903
119908) (23)
Mathematical Problems in Engineering 5
Equation (23) is similar to the productivity equation invertical plane of (D-3) derived by Joshi [7] which proves thecorrectness of the productivity equation in this paper
26 Horizontal Well Eccentricity Figure 5 and (lowast)-(lowastlowast) areobtained under the assumption that the horizontal well islocated at the center of the reservoir in the vertical planeAccording to Muskatrsquos [29] formulation for off-centeredwells the liquid production rate of a well placed at a distance120575 from the mid-height of the reservoir in a vertical plane is
119902119871119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)ℎ
sdot
120582119905(120585119908) sdot 1198661015840
sdot 120585119908
+[119890minus120572119896[119866
1015840120573minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
1015840120585119908]
(120572119896int120573
120585119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowast1015840
)
The oil production rate is
119902119900119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)ℎ120582
119900(120585119908)
times
11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840120573minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120582119905(120585119908) 120572119896int120573
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowastlowast1015840)
where 120575 is the vertical distance between the reservoir centerand horizontal well location m 120573 = 1 minus (2120575ℎ)
2
3 The Solving Method of 120582119905(119903)
In order to correctly calculate the productivity of horizontalwell with oilwater phase flow we must determinate theexpression of total mobility 120582
119905(119903) along the radius 119903 The
following steps are the procedure to calculate it
Step 1 According to the relative permeability curves and theviscosity of oil and water the relationship between 120582
119905(119903) and
119878119908can be obtained
Step 2 When the water displacement front breaks throughthe oil well the distribution of water saturation along the wellradius satisfies the following Buckley-Leverett equation
120587ℎ120601 (1199032
119890minus 1199032
) =d119891119908
d119904119908
sdot sum119876119897 (24)
where ℎ is the formation thickness m 120601 is the porosityfractionsum119876
119897is the cumulative fluid production m3d 119891
119908is
the water ratio fraction 119891119908= (119896119903119908120583119908)(119896119903119900120583119900) + (119896119903119908120583119908)
120582t(r) sim r
p1 larrminus pe p2 larrminus pwf
pc = (p1 + p2)2
qL eq (lowast)qHL eq (17)
If |qL minus qHL| gt eps
No
Yes If qL lt qHL p2 larrminus pc
If qL gt qHL p1 larrminus pc
0 re h Q1 120583o 120583w
kro sim Sw krw sim Sw
re h L rw G120572k pe pwf ki
Output the liquid and oil production rate
Figure 6 The productivity calculation diagram
In (24) the expressions of the d119891119908d119878119908versus 119904
119908can be
calculated by the relative permeability curves
Step 3 Combining the relationship of 120582119905(119903) versus 119878
119908and 119878119908
versus 119903 derived in Steps 1 and 2 the relationship of 120582119905(119903)with
119903 can be obtained
Statistical results show that the relationships of 119878119908versus
119903 120582119905(119903) versus 119878
119908 and 120582
119905(119903) versus 119903 can be approximated by
the following analytical functions
119878119908(119903) = 119886
1+ 11988711199032
120582119905(119878119908) = 1198862+ 1198872119878119908+ 11988821198782
119908
120582119905(119903) = 119886
3+ 11988731199032
+ 11988831199034
(25)
4 The Productivity Calculation Process
Using the electrical analog concept the well production ratein the horizontal plane must equal the production rate in thevertical plane Because of the nonlinearity of (17)-(18) and(lowast)-(lowastlowast) we cannot obtain productivity expressions similarto Joshi [7] With the aid of computer programs the resultscan be obtained and the calculation diagram is showed inFigure 6
5 Results and Their Sensitive Analysis
In this section the liquid and oil production rate are calcu-lated and the essential parameters of well reservoir and fluidproperties are listed in Table 1 and the relative permeabilitycurves are showed in Figure 7
According to the relative permeability curves and theparameters in Table 1 the relationship of 120582
119905(119878119908) versus 119878
119908
6 Mathematical Problems in Engineering
00
02
04
06
08
10
00 01 02 03 04 05 06 07 08 09 10
krok
rw
krwkro
Sw
Figure 7 The relative permeability curves
00
02
04
06
08
10
00 01 02 03 04 05 06 07 08 09 10Sw
120582t
120582t = 32547S2w minus 23096Sw + 04917
R2 = 1
Figure 8 The relationship between 120582119905(119878119908) and 119878
119908
Table 1 The parameters of reservoir and fluid properties
Variables valuePorosity 0 (fraction) 005Drainage radius 119903
119890(m) 600
Horizontal well length 119871 (m) 300Initial permeability of the reservoir 119896
119894(D) 2 times 10minus3
Cumulative fluid production sum119876119897(m3) 114 times 106
Oil viscosity in the reservoir 120583119900(mPasdots) 3
Formation thickness ℎ (m) 20Initial reservoir pressure 119901
119890(MPa) 30
Permeability modulus 120572119896(MPaminus1) 01
Water saturation in the bottomhole 119878119908119861
(fraction) 06Threshold pressure gradient 119866 (MPam) 0001Water viscosity in the reservoir 120583
119908(mPasdots) 1
and d119891119908d119878119908versus 119878
119908can be plotted when water cut ratio
reaches 06 (as shown in Figures 8 and 9)The correspondingregression curve equations can be obtained as follows
120582119905= 04917 minus 23096119878
119908+ 32547119878
2
119908
d119891119908
d119878119908
= minus60226119878119908+ 43193 119878
119908gt 06
(26)
0
1
2
3
4
5
6
02 03 04 05 06 07 08 09 10
000204060810
060 062 064 066 068 070dfwdS w df
wdS w
dfwdSw = minus60226Sw + 43193
R2 = 09716
Sw
Sw
Figure 9 The relationship between d119891119908d119878119908and 119878
119908
0
5
10
15
20
25
0 005 01 015 02 025 03
Liquid production rateOil production rate
Rate
(m3d
)
120572k (MPaminus1)
Figure 10 The effect of permeability modulus (120572119896) on liquid
production rate
Taking (26) as well as the cumulative fluid productioninto (24) the expressions between the 120582
119905(119903) and 119903 can be
obtained which is
120582119905= 02 + 153 times 10
minus6
1199032
+ 68 times 10minus13
1199034
(27)
Combining (27) and other parameters in Table 1 with(17)-(18) and (lowast)-(lowastlowast) the steady-state fluid productivity canbe calculated
Figure 10 shows the effect of permeability modulus 120572119896
on liquid and oil productivity of horizontal well in lowpermeability reservoir It can be seen from the figure thatthe permeability stress-sensitive has a significant effect on theproductivity the bigger the 120572
119896is the smaller the liquid and
oil productivity are which is mainly because with the samepressure drop of the reservoir big 120572
119896will lead to a serious
permeability decreasing When we do not take into accountthe permeability stress sensitive (120572
119896= 0) the liquid and oil
productivity can be calculated with the limit of 120572119896tending to
zero for (17)-(18) and (lowast)-(lowastlowast)Figures 11 and 12 show the effect of threshold pressure
gradient (119866) and well length (119871) on liquid and oil pro-ductivity It can be seen from the chart that the thresholdpressure gradient has small effect on the productivity ofhorizontal well for big drainage volume In general the bigger
Mathematical Problems in Engineering 7
Table 2 The productivity of liquid and oil in different bottomhole pressure
119901119908119891
(MPa)120572119896= 01 119866 = 0001 120572
119896= 0 119866 = 0
Liquid production rate(m3d)
Oil production rate(m3d)
Liquid production rate(m3d)
Oil production rate(m3d)
3000 000 000 000 0002980 000 000 039 0192960 000 000 079 0372950 033 016 098 0472700 458 218 589 2802400 861 409 1177 5602100 1171 556 1766 84018 1407 669 235 111915 1585 754 2944 13992 1720 818 3532 1679
0
2
4
6
8
10
12
14
0 0003 0006 0009 0012 0015
Liquid production rateOil production rate
G (MPaminus1m)
Rate
(m3d
)
Figure 11 The effect of threshold pressure gradient (119866) on liquidproduction rate
0
5
10
15
20
100 150 200 250 300 350 400 450
Liquid production rateOil production rate
Rate
(m3d
)
L (m)
Figure 12 The effect of well length (119871) on liquid production rate
the 119866 is the smaller the liquid and oil productivity areWhen the reservoir has both threshold pressure gradient andpermeability stress sensitive the longer the well length is thebigger the productivity is
048
12162024283236
12 14 16 18 20 22 24 26 28 30
Liquid production rate Oil production rate
Prod
uctio
n ra
te (m
3d
)
Liquid production rate Oil production rate
pwf (MPa)
(120572k = G = 0) (120572k = G = 0)
Figure 13 The productivity of horizontal well in different bottom-hole pressure
Figure 13 shows the liquid productivity with differentbottomhole pressure when 120572
119896= 01 119866 = 0001 and 120572
