Research Article Nonlinear Seepage Model of Gas Transport...
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Research Article Nonlinear Seepage Model of Gas Transport in Multiscale Shale Gas Reservoirs and Productivity Analysis of Fractured Well Ting Huang, 1 Xiao Guo, 1 and Kun Wang 2 1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Xindu Road 8, Chengdu 610500, China 2 Research Institute of CNOOC Ltd., Shenzhen Branch, Guangzhou 510240, China Correspondence should be addressed to Ting Huang; [email protected] Received 21 November 2014; Revised 3 March 2015; Accepted 7 March 2015 Academic Editor: Agus Sasmito Copyright © 2015 Ting Huang 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. Shale is abundant in nanoscale pores, so gas flow in shales cannot be simply represented by Darcy formula anymore. It is crucial to figure out the influence of gas flow in nano/micro pores on actual productivity, which can provide basic theories for optimizing parameters and improving the gas production from engineering perspective. is paper considers the effects of slippage and diffusion in nanoscale based on Beskok-Karniadakis (BK) equation, which can be applicable for different flow regimes including continuum flow, slip flow, transition flow, and free-molecule flow. A new non-Darcy equation was developed based on the analysis of effects of high order terms of BK equation on permeability correction factor. By using the conformal transformation principle and pressure coupling method, we established the productivity formula of fractured well (infinite and limited conductivity) satisfying mass variable seepage flowing in fractures. e simulation results have been compared with field data and influencing parameters are analyzed thoroughly. It is concluded that slippage effect affects gas production of fractured well when wellbore pressure is less than 15 MPa, and the effects of slippage and diffusion have greater influence on gas production of fractured well for reservoir with smaller permeability, especially when permeability is at nano-Darcy scale. 1. Introduction Shale has been found to contain numerous well-developed nanoscale pores in recent researches: pore diameter of Hay- nesville basin in North America mainly distributes from 2 nm to 20 nm [1]; pore diameter of Mississippian basin is between 5 nm and 750 nm [2]; pore diameter of Sichuan basin in China is around 100 nm [3]. Core analysis shows that the permeability of shale is in a range of 1 × 10 −9 –1 × 10 −3 m 2 [4], and flow in extremely low permeability shales undergoes a transition from Darcy regime to other regimes owing to the significant effect of collisions between molecules and pore walls on gas transport. Darcy Formula cannot describe all the flow regimes in shale gas reservoirs, so it is necessary to establish a transport equation valid for all flow regimes in multiscale shale gas reservoirs. Gas flow in shales cannot be simply represented by Darcy equation anymore owing to pore throat diameters on the order of nanometers [4–11]. Javadpour et al. [5] employed a group of Gaussian distribution curves to model nonlinear flow in a combination of nano/micropore networks and was validated with the results of canister desorption tests. Wang and Reed [6] proposed that gas flow in shale matrix can be non-Darcy as a result of the slippage effect but a Darcy type in fractures. Huang et al. [7, 8] used apparent permeability which combines Knudsen diffusion coefficient and slippage factor to describe gas flow in nanoscale pores within shale matrix. Michel et al. [9] established a model to correct the permeability to account for all flow regimes including continuum, such as slip, transition, and molecular flow in shales based on Beskok and Karniadakis equation [10, 11], but the slip coefficient is assumed to be constant. Deng et al. [4] developed a seepage model in regard to different slip coefficients on the basis of Michel’s model, and diffusion is considered as well. However, the slippage factor is only assigned to some different constants in previous articles which cannot really simulate the entire flow regimes for gas flow in shale gas Hindawi Publishing Corporation Journal of Chemistry Volume 2015, Article ID 349507, 10 pages http://dx.doi.org/10.1155/2015/349507
Transcript of Research Article Nonlinear Seepage Model of Gas Transport...
Research ArticleNonlinear Seepage Model of Gas Transport in Multiscale ShaleGas Reservoirs and Productivity Analysis of Fractured Well
Ting Huang1 Xiao Guo1 and Kun Wang2
1State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum University Xindu Road 8Chengdu 610500 China2Research Institute of CNOOC Ltd Shenzhen Branch Guangzhou 510240 China
Correspondence should be addressed to Ting Huang huangting331126com
Received 21 November 2014 Revised 3 March 2015 Accepted 7 March 2015
Academic Editor Agus Sasmito
Copyright copy 2015 Ting Huang 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
Shale is abundant in nanoscale pores so gas flow in shales cannot be simply represented by Darcy formula anymore It is crucialto figure out the influence of gas flow in nanomicro pores on actual productivity which can provide basic theories for optimizingparameters and improving the gas production from engineering perspective This paper considers the effects of slippage anddiffusion in nanoscale based on Beskok-Karniadakis (BK) equation which can be applicable for different flow regimes includingcontinuum flow slip flow transition flow and free-molecule flow A new non-Darcy equation was developed based on the analysisof effects of high order terms of BK equation on permeability correction factor By using the conformal transformation principle andpressure coupling method we established the productivity formula of fractured well (infinite and limited conductivity) satisfyingmass variable seepage flowing in fractures The simulation results have been compared with field data and influencing parametersare analyzed thoroughly It is concluded that slippage effect affects gas production of fractured well when wellbore pressure is lessthan 15MPa and the effects of slippage and diffusion have greater influence on gas production of fractured well for reservoir withsmaller permeability especially when permeability is at nano-Darcy scale
1 Introduction
Shale has been found to contain numerous well-developednanoscale pores in recent researches pore diameter of Hay-nesville basin inNorthAmericamainly distributes from2nmto 20 nm [1] pore diameter of Mississippian basin is between5 nm and 750 nm [2] pore diameter of Sichuan basin inChina is around 100 nm [3] Core analysis shows that thepermeability of shale is in a range of 1 times 10minus9ndash1 times 10minus3 120583m2[4] and flow in extremely low permeability shales undergoesa transition fromDarcy regime to other regimes owing to thesignificant effect of collisions between molecules and porewalls on gas transport Darcy Formula cannot describe allthe flow regimes in shale gas reservoirs so it is necessary toestablish a transport equation valid for all flow regimes inmultiscale shale gas reservoirs
Gas flow in shales cannot be simply represented by Darcyequation anymore owing to pore throat diameters on theorder of nanometers [4ndash11] Javadpour et al [5] employed
a group of Gaussian distribution curves to model nonlinearflow in a combination of nanomicropore networks and wasvalidated with the results of canister desorption tests Wangand Reed [6] proposed that gas flow in shale matrix can benon-Darcy as a result of the slippage effect but a Darcy typein fractures Huang et al [7 8] used apparent permeabilitywhich combines Knudsen diffusion coefficient and slippagefactor to describe gas flow in nanoscale pores within shalematrix Michel et al [9] established a model to correctthe permeability to account for all flow regimes includingcontinuum such as slip transition and molecular flow inshales based on Beskok and Karniadakis equation [10 11]but the slip coefficient 119887 is assumed to be constant Denget al [4] developed a seepage model in regard to different slipcoefficients on the basis of Michelrsquos model and diffusion isconsidered as well
However the slippage factor is only assigned to somedifferent constants in previous articles which cannot reallysimulate the entire flow regimes for gas flow in shale gas
Hindawi Publishing CorporationJournal of ChemistryVolume 2015 Article ID 349507 10 pageshttpdxdoiorg1011552015349507
2 Journal of Chemistry
Table 1 Classification of gas flow regimes based on Knudsen number
Knudsen number Kn le 0001 0001 lt Kn le 01 01 lt Kn le 10 Kn gt 10
Flow regime Continuum flow Slip flow Transition flow Free-molecule flow
reservoirs but actually slippage factor is changing withpressure pore radius and other parameters This paperconsiders the effects of slippage and diffusion based onBeskok-Karniadakis (BK) equation and slippage factor anddiffusion coefficient are varying with pressure and shalepermeability in the process of simulation A new non-Darcyequation of motion is obtained based on the analysis of theeffects of high order terms of BK equation on permeabilitycorrection factor By employing conformal transformationprinciple and pressure coupling method a second orderdifferential equation satisfying variable mass flowing in frac-tures is derived and productivity formula of fractured well(infinite and limited conductivity) for shale gas reservoir isestablished The simulation results are compared with fielddata and effects of slippage and diffusion on productivity offractured well have been analyzed thoroughly as well as otherparameters
2 Flow Regimes in Shale Gas Reservoirs
Knudsen number Kn is defined as the ratio of the fluid meanfree path 120582 and pore throat diameter 119903 which is a widelyrecognized dimensionless parameter to determine the degreeof appropriateness of continuum model
Kn = 120582
119903
(1)
Gas flow regimes can be classified into four categories [12]depending on the Knudsen number (Table 1) (1) continuumflow (2) slip flow (3) transition flow (4) and free-moleculeflow In continuum flow regime no-slip boundary conditionis valid and gas flow is linear As Kn increases the rarefac-tion effects become more pronounced and the continuumassumption breaks down eventually So in flow regimes otherthan continuum flow Darcy law is not applicable
To analyze the flow regimes in shale gas reservoirsFigure 1 presents the Knudsen number under different poresizes between 1 nm and 100 120583mand different pressure rangingfrom atmosphere to 100MPa where gas flow from formationto surface undergoes Knudsen number increases whenpressure drops and pore throat diameter decreases Thehorizontal plane in Figure 1 represents that Kn equals 0001Gas flow regime in shales is continuum flow when Knudsennumber is beneath the horizontal plane but it changes intoother flow regimes when Knudsen number is above thehorizontal plane
For example when formation pressure is 100MPa gasflow in shale gas reservoirs is slip flow when pore diameteris between 1 nm and 63 nm and is transition flow when porediameter is smaller than 1 nm but when formation pressuredrops to 10MPa gas flow in shale gas reservoirs is slip flowwhen pore diameter is between 10 nm and 631 nm and istransition flowwhen pore diameter is smaller than 10 nmGas