119896=
0 119866 = 0 the corresponding values are listed in Table 2It can be clearly seen that the permeability stress-sensitiveand threshold pressure gradient have significant effects onthe well productivity and the bigger 120572
119896and 119866 are the more
obvious the effect is And when the pressure drop is smallthe fluid cannot flow for the existing of threshold pressuregradient which is mainly because only the fluid can flowwhen the pressure drop overcomes the threshold pressure formultiphase flow
6 Conclusions
In this paper a semianalytical productivity equation of hor-izontal well in low-permeability oil reservoir with oilwatertwo-phase flow is established with the consideration ofpermeability stress-sensitive and threshold pressure gradientBased on the above study the following conclusions can besummarized
8 Mathematical Problems in Engineering
(1) The steady-state percolation mathematical modelsof horizontal well with oilwater two-phase floware established and the corresponding solutions aresolved by the method of separation of variables
(2) For low-permeability reservoir there always existsthe phenomenon of permeability stress-sensitive (120572
119896)
which has a significant influence on the well produc-tivity the bigger the 120572
119896is the smaller the productivity
is(3) Due to the existence of capillary pressure of two-
phase flow there always is threshold pressure gradient(119866) in the fluid seepage process Although the 119866 hasa smaller effect on the productivity of the horizontalwell for a big drainage volume we cannot neglect itseffect on the productivity
Appendix
From the expressions of (11) we have
119903120582119905119890minus120572119896(119901119894minus119901) (
d119901d119903
minus 1198661015840
) = 119888 (A1)
where 119888 is a constantWe define the following expression
119910 = 119890minus120572119896(119901119894minus119901) (A2)
Substituting (A2) into (A1) and (12)-(13) yields
d119910d119903
= 1205721198961198661015840
119910 +119888120572119896
119903120582119905
(A3)
1199101003816100381610038161003816119903=1 = 119890
minus120572119896(119901119894minus119901119888) (A4)
1199101003816100381610038161003816119903=119903119890119867
= 1 (A5)
According to the general solution of the Bernoulli differ-ential equation [30] the solution of (A3) can be obtained as
119910 (119903) = 1198901205721198961198661015840119903
sdot [119890minus120572119896119866
1015840
119890minus120572119896(119901119894minus119901119888) + int
119903
119903119908
119890minus120572119896119866
1015840119903
119888120572119896
119903120582119905(119903)
d119903] (A6)
Substituting (A6) into (A5) the value of constant 119888 canbe solved and then combining it with (A6) we have
119901 (119903)
= 119901119894+
1
120572119896
ln
1198901205721198961198661015840119903
sdot [
[
119890minus120572119896(119866
1015840+119901119894minus119901119888) +
119890minus1205721198961198661015840119903119890119867 minus 119890minus120572119896(119866
1015840+119901119894minus119901119888)
int119903119890119867
119903119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
sdot int119903
119903119908
119890minus120572119896119866
1015840119903
1
119903120582119905(119903)
d119903]
]
(A7)
Equation (A7) is the pressure distribution relation alongthe radius 119903 with the oilwater two-phase flows in thehorizontal plane of the horizontal well
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Natural Science Foundationof China (Grant no 51374181) and the project of NationalScience Fund for Distinguished Young Scholars of China(Grant no 51125019)The authors would also like to thank thereviewers and editors for their patience to read this paper andvaluable comments
References
[1] V P Merkulov ldquoLe debit des puits devies et horizontauxrdquo NeftKhoz vol 6 no 1 pp 51ndash56 1958
[2] J P Borisov Oil Production Using Horizontal and MultipleDeviation Wells The RampD Translation Company BartlesvilleOkla USA 1984
[3] F M Giger L H Reiss and A P Jourdan ldquoThe reservoir engi-neering aspects of horizontal drillingrdquo in Proceedings of the59th Annual Technical Conference and Exhibition Houston TexUSA 1984
[4] F M Giger ldquoHorizontal wells production techniques in het-erogeneous reservoirsrdquo in Proceedings of the Middle East OilTechnical Conference and Exhibition Bahrain 1985
[5] B J Karcher and F M Giger ldquoSome practical formulas to pre-dict horizontal well behaviorrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition New Orleans La USA1986
[6] L H Reiss ldquoProduction from horizontal wells after five yearsrdquoJournal of Petroleum Technology vol 39 no 11 pp 1411ndash14161987
[7] S D Joshi ldquoAugmentation of well productivity using slant andhorizontal wellsrdquo Journal of Petroleum Technology vol 40 no6 pp 729ndash739 1988
[8] D K Babu and A S Odeh ldquoProductivity of a horizontal wellrdquoSPE Reservoir Engineering vol 4 no 4 pp 417ndash421 1989
[9] G Renard and JM Dupuy ldquoFormation damage effects on hori-zontal-well flow efficiencyrdquo Journal of Petroleum Technologyvol 43 no 7 pp 786ndash789 1991
[10] M W Helmy and R A Wattenbarger ldquoSimplified productivityequations for horizontal wells producing at constant rate andconstant pressurerdquo in Proceedings of the SPE Technical Con-ference and Exhibition pp 379ndash388 New Orleans La USASeptember 1998
[11] T Billiter J Lee and R Chase ldquoDimensionless inflow-perform-ance-relationship curve for unfractured horizontal gas wellsrdquo inProceedings of the SPE Eastern Regional Meeting Canton OhioUSA 2001
[12] E G Anklam and M L Wiggins ldquoHorizontal well produc-tivity and wellbore pressure behavior incorporating wellborehydraulicsrdquo in Proceedings of the SPE Production andOperations
Mathematical Problems in Engineering 9
Symposium pp 565ndash584 Oklahoma City Okla USA April2005
[13] J Lu ldquoNewproductivity formulae of horizontal wellsrdquo Journal ofCanadian Petroleum Technology vol 40 no 10 pp 55ndash67 2001
[14] F Samaniego W E Brigham and F G Miller ldquoPerformance-prediction procedure for transient flow of fluids throughpressure-sensitive formationsrdquo Journal of PetroleumTechnologyvol 31 no 6 pp 779ndash786 1979
[15] G K Falade ldquoTransient flow of fluids in reservoirs with stresssensitive rock and fluid propertiesrdquo International Journal ofNon-Linear Mechanics vol 17 no 4 pp 277ndash283 1982
[16] R W Ostensen ldquoMicrocrack Permeability in tight gas sand-stonerdquo Society of Petroleum Engineers Journal vol 23 no 6 pp919ndash927 1983
[17] J Pedrosa and O A Petrobras ldquoPressure transient response instress-sensitive formationsrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA 1986
[18] D A Barry D A Lockington D-S Jeng J-Y Parlange L Liand F Stagnitti ldquoAnalytical approximations for flow in com-pressible saturated one-dimensional porous mediardquo Advancesin Water Resources vol 30 no 4 pp 927ndash936 2007
[19] T Friedel and H D Voigt ldquoAnalytical solutions for the radialflow equation with constant-rate and constant-pressure bound-ary conditions in reservoirs with pressure-sensitive perme-abilityrdquo in Proceedings of the SPE Rocky Mountain PetroleumTechnology Conference Denver Colo USA 2009
[20] B Ju YWu and T Fan ldquoStudy on fluid flow in nonlinear elasticporousmedia experimental andmodeling approachesrdquo Journalof Petroleum Science and Engineering vol 76 no 3-4 pp 205ndash211 2011
[21] D Swartzendruber ldquoNon-Darcy flow behavior in liquid satu-rated porousmediardquo Journal of Geophysical Research vol 67 no13 pp 5205ndash5213 1962
[22] R J Miller and F L Philip ldquoThreshold Gradient for water flowin clay systemrdquo Soil Science Society of America Journal vol 27no 6 pp 605ndash609 1963
[23] H W Olsen ldquoDeviations from Darcyrsquos law in saturated claysrdquoSoil Science Society of America Journal vol 29 no 2 pp 135ndash1401965
[24] H Pascal ldquoNonsteady flow through porous media in the pre-sence of a threshold gradientrdquo Acta Mechanica vol 39 no 3-4pp 207ndash224 1981
[25] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999
[26] J Lu and S Ghedan ldquoPressure behavior of vertical wells inlow-permeability reservoirs with threshold pressure gradientrdquoSpecial Topics and Reviews in Porous Media vol 2 no 3 pp157ndash169 2011
[27] Y L Zhao L H Zhang F Wu B N Zhang and Q G LiuldquoAnalysis of horizontal well pressure behaviour in fracturedlow permeability reservoirs with consideration of the thresholdpressure gradientrdquo Journal of Geophysics and Engineering vol10 no 3 pp 1ndash10 2013
[28] Y L Zhao L H Zhang J Z Zhao S Y Hu and B N ZhangldquoTransient pressure analysis of horizontal well in low per-meability oil reservoirrdquo International Journal of Oil Gas andCoal Technology 2014
[29] M Muskat The Flow of Homogeneous Fluids through a PorousMedia Intl Human Resources Development Corp BostonMass USA 1937
[30] I S Gradshteyn and I M Ryzhik Table of Integrals Seriesand Products Academic Press San Diego Calif USA Seventhedition 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal 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