Pore throat diameter (nm)Pressure (kPa)
Knud
sen
num
ber
Continuum flow
Slip flow
Transition flowFree-molecule flow104
102
102103
104105105
102
100
10minus2
10minus4
10minus6
10minus1
Figure 1 The relationship between Knudsen number pressure andpore radius
flow in shale gas reservoirs is free-molecule flow when porediameter is smaller than 63 nm and pressure is below 8MPaSo we can conclude that gas flow in shale gas reservoirs is amultiscale flow process which includes continuum flow slipflow transition flow and free-molecule flow
3 Multiscale Non-Darcy Seepage Model inShale Matrix
Volume flow rate of gas flow in reservoir can be representedby Darcy Law
V119892
= minus
119870
120583
119889119901
119889119909
(2)
According to the Beskok-Karniadakis equation whichaccounts for different flow regimes of the continuumflow slipflow transition flow and free-molecule flow the relationshipbetween flow rate and pressure gradient can be described as
V119892
= minus
1198700
120583
(1 + 120572Kn) (1 + 4Kn1 minus 119887Kn
)
119889119901
119889119909
(3)
Combining (2) and (3) permeability correction factor canbe written as
119870 = 1198700
120585
120585 = (1 + 120572Kn) (1 + 4Kn1 minus 119887Kn
)
(4)
We can see from (4) that the apparent permeabilityis close to the value of absolute permeability when Knapproaches 0 When Knudsen number gets bigger which
Journal of Chemistry 3
2E + 0
1E + 01E minus 3 1E minus 2 1E minus 1
Perm
eabi
lity
adju
stmen
t fac
tor
Knudsen number
120585 = 1 + 4Kn120585 = 1 + 4Kn + 512151205872Kn14
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2 + 2048151205872Kn24
(a) Slippage factor 119887 = 01
2E + 0
1E + 0Perm
eabi
lity
adju
stmen
t fac
tor
1E minus 3 1E minus 2 1E minus 1
Knudsen number
120585 = 1 + 4Kn120585 = 1 + 4Kn + 512151205872Kn14
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2 + 2048151205872Kn24
(b) Slippage factor 119887 = 5
Figure 2 Relationship between permeability correction factor and Knudsen number
means that the flow in microtube is no longer Darcy flow theapparent permeability should be corrected
The rarefaction coefficient is defined by Beskok andKarniadakis (1999) as
120572 =
128
151205872
tanminus1 (4Kn04) (5)
To simplify Beskok-Karniadakis equation [4 9 13] per-meability correction factor can be rewritten as
120585 = 1 + 120572Kn + 4Kn1 minus 119887Kn
+
4120572Kn2
1 minus 119887Kn (6)
The first two terms of (6) represent the first order termto no-slip flow in Beskok-Karniadakis equation When Kn lt01 the following approximations are valid [4 9]
120572 asymp
128
151205872
[4Kn04 minus 1
3
(4Kn04)3
] (7)
Kn1 minus 119887Kn
asymp Kn (1 + 119887Kn + 1198872Kn2) (8)
When Kn gt 01 the value of rarefaction coefficient cal-culated by (7) is negative which is very different fromrarefaction coefficient calculated by (5) so we use anotherapproximation instead
120572 asymp
128
151205872
(4Kn04) (9)
By combining (8) and (9) the permeability correctionfactor can be rewritten as
120585 = 1 + 4Kn + 512
151205872
Kn14 + 4119887Kn2 + 2048
151205872
Kn24
+ 41198872Kn3 + 119900 (Kn3)
(10)
Here the influence of high order terms of (10) on per-meability correction factor will be discussed In Figures 2(a)and 2(b) the term of Kn24 has little influence on permeabilitycorrection factor whether the value of slippage factor 119887 is bigor small the effect of term Kn2 on permeability correctionfactor can be neglected when the value of slippage factor issmall but this effect becomes greater when slippage factorgets bigger For slip flow and continuum flow the high order(gt2) correction term of (10) can be neglected and (3) can bederived as
V119892
= minus
1198700
120583
(1 + 4Kn + 512
151205872
Kn14 + 4119887Kn2)119889119901
119889119909
(11)
The definition of mean path is given as [14]
120582 = radic120587119911119877119879
2119872
120583
119901
(12)
Knudsen diffusion coefficient is mainly changing withpore diameter which is defined as [5 12 15]
119863K =2119903
3
(
8119877119879
120587119872
)
05
(13)
Slippage factor 119887 is changing with pressure and porediameter which is given by [16ndash18]
119887 = (
8120587119877119879
119872
)
05
120583
119901avg119903(
2
119891
minus 1) (14)
Knudsen number can be represented by Knudsen diffu-sion coefficient according to (12) and (13)
Kn = 3120587
81199032
120583
119901
119863K (15)
4 Journal of Chemistry
Taking (15) into (11) we can get
V119892
= minus
1198700
120583
[1 + 4
3120587
81199032
120583
119901
119863K +512
151205872
(
3120587
81199032
120583
119901
119863K)14
+ 4119887(
3120587
81199032
120583
119901
119863K)2
]
119889119901
119889119909
(16)
According to Hagen-Poiseuille model relationshipbetween permeability and pore diameter can be written as
1198700
=
1199032
8
(17)
Taking (17) into (16) we can derive
V119892
= minus
1198700
120583
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119889119901
119889119909
(18)
Considering both sides of (18) multiplied by seepage areaand divided by volume factor we can get the volume flow rateat standard condition
119902V =V119860119861119892
=
1198601198700
119861119892
120583119892
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119889119901
119889119909
(19)
where the gas volume factor 119861119892
can be derived according tothe equation of state
119861119892
=
119899119911119877119879119901
119899119877119879sc119901sc=
119901sc119901
119911119879
119879sc (20)