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Equation (23) is similar to the productivity equation invertical plane of (D-3) derived by Joshi [7] which proves thecorrectness of the productivity equation in this paper
26 Horizontal Well Eccentricity Figure 5 and (lowast)-(lowastlowast) areobtained under the assumption that the horizontal well islocated at the center of the reservoir in the vertical planeAccording to Muskatrsquos [29] formulation for off-centeredwells the liquid production rate of a well placed at a distance120575 from the mid-height of the reservoir in a vertical plane is
119902119871119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)ℎ
sdot
120582119905(120585119908) sdot 1198661015840
sdot 120585119908
+[119890minus120572119896[119866
1015840120573minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
1015840120585119908]
(120572119896int120573
120585119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowast1015840
)
The oil production rate is
119902119900119881
= 119862119896119894119890minus120572119896(119901119888minus119901119908119891)ℎ120582
119900(120585119908)
times
11986610158401015840
sdot 120585119908
+[119890minus120572119896[119866
10158401015840120573minus(119901119888minus119901119908119891)] minus 119890minus120572119896119866
10158401015840120585119908]
(120582119905(120585119908) 120572119896int120573
120585119908
119890minus12057211989611986610158401015840119903 sdot (1119903120582
119905(119903)) d119903)
(lowastlowast1015840)
where 120575 is the vertical distance between the reservoir centerand horizontal well location m 120573 = 1 minus (2120575ℎ)
2
3 The Solving Method of 120582119905(119903)
In order to correctly calculate the productivity of horizontalwell with oilwater phase flow we must determinate theexpression of total mobility 120582
119905(119903) along the radius 119903 The
following steps are the procedure to calculate it
Step 1 According to the relative permeability curves and theviscosity of oil and water the relationship between 120582
119905(119903) and
119878119908can be obtained
Step 2 When the water displacement front breaks throughthe oil well the distribution of water saturation along the wellradius satisfies the following Buckley-Leverett equation
120587ℎ120601 (1199032
119890minus 1199032
) =d119891119908
d119904119908
sdot sum119876119897 (24)
where ℎ is the formation thickness m 120601 is the porosityfractionsum119876
119897is the cumulative fluid production m3d 119891
119908is
the water ratio fraction 119891119908= (119896119903119908120583119908)(119896119903119900120583119900) + (119896119903119908120583119908)
120582t(r) sim r
p1 larrminus pe p2 larrminus pwf
pc = (p1 + p2)2
qL eq (lowast)qHL eq (17)
If |qL minus qHL| gt eps
No
Yes If qL lt qHL p2 larrminus pc
If qL gt qHL p1 larrminus pc
0 re h Q1 120583o 120583w
kro sim Sw krw sim Sw
re h L rw G120572k pe pwf ki
Output the liquid and oil production rate
Figure 6 The productivity calculation diagram
In (24) the expressions of the d119891119908d119878119908versus 119904
119908can be
calculated by the relative permeability curves
Step 3 Combining the relationship of 120582119905(119903) versus 119878
119908and 119878119908
versus 119903 derived in Steps 1 and 2 the relationship of 120582119905(119903)with
119903 can be obtained
Statistical results show that the relationships of 119878119908versus
119903 120582119905(119903) versus 119878
119908 and 120582
119905(119903) versus 119903 can be approximated by
the following analytical functions
119878119908(119903) = 119886
1+ 11988711199032
120582119905(119878119908) = 1198862+ 1198872119878119908+ 11988821198782
119908
120582119905(119903) = 119886
3+ 11988731199032
+ 11988831199034
(25)
4 The Productivity Calculation Process
Using the electrical analog concept the well production ratein the horizontal plane must equal the production rate in thevertical plane Because of the nonlinearity of (17)-(18) and(lowast)-(lowastlowast) we cannot obtain productivity expressions similarto Joshi [7] With the aid of computer programs the resultscan be obtained and the calculation diagram is showed inFigure 6
5 Results and Their Sensitive Analysis
In this section the liquid and oil production rate are calcu-lated and the essential parameters of well reservoir and fluidproperties are listed in Table 1 and the relative permeabilitycurves are showed in Figure 7
According to the relative permeability curves and theparameters in Table 1 the relationship of 120582
119905(119878119908) versus 119878
119908
6 Mathematical Problems in Engineering
00
02
04
06
08
10
00 01 02 03 04 05 06 07 08 09 10
krok
rw
krwkro
Sw
Figure 7 The relative permeability curves
00
02
04
06
08
10
00 01 02 03 04 05 06 07 08 09 10Sw
120582t
120582t = 32547S2w minus 23096Sw + 04917
R2 = 1
Figure 8 The relationship between 120582119905(119878119908) and 119878
119908
Table 1 The parameters of reservoir and fluid properties
Variables valuePorosity 0 (fraction) 005Drainage radius 119903
119890(m) 600
Horizontal well length 119871 (m) 300Initial permeability of the reservoir 119896
119894(D) 2 times 10minus3
Cumulative fluid production sum119876119897(m3) 114 times 106
Oil viscosity in the reservoir 120583119900(mPasdots) 3
Formation thickness ℎ (m) 20Initial reservoir pressure 119901
119890(MPa) 30
Permeability modulus 120572119896(MPaminus1) 01
Water saturation in the bottomhole 119878119908119861
(fraction) 06Threshold pressure gradient 119866 (MPam) 0001Water viscosity in the reservoir 120583
119908(mPasdots) 1
and d119891119908d119878119908versus 119878
119908can be plotted when water cut ratio
reaches 06 (as shown in Figures 8 and 9)The correspondingregression curve equations can be obtained as follows
120582119905= 04917 minus 23096119878
119908+ 32547119878
2
119908
d119891119908
d119878119908
= minus60226119878119908+ 43193 119878
119908gt 06
(26)
0
1
2
3
4
5
6
02 03 04 05 06 07 08 09 10
000204060810
060 062 064 066 068 070dfwdS w df
wdS w
dfwdSw = minus60226Sw + 43193
R2 = 09716
Sw
Sw
Figure 9 The relationship between d119891119908d119878119908and 119878
119908
0
5
10
15
20
25
0 005 01 015 02 025 03
Liquid production rateOil production rate
Rate
(m3d
)
120572k (MPaminus1)
Figure 10 The effect of permeability modulus (120572119896) on liquid
production rate
Taking (26) as well as the cumulative fluid productioninto (24) the expressions between the 120582
119905(119903) and 119903 can be
obtained which is
120582119905= 02 + 153 times 10
minus6
1199032
+ 68 times 10minus13
1199034
(27)
Combining (27) and other parameters in Table 1 with(17)-(18) and (lowast)-(lowastlowast) the steady-state fluid productivity canbe calculated
Figure 10 shows the effect of permeability modulus 120572119896
on liquid and oil productivity of horizontal well in lowpermeability reservoir It can be seen from the figure thatthe permeability stress-sensitive has a significant effect on theproductivity the bigger the 120572
119896is the smaller the liquid and
oil productivity are which is mainly because with the samepressure drop of the reservoir big 120572
119896will lead to a serious
permeability decreasing When we do not take into accountthe permeability stress sensitive (120572
119896= 0) the liquid and oil
productivity can be calculated with the limit of 120572119896tending to
zero for (17)-(18) and (lowast)-(lowastlowast)Figures 11 and 12 show the effect of threshold pressure
gradient (119866) and well length (119871) on liquid and oil pro-ductivity It can be seen from the chart that the thresholdpressure gradient has small effect on the productivity ofhorizontal well for big drainage volume In general the bigger
Mathematical Problems in Engineering 7
Table 2 The productivity of liquid and oil in different bottomhole pressure
119901119908119891
(MPa)120572119896= 01 119866 = 0001 120572
119896= 0 119866 = 0
Liquid production rate(m3d)
Oil production rate(m3d)
Liquid production rate(m3d)
Oil production rate(m3d)
3000 000 000 000 0002980 000 000 039 0192960 000 000 079 0372950 033 016 098 0472700 458 218 589 2802400 861 409 1177 5602100 1171 556 1766 84018 1407 669 235 111915 1585 754 2944 13992 1720 818 3532 1679
0
2
4
6
8
10
12
14
0 0003 0006 0009 0012 0015
Liquid production rateOil production rate
G (MPaminus1m)
Rate
(m3d
)
Figure 11 The effect of threshold pressure gradient (119866) on liquidproduction rate
0
5
10
15
20
100 150 200 250 300 350 400 450
Liquid production rateOil production rate
Rate
(m3d
)
L (m)
Figure 12 The effect of well length (119871) on liquid production rate
the 119866 is the smaller the liquid and oil productivity areWhen the reservoir has both threshold pressure gradient andpermeability stress sensitive the longer the well length is thebigger the productivity is
048
12162024283236
12 14 16 18 20 22 24 26 28 30
Liquid production rate Oil production rate
Prod
uctio
n ra
te (m
3d
)
Liquid production rate Oil production rate