4 Productivity Model of Fractured Well
The natural productivity of shale gas reservoirs is very lowso formation has to be artificially stimulated (hydraulicallyfractured) to render commercial production So this paperestablishes steady seepage model of fractured well for shalegas reservoirs by conformal transformation According to theprinciple of conformal transformation the gas productionand pressure of the boundary remains the same only thelength of line and flow forms change [19 20]
41 Physical Assumptions To make this mathematical modelmore tractable the following simplified assumptions aremade
(1) The reservoir is isotropic and gas flow in shale gasreservoir is planar (steady state)
(2) Flow in nanopores of shale matrix is assumed to benon-Darcy and fluid in shale gas reservoirs is singlephase
Table 2 Conformal transformations of the fractured well
Location ofpoint 119882 plane 119885 plane
A 119906 = 0 V = 0 119909 = 119871119891
119910 = 0
B 119906 = 0 V = 1205872 119909 = 0 119910 = 0
C 119906 = 0 V = 120587 119909 = minus119871119891
119910 = 0
D 119906 = 1199060
V = 0 119909 = 119871119891
ch1199060
119910 = 0
E 119906 = 1199060
V = 1205872 119909 = 0 119910 = 119871119891
sh1199060
F 119906 = 1199060
V = 120587 119909 = minus119871119891
ch1199060
119910 = 0
(3) Fractured vertical well is produced at a constantwellbore pressure and is penetrated fully in shale gasreservoir The height of hydraulic fractures is equal tothe thickness of formation
(4) The fractures are vertical and symmetrically locatedat two sides of wellbore Fracture half-length is 119871
119891
fracture width is negligible when the vertical fracturesare assumed to have infinite conductivity and is 119882
119891
when vertical fractures are assumed to have finiteconductivity
42 Fractured Well with Infinite Conductivity Conformaltransformation function is defined as
119885 = 119871119891
ch119882 (21)
Taking 119885 = 119909 + 119894119910119882 = 119906 + 119894V into (21) we can get
119909 = 119871119891
ch119906 cos V
119910 = 119871119891
sh119906 sin V(22)
We can see from Table 2 that the upper half planeformation in 119885 plane (Figure 3(a)) is transformed to a halfinfinite formationwithwidth of120587 at the right side of fracturedwell in 119882 plane (Figure 3(b)) elliptical constant pressureboundarywith radius of119877
119890
is transformed to a linear onewithlength of 120587 fractured well with length of 2119871
119891
is transformedto a line sink in119882 plane
Equipotential line equation of gas flow in 119885 plane can beobtained through (22)
1199092
1198712
119891
ch2119906+
1199102
1198712
119891
sh2119906= cos2V + sin2V = 1 (23)
The boundary can be viewed as circular when the bound-ary is far from fractured well so
ch1199060
asymp sh1199060
asymp
1
2
1198901199060 (24)
Then the equipotential line equation of boundary can berewritten as
1199092
+ 1199102
= 1198712
119891
(
1
2
1198901199060)
2
= 1198772
119890
(25)
Journal of Chemistry 5
y
xA BO
v
Hydraulic fracture
Wellbore C
6 3
3
22
1 4
5
1
y
Z = LfchW5
4
6
Q
2
O998400 u0 u
(a) Z plane (b) W plane
Figure 3 Sketch map of conformal transformation to fractured well
Solving (25) we can get the relationship between thelength of drainage area 119906
0
in 119882 plane and the radius ofboundary 119877
119890
in 119885 plane
1199060
= ln2119877119890
119871119891
(26)
Combined with (19) production rate of drainage area in119882 plane can be derived
119876V =21205871198700
ℎ119879sc119901sc1198791205831199111199060
[
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(27)
Gas production of fractured well in 119885 plane can beobtained according to (26)
119876V =21205871198700
ℎ119879sc
119901sc119879120583119911 ln (2119877119890119871119891)
sdot [
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(28)
43 Fractured Well with Finite Conductivity Compared withhydraulic fracture with infinite conductivity fractured ver-tical well with finite conductivity is more close to realcondition and the effects of parameters of hydraulic fractureson well performance can be analyzed as well
Considering an element of hydraulic fracture in119882 planein Figure 3(b) we assume that gas only flows in V direction(flows to wellbore) in hydraulic fracture (Figure 4) So flow
u
d
u
Wf
Figure 4 Sketch map of hydraulic fracture element
rate in fracture VV is a constant value in 119906 direction and avariable value in V direction flow rate from reservoir to theelement V
119906
equals a constant valueThe mass balance equation of gas flow in hydraulic frac-
ture can be obtained by principle of mass conservation
minus (VV1003816100381610038161003816V+ΔV minus VV
1003816100381610038161003816V) sdot
1
2
119882119891
ℎ + V119906
sdot ΔVℎ = 0 (29)
Considering both sides of (29) divided by ΔV the partialdifference equation can be derivedwhenΔV takes an infinites-imal
119889VV119889V
sdot
1
2
119882119891
= V119906
(30)
Pseudopressure function of gas flow in hydraulic fractureis defined as
119898(119901) = 2int
119901
119901119890
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119901
120583119911
119889119901
(31)
Equation (30) can be
1205972
119898
120597V2minus
1198700
(12) 119896119891
119882119891
1
ln (2119877119890
119871119891
)
119898
= minus
1198700
(12) 119896119891
119882119891
1
ln (2119877119890
119871119891
)
119898119890
(32)
6 Journal of Chemistry
The boundary conditions of hydraulic fracture are givenas follows
119889119898
119889V= 0 V = 0
119898 = 119898119908
V =120587
2
(33)
Combining with (33) we can get the solution of (32)
119898(119901) = 1198881
119890120582V+ 1198882
119890minus120582V
+ 119898119890
(34)
where the parameters 1198881
1198882
and 120582 are defined as follows
120582 = radic
21198700
119896119891
119882119891
1
ln 2119877119890
119871119891
1198881
= 1198882
=
119898119908
minus 119898119890
119890(1205872)120582
+ 119890minus(1205872)120582
(35)
The gas production of fractured vertical well can be writ-ten as
119876119891
= minus
119896119891
119882119891
ℎ119879sc
119901sc119879
119889119898
119889V
10038161003816100381610038161003816100381610038161003816V=1205872
=
119896119891
119882119891
ℎ119879sc
119901sc119879120582 (119898119890
minus 119898119908
)
119890(1205872)120582
minus 119890minus(1205872)120582
119890(1205872)120582
+ 119890minus(1205872)120582
(36)
Combined with (31) we can get the productivity formulaof fractured vertical well with finite conductivity
119876119891
=
2119896119891
119882119891
ℎ120582119879sc
119901sc119879120583119911tanh 120587120582
2
sdot [
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(37)
where the first term in the bracket is the gas productioncalculated by Darcy formula which is defined as 119876
119863
5 Results and Discussion
According to the productivity formula of fractured well withfinite conductivity the effects of permeability correctionfactor slippage factor Knudsen diffusion coefficient matrixpermeability fracture half-length and fracture conductivityon productivity of fractured well are analyzed by combiningthe data from a single well of shale gas reservoirs in ChinaThe data for production simulation are presented in Table 3
51 Effect of Permeability Correction Factor on ProductivityFigure 5 presents the effect of permeability correction factoron productivity of fractured well and we can see fromthis figure that permeability correction factor has a great
Table 3 Data of shale gas reservoir for production simulation
Parameter Value UnitsPermeability 119870 0005 mDPorosity 120601 007 mdashFormation temperature 119879 36615 KSurface temperature 119879sc 293 KFormation pressure 119901
119890
28 MPaFormation thickness ℎ 305 mPressure relief radius 119877
119890
400 mGas viscosity 120583 0014 mPasdotsGas compressibility factor 119911 089 mdashRadius of wellbore 119903
119908
01 m
120585 = 1 + 4Kn + 512151205872Kn14
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
Field data
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
120585 = 1 (calculated by Darcy formula)120585 = 1 + 4Kn
Figure 5 Effect of permeability correction factor 120585 on productivity
influence on productivity Production of fractured well isvery small when our model only considers Darcy flow innanomicroscale pores (120585 = 1) but gas production increasesa lot when matrix apparent permeability is corrected bypermeability correction factor 120585 Field data shows that gasproduction of a fractured vertical well is 12 times 104m3 whenproduction pressure drop keeps 7MPa and fracture half-length is 200m [4] The calculation results of multiscalenon-Darcy seepage model are much more close to thefield data compared with the production rate calculated byDarcy formula andproduction rate calculated by productivityformula only considering first order term of permeabilitycorrection factor
52 Effects of Formation and Fracture Parameters on Produc-tivity The effect of matrix permeability 119870 on productivityof fractured well is illustrated in Figure 6 Gas productionof fractured well increases with the matrix permeabilityand gas production increases to 171 times 232 times and287 times when matrix permeability increases to 2 times 3times and 4 times respectively Figure 7 reflects productivity
Journal of Chemistry 7
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
K = 0005mDK = 0010mD
K = 0015mDK = 0020mD
Gas production rate (times104 m3d)
Figure 6 Effect of matrix permeability 119870 on productivity
0
5
10
15
20
25
30
0 05 1 15 2 25 3
Well
bore
pre
ssur
e (M
Pa)
Gas production rate (times104 m3d)
Lf = 50mLf = 150m
Lf = 250mLf = 350m
Figure 7 Effect of fracture half-length 119871119891
on productivity
of fractured well affected by fracture half-length Gas pro-duction increases to 154 times 206 times and 265 timeswhen fracture half-length increases to 3 times 4 times and5 times respectively We can find that the increase of gasproduction rate affected by matrix permeability is still morethan production rate affected by fracture half-length even ifthe increase ratio of fracture half-length is bigger thanmatrixpermeability which means that longer fracture half-lengthonly can improve local seepage ability of formation adjacentto hydraulic fractures but improvement of flow capacityin the whole formation can increase the gas productionsubstantially
Effect of fracture conductivity on productivity of frac-tured well has been analyzed under different fracture pene-tration ratio (119871
119891
119877119890
) We can see from Figure 8 that gas pro-duction increases with the fracture conductivity and fracturepenetration ratio But gas production increase becomes slowwhen the fracture conductivity increases to a certain valueand this value becomes bigger when fracture penetrationratio increases For examples this value will be 2 120583m2sdotcmwhen fracture penetration ratio is 01 but this value willincrease to 4120583m2sdotcm when fracture penetration ratio is 03
0
05
1
15
2
25
3
0 1 2 3 4 5 6
Gas
pro
duct
ion
rate
(times10
4m
3d
)
LfRe = 01
LfRe = 03LfRe = 05LfRe = 07
kf middot Wf (120583m2middotcm)
Figure 8 Effect of fracture conductivity 119896119891
sdot 119882119891
on productivity
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 1 2 3 4 5