pwf (MPa)
(120572k = G = 0) (120572k = G = 0)
Figure 13 The productivity of horizontal well in different bottom-hole pressure
Figure 13 shows the liquid productivity with differentbottomhole pressure when 120572
119896= 01 119866 = 0001 and 120572
119896=
0 119866 = 0 the corresponding values are listed in Table 2It can be clearly seen that the permeability stress-sensitiveand threshold pressure gradient have significant effects onthe well productivity and the bigger 120572
119896and 119866 are the more
obvious the effect is And when the pressure drop is smallthe fluid cannot flow for the existing of threshold pressuregradient which is mainly because only the fluid can flowwhen the pressure drop overcomes the threshold pressure formultiphase flow
6 Conclusions
In this paper a semianalytical productivity equation of hor-izontal well in low-permeability oil reservoir with oilwatertwo-phase flow is established with the consideration ofpermeability stress-sensitive and threshold pressure gradientBased on the above study the following conclusions can besummarized
8 Mathematical Problems in Engineering
(1) The steady-state percolation mathematical modelsof horizontal well with oilwater two-phase floware established and the corresponding solutions aresolved by the method of separation of variables
(2) For low-permeability reservoir there always existsthe phenomenon of permeability stress-sensitive (120572
119896)
which has a significant influence on the well produc-tivity the bigger the 120572
119896is the smaller the productivity
is(3) Due to the existence of capillary pressure of two-
phase flow there always is threshold pressure gradient(119866) in the fluid seepage process Although the 119866 hasa smaller effect on the productivity of the horizontalwell for a big drainage volume we cannot neglect itseffect on the productivity
Appendix
From the expressions of (11) we have
119903120582119905119890minus120572119896(119901119894minus119901) (
d119901d119903
minus 1198661015840
) = 119888 (A1)
where 119888 is a constantWe define the following expression
119910 = 119890minus120572119896(119901119894minus119901) (A2)
Substituting (A2) into (A1) and (12)-(13) yields
d119910d119903
= 1205721198961198661015840
119910 +119888120572119896
119903120582119905
(A3)
1199101003816100381610038161003816119903=1 = 119890
minus120572119896(119901119894minus119901119888) (A4)
1199101003816100381610038161003816119903=119903119890119867
= 1 (A5)
According to the general solution of the Bernoulli differ-ential equation [30] the solution of (A3) can be obtained as
119910 (119903) = 1198901205721198961198661015840119903
sdot [119890minus120572119896119866
1015840
119890minus120572119896(119901119894minus119901119888) + int
119903
119903119908
119890minus120572119896119866
1015840119903
119888120572119896
119903120582119905(119903)
d119903] (A6)
Substituting (A6) into (A5) the value of constant 119888 canbe solved and then combining it with (A6) we have
119901 (119903)
= 119901119894+
1
120572119896
ln
1198901205721198961198661015840119903
sdot [
[
119890minus120572119896(119866
1015840+119901119894minus119901119888) +
119890minus1205721198961198661015840119903119890119867 minus 119890minus120572119896(119866
1015840+119901119894minus119901119888)
int119903119890119867
119903119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
sdot int119903
119903119908
119890minus120572119896119866
1015840119903
1
119903120582119905(119903)
d119903]
]
(A7)
Equation (A7) is the pressure distribution relation alongthe radius 119903 with the oilwater two-phase flows in thehorizontal plane of the horizontal well
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Natural Science Foundationof China (Grant no 51374181) and the project of NationalScience Fund for Distinguished Young Scholars of China(Grant no 51125019)The authors would also like to thank thereviewers and editors for their patience to read this paper andvaluable comments
References
[1] V P Merkulov ldquoLe debit des puits devies et horizontauxrdquo NeftKhoz vol 6 no 1 pp 51ndash56 1958
[2] J P Borisov Oil Production Using Horizontal and MultipleDeviation Wells The RampD Translation Company BartlesvilleOkla USA 1984
[3] F M Giger L H Reiss and A P Jourdan ldquoThe reservoir engi-neering aspects of horizontal drillingrdquo in Proceedings of the59th Annual Technical Conference and Exhibition Houston TexUSA 1984
[4] F M Giger ldquoHorizontal wells production techniques in het-erogeneous reservoirsrdquo in Proceedings of the Middle East OilTechnical Conference and Exhibition Bahrain 1985
[5] B J Karcher and F M Giger ldquoSome practical formulas to pre-dict horizontal well behaviorrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition New Orleans La USA1986
[6] L H Reiss ldquoProduction from horizontal wells after five yearsrdquoJournal of Petroleum Technology vol 39 no 11 pp 1411ndash14161987
[7] S D Joshi ldquoAugmentation of well productivity using slant andhorizontal wellsrdquo Journal of Petroleum Technology vol 40 no6 pp 729ndash739 1988
[8] D K Babu and A S Odeh ldquoProductivity of a horizontal wellrdquoSPE Reservoir Engineering vol 4 no 4 pp 417ndash421 1989
[9] G Renard and JM Dupuy ldquoFormation damage effects on hori-zontal-well flow efficiencyrdquo Journal of Petroleum Technologyvol 43 no 7 pp 786ndash789 1991
[10] M W Helmy and R A Wattenbarger ldquoSimplified productivityequations for horizontal wells producing at constant rate andconstant pressurerdquo in Proceedings of the SPE Technical Con-ference and Exhibition pp 379ndash388 New Orleans La USASeptember 1998
[11] T Billiter J Lee and R Chase ldquoDimensionless inflow-perform-ance-relationship curve for unfractured horizontal gas wellsrdquo inProceedings of the SPE Eastern Regional Meeting Canton OhioUSA 2001
[12] E G Anklam and M L Wiggins ldquoHorizontal well produc-tivity and wellbore pressure behavior incorporating wellborehydraulicsrdquo in Proceedings of the SPE Production andOperations
Mathematical Problems in Engineering 9
Symposium pp 565ndash584 Oklahoma City Okla USA April2005
[13] J Lu ldquoNewproductivity formulae of horizontal wellsrdquo Journal ofCanadian Petroleum Technology vol 40 no 10 pp 55ndash67 2001
[14] F Samaniego W E Brigham and F G Miller ldquoPerformance-prediction procedure for transient flow of fluids throughpressure-sensitive formationsrdquo Journal of PetroleumTechnologyvol 31 no 6 pp 779ndash786 1979
[15] G K Falade ldquoTransient flow of fluids in reservoirs with stresssensitive rock and fluid propertiesrdquo International Journal ofNon-Linear Mechanics vol 17 no 4 pp 277ndash283 1982
[16] R W Ostensen ldquoMicrocrack Permeability in tight gas sand-stonerdquo Society of Petroleum Engineers Journal vol 23 no 6 pp919ndash927 1983
[17] J Pedrosa and O A Petrobras ldquoPressure transient response instress-sensitive formationsrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA 1986
[18] D A Barry D A Lockington D-S Jeng J-Y Parlange L Liand F Stagnitti ldquoAnalytical approximations for flow in com-pressible saturated one-dimensional porous mediardquo Advancesin Water Resources vol 30 no 4 pp 927ndash936 2007
[19] T Friedel and H D Voigt ldquoAnalytical solutions for the radialflow equation with constant-rate and constant-pressure bound-ary conditions in reservoirs with pressure-sensitive perme-abilityrdquo in Proceedings of the SPE Rocky Mountain PetroleumTechnology Conference Denver Colo USA 2009
[20] B Ju YWu and T Fan ldquoStudy on fluid flow in nonlinear elasticporousmedia experimental andmodeling approachesrdquo Journalof Petroleum Science and Engineering vol 76 no 3-4 pp 205ndash211 2011
[21] D Swartzendruber ldquoNon-Darcy flow behavior in liquid satu-rated porousmediardquo Journal of Geophysical Research vol 67 no13 pp 5205ndash5213 1962
[22] R J Miller and F L Philip ldquoThreshold Gradient for water flowin clay systemrdquo Soil Science Society of America Journal vol 27no 6 pp 605ndash609 1963
[23] H W Olsen ldquoDeviations from Darcyrsquos law in saturated claysrdquoSoil Science Society of America Journal vol 29 no 2 pp 135ndash1401965
[24] H Pascal ldquoNonsteady flow through porous media in the pre-sence of a threshold gradientrdquo Acta Mechanica vol 39 no 3-4pp 207ndash224 1981
[25] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999
[26] J Lu and S Ghedan ldquoPressure behavior of vertical wells inlow-permeability reservoirs with threshold pressure gradientrdquoSpecial Topics and Reviews in Porous Media vol 2 no 3 pp157ndash169 2011
[27] Y L Zhao L H Zhang F Wu B N Zhang and Q G LiuldquoAnalysis of horizontal well pressure behaviour in fracturedlow permeability reservoirs with consideration of the thresholdpressure gradientrdquo Journal of Geophysics and Engineering vol10 no 3 pp 1ndash10 2013
[28] Y L Zhao L H Zhang J Z Zhao S Y Hu and B N ZhangldquoTransient pressure analysis of horizontal well in low per-meability oil reservoirrdquo International Journal of Oil Gas andCoal Technology 2014
[29] M Muskat The Flow of Homogeneous Fluids through a PorousMedia Intl Human Resources Development Corp BostonMass USA 1937
[30] I S Gradshteyn and I M Ryzhik Table of Integrals Seriesand Products Academic Press San Diego Calif USA Seventhedition 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
00
02
04
06
08
10
00 01 02 03 04 05 06 07 08 09 10
krok
rw
krwkro
Sw
Figure 7 The relative permeability curves
00
02
04
06
08
10
00 01 02 03 04 05 06 07 08 09 10Sw
120582t
120582t = 32547S2w minus 23096Sw + 04917
R2 = 1
Figure 8 The relationship between 120582119905(119878119908) and 119878
119908
Table 1 The parameters of reservoir and fluid properties
Variables valuePorosity 0 (fraction) 005Drainage radius 119903
119890(m) 600
Horizontal well length 119871 (m) 300Initial permeability of the reservoir 119896
119894(D) 2 times 10minus3
Cumulative fluid production sum119876119897(m3) 114 times 106
Oil viscosity in the reservoir 120583119900(mPasdots) 3
Formation thickness ℎ (m) 20Initial reservoir pressure 119901
119890(MPa) 30
Permeability modulus 120572119896(MPaminus1) 01
Water saturation in the bottomhole 119878119908119861
(fraction) 06Threshold pressure gradient 119866 (MPam) 0001Water viscosity in the reservoir 120583
119908(mPasdots) 1
and d119891119908d119878119908versus 119878
119908can be plotted when water cut ratio
reaches 06 (as shown in Figures 8 and 9)The correspondingregression curve equations can be obtained as follows
120582119905= 04917 minus 23096119878
119908+ 32547119878
2
119908
d119891119908
d119878119908
= minus60226119878119908+ 43193 119878
119908gt 06
(26)
0
1
2
3
4
5
6
02 03 04 05 06 07 08 09 10
000204060810
060 062 064 066 068 070dfwdS w df
wdS w
dfwdSw = minus60226Sw + 43193
R2 = 09716
Sw
Sw
Figure 9 The relationship between d119891119908d119878119908and 119878
119908
0
5
10
15
20
25
0 005 01 015 02 025 03
Liquid production rateOil production rate
Rate
(m3d
)
120572k (MPaminus1)
Figure 10 The effect of permeability modulus (120572119896) on liquid
production rate
Taking (26) as well as the cumulative fluid productioninto (24) the expressions between the 120582
119905(119903) and 119903 can be
obtained which is
120582119905= 02 + 153 times 10
minus6
1199032
+ 68 times 10minus13
1199034
(27)
Combining (27) and other parameters in Table 1 with(17)-(18) and (lowast)-(lowastlowast) the steady-state fluid productivity canbe calculated
Figure 10 shows the effect of permeability modulus 120572119896
on liquid and oil productivity of horizontal well in lowpermeability reservoir It can be seen from the figure thatthe permeability stress-sensitive has a significant effect on theproductivity the bigger the 120572
119896is the smaller the liquid and
oil productivity are which is mainly because with the samepressure drop of the reservoir big 120572
119896will lead to a serious
permeability decreasing When we do not take into accountthe permeability stress sensitive (120572
119896= 0) the liquid and oil
productivity can be calculated with the limit of 120572119896tending to
zero for (17)-(18) and (lowast)-(lowastlowast)Figures 11 and 12 show the effect of threshold pressure
gradient (119866) and well length (119871) on liquid and oil pro-ductivity It can be seen from the chart that the thresholdpressure gradient has small effect on the productivity ofhorizontal well for big drainage volume In general the bigger
Mathematical Problems in Engineering 7
Table 2 The productivity of liquid and oil in different bottomhole pressure
119901119908119891
(MPa)120572119896= 01 119866 = 0001 120572
119896= 0 119866 = 0
Liquid production rate(m3d)
Oil production rate(m3d)
Liquid production rate(m3d)
Oil production rate(m3d)
3000 000 000 000 0002980 000 000 039 0192960 000 000 079 0372950 033 016 098 0472700 458 218 589 2802400 861 409 1177 5602100 1171 556 1766 84018 1407 669 235 111915 1585 754 2944 13992 1720 818 3532 1679
0
2
4
6
8
10
12
14
0 0003 0006 0009 0012 0015
Liquid production rateOil production rate
G (MPaminus1m)
Rate
(m3d
)
Figure 11 The effect of threshold pressure gradient (119866) on liquidproduction rate
0
5
10
15
20
100 150 200 250 300 350 400 450
Liquid production rateOil production rate
Rate
(m3d
)
L (m)
Figure 12 The effect of well length (119871) on liquid production rate
the 119866 is the smaller the liquid and oil productivity areWhen the reservoir has both threshold pressure gradient andpermeability stress sensitive the longer the well length is thebigger the productivity is
048
12162024283236
12 14 16 18 20 22 24 26 28 30
Liquid production rate Oil production rate
Prod
uctio
n ra
te (m
3d
)
Liquid production rate Oil production rate
pwf (MPa)
(120572k = G = 0) (120572k = G = 0)
Figure 13 The productivity of horizontal well in different bottom-hole pressure
Figure 13 shows the liquid productivity with differentbottomhole pressure when 120572
119896= 01 119866 = 0001 and 120572
119896=
0 119866 = 0 the corresponding values are listed in Table 2It can be clearly seen that the permeability stress-sensitiveand threshold pressure gradient have significant effects onthe well productivity and the bigger 120572
119896and 119866 are the more
obvious the effect is And when the pressure drop is smallthe fluid cannot flow for the existing of threshold pressuregradient which is mainly because only the fluid can flowwhen the pressure drop overcomes the threshold pressure formultiphase flow
6 Conclusions
In this paper a semianalytical productivity equation of hor-izontal well in low-permeability oil reservoir with oilwatertwo-phase flow is established with the consideration ofpermeability stress-sensitive and threshold pressure gradientBased on the above study the following conclusions can besummarized
8 Mathematical Problems in Engineering
(1) The steady-state percolation mathematical modelsof horizontal well with oilwater two-phase floware established and the corresponding solutions aresolved by the method of separation of variables
(2) For low-permeability reservoir there always existsthe phenomenon of permeability stress-sensitive (120572
119896)
which has a significant influence on the well produc-tivity the bigger the 120572
119896is the smaller the productivity
is(3) Due to the existence of capillary pressure of two-
phase flow there always is threshold pressure gradient(119866) in the fluid seepage process Although the 119866 hasa smaller effect on the productivity of the horizontalwell for a big drainage volume we cannot neglect itseffect on the productivity
Appendix
From the expressions of (11) we have
119903120582119905119890minus120572119896(119901119894minus119901) (
d119901d119903
minus 1198661015840
) = 119888 (A1)
where 119888 is a constantWe define the following expression
119910 = 119890minus120572119896(119901119894minus119901) (A2)
Substituting (A2) into (A1) and (12)-(13) yields
d119910d119903
= 1205721198961198661015840
119910 +119888120572119896
119903120582119905
(A3)
1199101003816100381610038161003816119903=1 = 119890
minus120572119896(119901119894minus119901119888) (A4)
1199101003816100381610038161003816119903=119903119890119867
= 1 (A5)
According to the general solution of the Bernoulli differ-ential equation [30] the solution of (A3) can be obtained as
119910 (119903) = 1198901205721198961198661015840119903
sdot [119890minus120572119896119866
1015840
119890minus120572119896(119901119894minus119901119888) + int
119903
119903119908
119890minus120572119896119866
1015840119903
119888120572119896
119903120582119905(119903)
d119903] (A6)
Substituting (A6) into (A5) the value of constant 119888 canbe solved and then combining it with (A6) we have
119901 (119903)
= 119901119894+
1
120572119896
ln
1198901205721198961198661015840119903
sdot [
[
119890minus120572119896(119866
1015840+119901119894minus119901119888) +
119890minus1205721198961198661015840119903119890119867 minus 119890minus120572119896(119866
1015840+119901119894minus119901119888)
int119903119890119867
119903119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
sdot int119903
119903119908
119890minus120572119896119866
1015840119903
1
119903120582119905(119903)
d119903]
]
(A7)
Equation (A7) is the pressure distribution relation alongthe radius 119903 with the oilwater two-phase flows in thehorizontal plane of the horizontal well
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Natural Science Foundationof China (Grant no 51374181) and the project of NationalScience Fund for Distinguished Young Scholars of China(Grant no 51125019)The authors would also like to thank thereviewers and editors for their patience to read this paper andvaluable comments
References
[1] V P Merkulov ldquoLe debit des puits devies et horizontauxrdquo NeftKhoz vol 6 no 1 pp 51ndash56 1958
[2] J P Borisov Oil Production Using Horizontal and MultipleDeviation Wells The RampD Translation Company BartlesvilleOkla USA 1984
[3] F M Giger L H Reiss and A P Jourdan ldquoThe reservoir engi-neering aspects of horizontal drillingrdquo in Proceedings of the59th Annual Technical Conference and Exhibition Houston TexUSA 1984