Gas production rate (times104 m3d)
b = 0b = 1
b = 2
b = 3
Figure 9 Effect of slippage factor 119887 on productivity
This chart can help to optimize fracture conductivity underdifferent fracture penetration ratio
53 Effects of Slippage and Knudsen Diffusion on ProductivityThe relationship between wellbore pressure of fractured wellwith finite conductivity and gas production under differentslippage factor 119887 has been shown in Figure 9 Slippage effecthas negligible influence on productivity of fractured wellwhen wellbore pressure is high but slippage effect begins toaffect gas production when wellbore pressure is lower than15MPa and productivity of the fractured well increases withthe slippage factor So decreasing the wellbore pressure offractured well properly can improve gas production whenexploiting shale gas reservoirs We can see from Figure 10that gas production rate increases with Knudsen diffusioncoefficient which has a greater impact on gas production thanslippage effect
As we can see in Figure 11 the productivity of fracturedwell derived in this paper (119876
119891
) has been compared withproduction calculated by Darcy formula (119876
119863
) under differ-ent matrix permeability and wellbore pressure Figure 11(a)shows that bigger matrix permeability and a lower wellbore
8 Journal of Chemistry
Table 4 Comparison of production at turning point and production at minimum permeability under different wellbore pressure (the gasproduction rate unit times104 m3d)
119875119908
MPa 1 3 5 7 9 11 13 15 17 19 21 23 25 27119876119870min 0505 0333 0250 0197 0158 0127 0103 0082 0065 0050 0037 0025 0014 0005
119876TP 0315 0238 0193 0160 0134 0112 0093 0076 0061 0048 0035 0024 0014 0004119876119870min119876TP 1601 1398 1296 1228 1178 1138 1109 1082 1064 1046 1034 1024 1014 1010
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
DK = 1 times 10minus6 m2sDK = 3 times 10minus6 m2s
DK = 5 times 10minus6 m2sDK = 7 times 10minus6 m2s
Figure 10 Effect of Knudsen diffusion coefficient119863K on productiv-ity
pressure lead to a higher productivity of fractured well andgas production 119876
119891
is much bigger than Darcy productionrate 119876
119863
The effects of Knudsen diffusion and slip flow onproductivity of fractured well can be observed by taking theratio of119876
119891
and119876119863
in Figure 11(b)The effects on gas produc-tion increase whenmatrix permeability becomes smaller andwellbore pressure declines which become most significantunder the smallest matrix permeability (1 times 10minus5mD) and thelowest wellbore pressure (1MPa) in Figure 11(b)
Figure 12 represents the relationship of gas productionrate119876
119891
and matrix permeability119870 at different wellbore pres-sure For conventional reservoirs production rate decreaseswith formation permeability for shale gas reservoirs gasproduction rate decreases with matrix permeability first butthen increases with permeability owing to the effects ofKnudsen diffusion and slip flow after a certain permeabilityvalue which is called turning point in this paperThe value ofturning point decreases when wellbore pressure rises whichis because Knudsen diffusion and slippage effect are morelikely to occur in formation with smaller permeability andlower pressure so a smaller turning point would be requiredwhen pressure rises
Table 4 shows the ratio of gas production rate at min-imum permeability 119876
119870min in Figure 12 and production atturning point under different wellbore pressures 119876TP Theratios decrease when wellbore pressure increases whichmeans that Knudsen diffusion and slippage effects have agreater influence on productivity of fractured well under alower wellbore pressure But the gas production rate119876
119891
(119870 =1 times 10minus5mD 119875
119908
= 1MPa) is only 16 times the production
119876119891
at turning point (Table 4) even though the production119876119891
is almost 150 times the Darcy production rate 119876119863
in Figure 11(b) Knudsen diffusion and slippage effects canimprove the productivity at extremely low permeabilitysignificantly compared with gas production calculated byDarcy formula (without considering effects of diffusion andslip flow) but the improvement of productivity is still smallcompared with higher permeability when Knudsen diffusionand slippage effects are considered in both situations
6 Conclusions
In this paper a multiscale comprehensive mathematicalmodel had been established which can simulate differentflow regimes including continuum flow slip flow transitionflow and free-molecule flow We deduced the productivityformula for fractured well with limited conductivity for shalegas reservoirs and the influencing parameters were analyzedthoroughly The following conclusions can be drawn
(1) Flow regimes in shale gas reservoir were analyzedby calculating Knudsen number under different pres-sure and different pore throat diameter Gas flow inshale gas reservoirs with nanomicroscale pores is acombination of continuum flow slip flow transitionflow and free-molecule flow which cannot simply berepresented by Darcy formula anymore
(2) A new non-Darcy equation was developed based onBeskok-Karniadakis equation which is applicable forall flow regimes and effects of high order terms ofBK equation on permeability correction factor wereanalyzed under different Knudsen number Perme-ability correction factor is increasingly deviating froma Darcy type as Knudsen number increases