[4] F M Giger ldquoHorizontal wells production techniques in het-erogeneous reservoirsrdquo in Proceedings of the Middle East OilTechnical Conference and Exhibition Bahrain 1985
[5] B J Karcher and F M Giger ldquoSome practical formulas to pre-dict horizontal well behaviorrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition New Orleans La USA1986
[6] L H Reiss ldquoProduction from horizontal wells after five yearsrdquoJournal of Petroleum Technology vol 39 no 11 pp 1411ndash14161987
[7] S D Joshi ldquoAugmentation of well productivity using slant andhorizontal wellsrdquo Journal of Petroleum Technology vol 40 no6 pp 729ndash739 1988
[8] D K Babu and A S Odeh ldquoProductivity of a horizontal wellrdquoSPE Reservoir Engineering vol 4 no 4 pp 417ndash421 1989
[9] G Renard and JM Dupuy ldquoFormation damage effects on hori-zontal-well flow efficiencyrdquo Journal of Petroleum Technologyvol 43 no 7 pp 786ndash789 1991
[10] M W Helmy and R A Wattenbarger ldquoSimplified productivityequations for horizontal wells producing at constant rate andconstant pressurerdquo in Proceedings of the SPE Technical Con-ference and Exhibition pp 379ndash388 New Orleans La USASeptember 1998
[11] T Billiter J Lee and R Chase ldquoDimensionless inflow-perform-ance-relationship curve for unfractured horizontal gas wellsrdquo inProceedings of the SPE Eastern Regional Meeting Canton OhioUSA 2001
[12] E G Anklam and M L Wiggins ldquoHorizontal well produc-tivity and wellbore pressure behavior incorporating wellborehydraulicsrdquo in Proceedings of the SPE Production andOperations
Mathematical Problems in Engineering 9
Symposium pp 565ndash584 Oklahoma City Okla USA April2005
[13] J Lu ldquoNewproductivity formulae of horizontal wellsrdquo Journal ofCanadian Petroleum Technology vol 40 no 10 pp 55ndash67 2001
[14] F Samaniego W E Brigham and F G Miller ldquoPerformance-prediction procedure for transient flow of fluids throughpressure-sensitive formationsrdquo Journal of PetroleumTechnologyvol 31 no 6 pp 779ndash786 1979
[15] G K Falade ldquoTransient flow of fluids in reservoirs with stresssensitive rock and fluid propertiesrdquo International Journal ofNon-Linear Mechanics vol 17 no 4 pp 277ndash283 1982
[16] R W Ostensen ldquoMicrocrack Permeability in tight gas sand-stonerdquo Society of Petroleum Engineers Journal vol 23 no 6 pp919ndash927 1983
[17] J Pedrosa and O A Petrobras ldquoPressure transient response instress-sensitive formationsrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA 1986
[18] D A Barry D A Lockington D-S Jeng J-Y Parlange L Liand F Stagnitti ldquoAnalytical approximations for flow in com-pressible saturated one-dimensional porous mediardquo Advancesin Water Resources vol 30 no 4 pp 927ndash936 2007
[19] T Friedel and H D Voigt ldquoAnalytical solutions for the radialflow equation with constant-rate and constant-pressure bound-ary conditions in reservoirs with pressure-sensitive perme-abilityrdquo in Proceedings of the SPE Rocky Mountain PetroleumTechnology Conference Denver Colo USA 2009
[20] B Ju YWu and T Fan ldquoStudy on fluid flow in nonlinear elasticporousmedia experimental andmodeling approachesrdquo Journalof Petroleum Science and Engineering vol 76 no 3-4 pp 205ndash211 2011
[21] D Swartzendruber ldquoNon-Darcy flow behavior in liquid satu-rated porousmediardquo Journal of Geophysical Research vol 67 no13 pp 5205ndash5213 1962
[22] R J Miller and F L Philip ldquoThreshold Gradient for water flowin clay systemrdquo Soil Science Society of America Journal vol 27no 6 pp 605ndash609 1963
[23] H W Olsen ldquoDeviations from Darcyrsquos law in saturated claysrdquoSoil Science Society of America Journal vol 29 no 2 pp 135ndash1401965
[24] H Pascal ldquoNonsteady flow through porous media in the pre-sence of a threshold gradientrdquo Acta Mechanica vol 39 no 3-4pp 207ndash224 1981
[25] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999
[26] J Lu and S Ghedan ldquoPressure behavior of vertical wells inlow-permeability reservoirs with threshold pressure gradientrdquoSpecial Topics and Reviews in Porous Media vol 2 no 3 pp157ndash169 2011
[27] Y L Zhao L H Zhang F Wu B N Zhang and Q G LiuldquoAnalysis of horizontal well pressure behaviour in fracturedlow permeability reservoirs with consideration of the thresholdpressure gradientrdquo Journal of Geophysics and Engineering vol10 no 3 pp 1ndash10 2013
[28] Y L Zhao L H Zhang J Z Zhao S Y Hu and B N ZhangldquoTransient pressure analysis of horizontal well in low per-meability oil reservoirrdquo International Journal of Oil Gas andCoal Technology 2014
[29] M Muskat The Flow of Homogeneous Fluids through a PorousMedia Intl Human Resources Development Corp BostonMass USA 1937
[30] I S Gradshteyn and I M Ryzhik Table of Integrals Seriesand Products Academic Press San Diego Calif USA Seventhedition 2007
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
Table 2 The productivity of liquid and oil in different bottomhole pressure
119901119908119891
(MPa)120572119896= 01 119866 = 0001 120572
119896= 0 119866 = 0
Liquid production rate(m3d)
Oil production rate(m3d)
Liquid production rate(m3d)
Oil production rate(m3d)
3000 000 000 000 0002980 000 000 039 0192960 000 000 079 0372950 033 016 098 0472700 458 218 589 2802400 861 409 1177 5602100 1171 556 1766 84018 1407 669 235 111915 1585 754 2944 13992 1720 818 3532 1679
0
2
4
6
8
10
12
14
0 0003 0006 0009 0012 0015
Liquid production rateOil production rate
G (MPaminus1m)
Rate
(m3d
)
Figure 11 The effect of threshold pressure gradient (119866) on liquidproduction rate
0
5
10
15
20
100 150 200 250 300 350 400 450
Liquid production rateOil production rate
Rate
(m3d
)
L (m)
Figure 12 The effect of well length (119871) on liquid production rate
the 119866 is the smaller the liquid and oil productivity areWhen the reservoir has both threshold pressure gradient andpermeability stress sensitive the longer the well length is thebigger the productivity is
048
12162024283236
12 14 16 18 20 22 24 26 28 30
Liquid production rate Oil production rate
Prod
uctio
n ra
te (m
3d
)
Liquid production rate Oil production rate
pwf (MPa)
(120572k = G = 0) (120572k = G = 0)
Figure 13 The productivity of horizontal well in different bottom-hole pressure
Figure 13 shows the liquid productivity with differentbottomhole pressure when 120572
119896= 01 119866 = 0001 and 120572
119896=
0 119866 = 0 the corresponding values are listed in Table 2It can be clearly seen that the permeability stress-sensitiveand threshold pressure gradient have significant effects onthe well productivity and the bigger 120572
119896and 119866 are the more
obvious the effect is And when the pressure drop is smallthe fluid cannot flow for the existing of threshold pressuregradient which is mainly because only the fluid can flowwhen the pressure drop overcomes the threshold pressure formultiphase flow
6 Conclusions
In this paper a semianalytical productivity equation of hor-izontal well in low-permeability oil reservoir with oilwatertwo-phase flow is established with the consideration ofpermeability stress-sensitive and threshold pressure gradientBased on the above study the following conclusions can besummarized
8 Mathematical Problems in Engineering
(1) The steady-state percolation mathematical modelsof horizontal well with oilwater two-phase floware established and the corresponding solutions aresolved by the method of separation of variables
(2) For low-permeability reservoir there always existsthe phenomenon of permeability stress-sensitive (120572
119896)
which has a significant influence on the well produc-tivity the bigger the 120572
119896is the smaller the productivity
is(3) Due to the existence of capillary pressure of two-
phase flow there always is threshold pressure gradient(119866) in the fluid seepage process Although the 119866 hasa smaller effect on the productivity of the horizontalwell for a big drainage volume we cannot neglect itseffect on the productivity
Appendix
From the expressions of (11) we have
119903120582119905119890minus120572119896(119901119894minus119901) (
d119901d119903
minus 1198661015840
) = 119888 (A1)
where 119888 is a constantWe define the following expression
119910 = 119890minus120572119896(119901119894minus119901) (A2)
Substituting (A2) into (A1) and (12)-(13) yields
d119910d119903
= 1205721198961198661015840
119910 +119888120572119896
119903120582119905
(A3)
1199101003816100381610038161003816119903=1 = 119890
minus120572119896(119901119894minus119901119888) (A4)
1199101003816100381610038161003816119903=119903119890119867
= 1 (A5)
According to the general solution of the Bernoulli differ-ential equation [30] the solution of (A3) can be obtained as
119910 (119903) = 1198901205721198961198661015840119903
sdot [119890minus120572119896119866
1015840
119890minus120572119896(119901119894minus119901119888) + int
119903
119903119908
119890minus120572119896119866
1015840119903
119888120572119896
119903120582119905(119903)
d119903] (A6)
Substituting (A6) into (A5) the value of constant 119888 canbe solved and then combining it with (A6) we have
119901 (119903)
= 119901119894+
1
120572119896
ln
1198901205721198961198661015840119903
sdot [
[
119890minus120572119896(119866
1015840+119901119894minus119901119888) +
119890minus1205721198961198661015840119903119890119867 minus 119890minus120572119896(119866
1015840+119901119894minus119901119888)
int119903119890119867
119903119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
sdot int119903
119903119908
119890minus120572119896119866
1015840119903
1
119903120582119905(119903)
d119903]
]
(A7)
Equation (A7) is the pressure distribution relation alongthe radius 119903 with the oilwater two-phase flows in thehorizontal plane of the horizontal well
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Natural Science Foundationof China (Grant no 51374181) and the project of NationalScience Fund for Distinguished Young Scholars of China(Grant no 51125019)The authors would also like to thank thereviewers and editors for their patience to read this paper andvaluable comments
References
[1] V P Merkulov ldquoLe debit des puits devies et horizontauxrdquo NeftKhoz vol 6 no 1 pp 51ndash56 1958
[2] J P Borisov Oil Production Using Horizontal and MultipleDeviation Wells The RampD Translation Company BartlesvilleOkla USA 1984
[3] F M Giger L H Reiss and A P Jourdan ldquoThe reservoir engi-neering aspects of horizontal drillingrdquo in Proceedings of the59th Annual Technical Conference and Exhibition Houston TexUSA 1984
[4] F M Giger ldquoHorizontal wells production techniques in het-erogeneous reservoirsrdquo in Proceedings of the Middle East OilTechnical Conference and Exhibition Bahrain 1985
[5] B J Karcher and F M Giger ldquoSome practical formulas to pre-dict horizontal well behaviorrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition New Orleans La USA1986
[6] L H Reiss ldquoProduction from horizontal wells after five yearsrdquoJournal of Petroleum Technology vol 39 no 11 pp 1411ndash14161987
[7] S D Joshi ldquoAugmentation of well productivity using slant andhorizontal wellsrdquo Journal of Petroleum Technology vol 40 no6 pp 729ndash739 1988
[8] D K Babu and A S Odeh ldquoProductivity of a horizontal wellrdquoSPE Reservoir Engineering vol 4 no 4 pp 417ndash421 1989
[9] G Renard and JM Dupuy ldquoFormation damage effects on hori-zontal-well flow efficiencyrdquo Journal of Petroleum Technologyvol 43 no 7 pp 786ndash789 1991
[10] M W Helmy and R A Wattenbarger ldquoSimplified productivityequations for horizontal wells producing at constant rate andconstant pressurerdquo in Proceedings of the SPE Technical Con-ference and Exhibition pp 379ndash388 New Orleans La USASeptember 1998
[11] T Billiter J Lee and R Chase ldquoDimensionless inflow-perform-ance-relationship curve for unfractured horizontal gas wellsrdquo inProceedings of the SPE Eastern Regional Meeting Canton OhioUSA 2001
[12] E G Anklam and M L Wiggins ldquoHorizontal well produc-tivity and wellbore pressure behavior incorporating wellborehydraulicsrdquo in Proceedings of the SPE Production andOperations
Mathematical Problems in Engineering 9
Symposium pp 565ndash584 Oklahoma City Okla USA April2005
[13] J Lu ldquoNewproductivity formulae of horizontal wellsrdquo Journal ofCanadian Petroleum Technology vol 40 no 10 pp 55ndash67 2001
[14] F Samaniego W E Brigham and F G Miller ldquoPerformance-prediction procedure for transient flow of fluids throughpressure-sensitive formationsrdquo Journal of PetroleumTechnologyvol 31 no 6 pp 779ndash786 1979
[15] G K Falade ldquoTransient flow of fluids in reservoirs with stresssensitive rock and fluid propertiesrdquo International Journal ofNon-Linear Mechanics vol 17 no 4 pp 277ndash283 1982
[16] R W Ostensen ldquoMicrocrack Permeability in tight gas sand-stonerdquo Society of Petroleum Engineers Journal vol 23 no 6 pp919ndash927 1983
[17] J Pedrosa and O A Petrobras ldquoPressure transient response instress-sensitive formationsrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA 1986
[18] D A Barry D A Lockington D-S Jeng J-Y Parlange L Liand F Stagnitti ldquoAnalytical approximations for flow in com-pressible saturated one-dimensional porous mediardquo Advancesin Water Resources vol 30 no 4 pp 927ndash936 2007
[19] T Friedel and H D Voigt ldquoAnalytical solutions for the radialflow equation with constant-rate and constant-pressure bound-ary conditions in reservoirs with pressure-sensitive perme-abilityrdquo in Proceedings of the SPE Rocky Mountain PetroleumTechnology Conference Denver Colo USA 2009
[20] B Ju YWu and T Fan ldquoStudy on fluid flow in nonlinear elasticporousmedia experimental andmodeling approachesrdquo Journalof Petroleum Science and Engineering vol 76 no 3-4 pp 205ndash211 2011
[21] D Swartzendruber ldquoNon-Darcy flow behavior in liquid satu-rated porousmediardquo Journal of Geophysical Research vol 67 no13 pp 5205ndash5213 1962
[22] R J Miller and F L Philip ldquoThreshold Gradient for water flowin clay systemrdquo Soil Science Society of America Journal vol 27no 6 pp 605ndash609 1963
[23] H W Olsen ldquoDeviations from Darcyrsquos law in saturated claysrdquoSoil Science Society of America Journal vol 29 no 2 pp 135ndash1401965
[24] H Pascal ldquoNonsteady flow through porous media in the pre-sence of a threshold gradientrdquo Acta Mechanica vol 39 no 3-4pp 207ndash224 1981
[25] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999
[26] J Lu and S Ghedan ldquoPressure behavior of vertical wells inlow-permeability reservoirs with threshold pressure gradientrdquoSpecial Topics and Reviews in Porous Media vol 2 no 3 pp157ndash169 2011
[27] Y L Zhao L H Zhang F Wu B N Zhang and Q G LiuldquoAnalysis of horizontal well pressure behaviour in fracturedlow permeability reservoirs with consideration of the thresholdpressure gradientrdquo Journal of Geophysics and Engineering vol10 no 3 pp 1ndash10 2013
[28] Y L Zhao L H Zhang J Z Zhao S Y Hu and B N ZhangldquoTransient pressure analysis of horizontal well in low per-meability oil reservoirrdquo International Journal of Oil Gas andCoal Technology 2014
[29] M Muskat The Flow of Homogeneous Fluids through a PorousMedia Intl Human Resources Development Corp BostonMass USA 1937
[30] I S Gradshteyn and I M Ryzhik Table of Integrals Seriesand Products Academic Press San Diego Calif USA Seventhedition 2007
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 Engineering
(1) The steady-state percolation mathematical modelsof horizontal well with oilwater two-phase floware established and the corresponding solutions aresolved by the method of separation of variables
(2) For low-permeability reservoir there always existsthe phenomenon of permeability stress-sensitive (120572
119896)
which has a significant influence on the well produc-tivity the bigger the 120572
119896is the smaller the productivity
is(3) Due to the existence of capillary pressure of two-
phase flow there always is threshold pressure gradient(119866) in the fluid seepage process Although the 119866 hasa smaller effect on the productivity of the horizontalwell for a big drainage volume we cannot neglect itseffect on the productivity
Appendix
From the expressions of (11) we have
119903120582119905119890minus120572119896(119901119894minus119901) (
d119901d119903
minus 1198661015840
) = 119888 (A1)
where 119888 is a constantWe define the following expression
119910 = 119890minus120572119896(119901119894minus119901) (A2)
Substituting (A2) into (A1) and (12)-(13) yields
d119910d119903
= 1205721198961198661015840
119910 +119888120572119896
119903120582119905
(A3)
1199101003816100381610038161003816119903=1 = 119890
minus120572119896(119901119894minus119901119888) (A4)
1199101003816100381610038161003816119903=119903119890119867
= 1 (A5)
According to the general solution of the Bernoulli differ-ential equation [30] the solution of (A3) can be obtained as
119910 (119903) = 1198901205721198961198661015840119903
sdot [119890minus120572119896119866
1015840
119890minus120572119896(119901119894minus119901119888) + int
119903
119903119908
119890minus120572119896119866
1015840119903
119888120572119896
119903120582119905(119903)
d119903] (A6)
Substituting (A6) into (A5) the value of constant 119888 canbe solved and then combining it with (A6) we have
119901 (119903)
= 119901119894+
1
120572119896
ln
1198901205721198961198661015840119903
sdot [
[
119890minus120572119896(119866
1015840+119901119894minus119901119888) +
119890minus1205721198961198661015840119903119890119867 minus 119890minus120572119896(119866
1015840+119901119894minus119901119888)
int119903119890119867
119903119908
119890minus1205721198961198661015840119903 sdot (1119903120582
119905(119903)) d119903
sdot int119903
119903119908
119890minus120572119896119866
1015840119903
1
119903120582119905(119903)
d119903]
]
(A7)
Equation (A7) is the pressure distribution relation alongthe radius 119903 with the oilwater two-phase flows in thehorizontal plane of the horizontal well