(3) Productivity formula of fractured well satisfying vari-able mass flowing in fractures was obtained andthe simulated results showed that production ratecalculated by non-Darcy seepage model with higherorder terms of permeability correction factor is moreclose to the field production data compared withthe production rate calculated by Darcy formula andproduction rate calculated by productivity formulaonly considering first order term of permeabilitycorrection factor
(4) Effects of slippage and diffusion which are changingwith pressure and pore sizes in nanoscale pores wereconsidered in simulation Simulation results showedthat slippage effect affects gas production of fracturedwell only when wellbore pressure less than 15MPa
Journal of Chemistry 9
010
20300
05
1
15
2
Matrix permeability (mD)Wellbore pressure (MPa)
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
QD
Qf
times104
Prod
uctio
n (m
3d
)
(a) Comparison between 119876119863
and 119876119891
(at different permeability andwellbore pressure)
010
20300
50
100
150
Matrix permeability (mD)Wellbore pressure (MPa)
Prod
uctio
n co
mpa
rison
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
(b) The value of 119876119891
119876
119863
at different matrix permeability and wellborepressure
Figure 11 Effects of Knudsen diffusion coefficient119863K and slip factor 119887 on productivity
0
04
08
12
16
2
Turning point
Effects of slip flow andKnudsen diffusion
Matrix permeability (mD)1E minus 5 1E minus 4 1E minus 3 1E minus 2
Pw = 1Pw = 3Pw = 5Pw = 7Pw = 9Pw = 11
(MPa
)
Pw = 13
Pw = 15
Pw = 17
Pw = 19
Pw = 21
Pw = 23
Pw = 25
Pw = 27
Gas
pro
duct
ion
rate
(times10
4m
3d
)
Figure 12 The relationship between gas production and matrix permeability at different wellbore pressure
and the effects of slippage and diffusion have a greaterinfluence on gas production of reservoirs with smallermatrix permeability especially when permeability isat nano-Darcy scale Matrix permeability fracturehalf length fracture penetration ratio and fractureconductivity can improve gas production at differentdegrees as well
Nomenclature
Latin
119887 Slippage coefficient119861119892
Gas volume factor119863K Knudsen diffusion coefficient m2s119891 Fraction of molecules striking pore wall which are
diffusely reflected
ℎ Thickness of shale gas reservoir m119870 Apparent permeabilityKn Knudsen number119896119891
Permeability of hydraulic fracture1198700
Absolute permeability119871119891
Fracture half length119872 Molecular mass kgmol119898 Pseudopressure119898119890
Initial pseudopressure119898119908
Pseudopressure of fractured well119901 Pressure Pa119901119890
Formation pressure Pa119901119908
Wellbore pressure Pa119901avg Average pressure Pa119901sc Pressure at standard condition Pa119876119863
Volume flow rate calculated by Darcy formula m3s
10 Journal of Chemistry
119876119891
Volume flow rate of fractured well with finiteconductivity m3s
119876TP Volume flow rate of fractured well at turning pointunder different matrix permeability 104m3d
119876V Volume flow rate of fractured well with infiniteconductivity m3s
119902V Volume flow rate m3s119877 Universal gas constant 8314 JKmol119877119890
Radius of boundary m119903 Pore radius119879 Temperature at formation condition K119879sc Temperature at standard condition K119906 Coordinate in119882 plane1199060
Distance between boundary and line sink in119882 planeV Coordinate in119882 planeV119892
Gas flow rate msV119906
Flow rate in 119906 direction of119882 planeVV Flow rate in V direction of119882 plane119882 119882 plane119882119891
Fracture width119909 Coordinate in 119885 plane119884 Coordinate in 119885 plane119885 119885 plane119911 Gas compressibility factor
Greek
120572 Rarefaction coefficient120585 Permeability correction factor120582 Gas molecule mean free path m120583 Gas viscosity Pasdots
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportof the National Program on Key Basic Research Project (973Program) (Grant no 2013CB228002) through the effectivedevelopment research of the Southern Marine Shale GasReservoirs in China
References
[1] M Elgmati Shale Gas Rock Characterization and 3D SubmicronPore Network Reconstruction Missouri University of Scienceand Technology Rolla Mo USA 2011
[2] R G Loucks R M Reed S C Ruppel and D M JarvieldquoMorphology genesis and distribution of nanometer-scalepores in siliceousmudstones of themississippian barnett shalerdquoJournal of Sedimentary Research vol 79 no 12 pp 848ndash8612009
[3] C Zou R Zhu B Bai et al ldquoFirst discovery of nano-pore throatin oil and gas reservoir in China and its scientific valuerdquo ActaPetrologica Sinica vol 27 no 6 pp 1857ndash1864 2011
[4] J DengW Zhu and QMa ldquoA new seepage model for shale gasreservoir and productivity analysis of fractured wellrdquo Fuel vol124 pp 232ndash240 2014
[5] F Javadpour D Fisher and M Unsworth ldquoNanoscale gasflow in shale gas sedimentsrdquo Journal of Canadian PetroleumTechnology vol 46 no 10 pp 55ndash61 2007
[6] F P Wang and R M Reed ldquoPore networks and fluid flow in gasshalesrdquo in Proceedings of the SPE Annual Technical Conferenceand Exhibition (ATCE rsquo09) SPE 124253 pp 1550ndash1557 NewOrleans La USA October 2009
[7] T Huang X Guo and F Chen ldquoModeling transient flowbehavior of a multiscale triple porosity model for shale gasreservoirsrdquo Journal of Natural Gas Science and Engineering vol23 pp 33ndash46 2015
[8] T Huang X Guo and F Chen ldquoModeling transient pressurebehavior of a fractured well for shale gas reservoirs based onthe properties of nanoporesrdquo Journal of Natural Gas Science andEngineering vol 23 pp 387ndash398 2015