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Natural Science Foundationof China (Grant no 51374181) and the project of NationalScience Fund for Distinguished Young Scholars of China(Grant no 51125019)The authors would also like to thank thereviewers and editors for their patience to read this paper andvaluable comments
References
[1] V P Merkulov ldquoLe debit des puits devies et horizontauxrdquo NeftKhoz vol 6 no 1 pp 51ndash56 1958
[2] J P Borisov Oil Production Using Horizontal and MultipleDeviation Wells The RampD Translation Company BartlesvilleOkla USA 1984
[3] F M Giger L H Reiss and A P Jourdan ldquoThe reservoir engi-neering aspects of horizontal drillingrdquo in Proceedings of the59th Annual Technical Conference and Exhibition Houston TexUSA 1984
[4] F M Giger ldquoHorizontal wells production techniques in het-erogeneous reservoirsrdquo in Proceedings of the Middle East OilTechnical Conference and Exhibition Bahrain 1985
[5] B J Karcher and F M Giger ldquoSome practical formulas to pre-dict horizontal well behaviorrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition New Orleans La USA1986
[6] L H Reiss ldquoProduction from horizontal wells after five yearsrdquoJournal of Petroleum Technology vol 39 no 11 pp 1411ndash14161987
[7] S D Joshi ldquoAugmentation of well productivity using slant andhorizontal wellsrdquo Journal of Petroleum Technology vol 40 no6 pp 729ndash739 1988
[8] D K Babu and A S Odeh ldquoProductivity of a horizontal wellrdquoSPE Reservoir Engineering vol 4 no 4 pp 417ndash421 1989
[9] G Renard and JM Dupuy ldquoFormation damage effects on hori-zontal-well flow efficiencyrdquo Journal of Petroleum Technologyvol 43 no 7 pp 786ndash789 1991
[10] M W Helmy and R A Wattenbarger ldquoSimplified productivityequations for horizontal wells producing at constant rate andconstant pressurerdquo in Proceedings of the SPE Technical Con-ference and Exhibition pp 379ndash388 New Orleans La USASeptember 1998
[11] T Billiter J Lee and R Chase ldquoDimensionless inflow-perform-ance-relationship curve for unfractured horizontal gas wellsrdquo inProceedings of the SPE Eastern Regional Meeting Canton OhioUSA 2001
[12] E G Anklam and M L Wiggins ldquoHorizontal well produc-tivity and wellbore pressure behavior incorporating wellborehydraulicsrdquo in Proceedings of the SPE Production andOperations
Mathematical Problems in Engineering 9
Symposium pp 565ndash584 Oklahoma City Okla USA April2005
[13] J Lu ldquoNewproductivity formulae of horizontal wellsrdquo Journal ofCanadian Petroleum Technology vol 40 no 10 pp 55ndash67 2001
[14] F Samaniego W E Brigham and F G Miller ldquoPerformance-prediction procedure for transient flow of fluids throughpressure-sensitive formationsrdquo Journal of PetroleumTechnologyvol 31 no 6 pp 779ndash786 1979
[15] G K Falade ldquoTransient flow of fluids in reservoirs with stresssensitive rock and fluid propertiesrdquo International Journal ofNon-Linear Mechanics vol 17 no 4 pp 277ndash283 1982
[16] R W Ostensen ldquoMicrocrack Permeability in tight gas sand-stonerdquo Society of Petroleum Engineers Journal vol 23 no 6 pp919ndash927 1983
[17] J Pedrosa and O A Petrobras ldquoPressure transient response instress-sensitive formationsrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA 1986
[18] D A Barry D A Lockington D-S Jeng J-Y Parlange L Liand F Stagnitti ldquoAnalytical approximations for flow in com-pressible saturated one-dimensional porous mediardquo Advancesin Water Resources vol 30 no 4 pp 927ndash936 2007
[19] T Friedel and H D Voigt ldquoAnalytical solutions for the radialflow equation with constant-rate and constant-pressure bound-ary conditions in reservoirs with pressure-sensitive perme-abilityrdquo in Proceedings of the SPE Rocky Mountain PetroleumTechnology Conference Denver Colo USA 2009
[20] B Ju YWu and T Fan ldquoStudy on fluid flow in nonlinear elasticporousmedia experimental andmodeling approachesrdquo Journalof Petroleum Science and Engineering vol 76 no 3-4 pp 205ndash211 2011
[21] D Swartzendruber ldquoNon-Darcy flow behavior in liquid satu-rated porousmediardquo Journal of Geophysical Research vol 67 no13 pp 5205ndash5213 1962
[22] R J Miller and F L Philip ldquoThreshold Gradient for water flowin clay systemrdquo Soil Science Society of America Journal vol 27no 6 pp 605ndash609 1963
[23] H W Olsen ldquoDeviations from Darcyrsquos law in saturated claysrdquoSoil Science Society of America Journal vol 29 no 2 pp 135ndash1401965
[24] H Pascal ldquoNonsteady flow through porous media in the pre-sence of a threshold gradientrdquo Acta Mechanica vol 39 no 3-4pp 207ndash224 1981
[25] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999
[26] J Lu and S Ghedan ldquoPressure behavior of vertical wells inlow-permeability reservoirs with threshold pressure gradientrdquoSpecial Topics and Reviews in Porous Media vol 2 no 3 pp157ndash169 2011
[27] Y L Zhao L H Zhang F Wu B N Zhang and Q G LiuldquoAnalysis of horizontal well pressure behaviour in fracturedlow permeability reservoirs with consideration of the thresholdpressure gradientrdquo Journal of Geophysics and Engineering vol10 no 3 pp 1ndash10 2013
[28] Y L Zhao L H Zhang J Z Zhao S Y Hu and B N ZhangldquoTransient pressure analysis of horizontal well in low per-meability oil reservoirrdquo International Journal of Oil Gas andCoal Technology 2014
[29] M Muskat The Flow of Homogeneous Fluids through a PorousMedia Intl Human Resources Development Corp BostonMass USA 1937
[30] I S Gradshteyn and I M Ryzhik Table of Integrals Seriesand Products Academic Press San Diego Calif USA Seventhedition 2007
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
Symposium pp 565ndash584 Oklahoma City Okla USA April2005
[13] J Lu ldquoNewproductivity formulae of horizontal wellsrdquo Journal ofCanadian Petroleum Technology vol 40 no 10 pp 55ndash67 2001
[14] F Samaniego W E Brigham and F G Miller ldquoPerformance-prediction procedure for transient flow of fluids throughpressure-sensitive formationsrdquo Journal of PetroleumTechnologyvol 31 no 6 pp 779ndash786 1979
[15] G K Falade ldquoTransient flow of fluids in reservoirs with stresssensitive rock and fluid propertiesrdquo International Journal ofNon-Linear Mechanics vol 17 no 4 pp 277ndash283 1982
[16] R W Ostensen ldquoMicrocrack Permeability in tight gas sand-stonerdquo Society of Petroleum Engineers Journal vol 23 no 6 pp919ndash927 1983
[17] J Pedrosa and O A Petrobras ldquoPressure transient response instress-sensitive formationsrdquo in Proceedings of the SPE CaliforniaRegional Meeting Oakland Calif USA 1986
[18] D A Barry D A Lockington D-S Jeng J-Y Parlange L Liand F Stagnitti ldquoAnalytical approximations for flow in com-pressible saturated one-dimensional porous mediardquo Advancesin Water Resources vol 30 no 4 pp 927ndash936 2007
[19] T Friedel and H D Voigt ldquoAnalytical solutions for the radialflow equation with constant-rate and constant-pressure bound-ary conditions in reservoirs with pressure-sensitive perme-abilityrdquo in Proceedings of the SPE Rocky Mountain PetroleumTechnology Conference Denver Colo USA 2009
[20] B Ju YWu and T Fan ldquoStudy on fluid flow in nonlinear elasticporousmedia experimental andmodeling approachesrdquo Journalof Petroleum Science and Engineering vol 76 no 3-4 pp 205ndash211 2011
[21] D Swartzendruber ldquoNon-Darcy flow behavior in liquid satu-rated porousmediardquo Journal of Geophysical Research vol 67 no13 pp 5205ndash5213 1962
[22] R J Miller and F L Philip ldquoThreshold Gradient for water flowin clay systemrdquo Soil Science Society of America Journal vol 27no 6 pp 605ndash609 1963
[23] H W Olsen ldquoDeviations from Darcyrsquos law in saturated claysrdquoSoil Science Society of America Journal vol 29 no 2 pp 135ndash1401965
[24] H Pascal ldquoNonsteady flow through porous media in the pre-sence of a threshold gradientrdquo Acta Mechanica vol 39 no 3-4pp 207ndash224 1981
[25] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999
[26] J Lu and S Ghedan ldquoPressure behavior of vertical wells inlow-permeability reservoirs with threshold pressure gradientrdquoSpecial Topics and Reviews in Porous Media vol 2 no 3 pp157ndash169 2011
[27] Y L Zhao L H Zhang F Wu B N Zhang and Q G LiuldquoAnalysis of horizontal well pressure behaviour in fracturedlow permeability reservoirs with consideration of the thresholdpressure gradientrdquo Journal of Geophysics and Engineering vol10 no 3 pp 1ndash10 2013
[28] Y L Zhao L H Zhang J Z Zhao S Y Hu and B N ZhangldquoTransient pressure analysis of horizontal well in low per-meability oil reservoirrdquo International Journal of Oil Gas andCoal Technology 2014
[29] M Muskat The Flow of Homogeneous Fluids through a PorousMedia Intl Human Resources Development Corp BostonMass USA 1937
[30] I S Gradshteyn and I M Ryzhik Table of Integrals Seriesand Products Academic Press San Diego Calif USA Seventhedition 2007
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
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