[9] G G Michel R F Sigal F Civan and D Devegowda ldquoPara-metric investigation of shale gas production considering nano-scale pore size distribution formation factor and non-darcyflow mechanismsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE 147438 Denver Colo USAOctober-November 2011
[10] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Transactions ofthe ASME Journal of Fluids Engineering vol 118 no 3 pp 448ndash456 1996
[11] A Beskok and G E Karniadakis ldquoA model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[12] S Roy R Raju H F Chuang B A Cruden andMMeyyappanldquoModeling gas flow through microchannels and nanoporesrdquoJournal of Applied Physics vol 93 no 8 pp 4870ndash4879 2003
[13] F Civan ldquoEffective correlation of apparent gas permeability intight porous mediardquo Transport in Porous Media vol 82 no 2pp 375ndash384 2010
[14] E A Guggenheim Elements of the Kinetic Theory of GasesPergamon Press Oxford UK 1960
[15] M Knudsen ldquoThe law of the molecular flow and viscosity ofgases moving through tubesrdquo Annals of Physics vol 28 no 1pp 75ndash130 1909
[16] G P Brown A Dinardo G K Cheng and T K SherwoodldquoThe flow of gases in pipes at low pressuresrdquo Journal of AppliedPhysics vol 17 no 10 pp 802ndash813 1946
[17] G J I Igwe ldquoGas transport mechanism and slippage phe-nomenon in porous mediardquo Tech Rep SPE-16479-MS Societyof Petroleum Engineers 1987
[18] F Javadpour ldquoNanopores and apparent permeability of gasflow in mudrocks (shales and siltstone)rdquo Journal of CanadianPetroleum Technology vol 48 no 8 pp 16ndash21 2009
[19] T JiangW Shan and Y Yang ldquoCalculation of stable productioncapability of vertically fractured wellrdquo Petroleum Explorationand Development vol 28 no 2 p 61 2001
[20] Y Wang T Jiang and B Zeng ldquoProductivity performances ofhydraulically fractured gas wellrdquoActa Petrolei Sinica vol 24 no4 pp 65ndash68 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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CatalystsJournal of
2 Journal of Chemistry
Table 1 Classification of gas flow regimes based on Knudsen number
Knudsen number Kn le 0001 0001 lt Kn le 01 01 lt Kn le 10 Kn gt 10
Flow regime Continuum flow Slip flow Transition flow Free-molecule flow
reservoirs but actually slippage factor is changing withpressure pore radius and other parameters This paperconsiders the effects of slippage and diffusion based onBeskok-Karniadakis (BK) equation and slippage factor anddiffusion coefficient are varying with pressure and shalepermeability in the process of simulation A new non-Darcyequation of motion is obtained based on the analysis of theeffects of high order terms of BK equation on permeabilitycorrection factor By employing conformal transformationprinciple and pressure coupling method a second orderdifferential equation satisfying variable mass flowing in frac-tures is derived and productivity formula of fractured well(infinite and limited conductivity) for shale gas reservoir isestablished The simulation results are compared with fielddata and effects of slippage and diffusion on productivity offractured well have been analyzed thoroughly as well as otherparameters
2 Flow Regimes in Shale Gas Reservoirs
Knudsen number Kn is defined as the ratio of the fluid meanfree path 120582 and pore throat diameter 119903 which is a widelyrecognized dimensionless parameter to determine the degreeof appropriateness of continuum model
Kn = 120582
119903
(1)
Gas flow regimes can be classified into four categories [12]depending on the Knudsen number (Table 1) (1) continuumflow (2) slip flow (3) transition flow (4) and free-moleculeflow In continuum flow regime no-slip boundary conditionis valid and gas flow is linear As Kn increases the rarefac-tion effects become more pronounced and the continuumassumption breaks down eventually So in flow regimes otherthan continuum flow Darcy law is not applicable
To analyze the flow regimes in shale gas reservoirsFigure 1 presents the Knudsen number under different poresizes between 1 nm and 100 120583mand different pressure rangingfrom atmosphere to 100MPa where gas flow from formationto surface undergoes Knudsen number increases whenpressure drops and pore throat diameter decreases Thehorizontal plane in Figure 1 represents that Kn equals 0001Gas flow regime in shales is continuum flow when Knudsennumber is beneath the horizontal plane but it changes intoother flow regimes when Knudsen number is above thehorizontal plane
For example when formation pressure is 100MPa gasflow in shale gas reservoirs is slip flow when pore diameteris between 1 nm and 63 nm and is transition flow when porediameter is smaller than 1 nm but when formation pressuredrops to 10MPa gas flow in shale gas reservoirs is slip flowwhen pore diameter is between 10 nm and 631 nm and istransition flowwhen pore diameter is smaller than 10 nmGas
Pore throat diameter (nm)Pressure (kPa)
Knud
sen
num
ber
Continuum flow
Slip flow
Transition flowFree-molecule flow104
102
102103
104105105
102
100
10minus2
10minus4
10minus6
10minus1
Figure 1 The relationship between Knudsen number pressure andpore radius
flow in shale gas reservoirs is free-molecule flow when porediameter is smaller than 63 nm and pressure is below 8MPaSo we can conclude that gas flow in shale gas reservoirs is amultiscale flow process which includes continuum flow slipflow transition flow and free-molecule flow
3 Multiscale Non-Darcy Seepage Model inShale Matrix
Volume flow rate of gas flow in reservoir can be representedby Darcy Law
V119892
= minus
119870
120583
119889119901
119889119909
(2)
According to the Beskok-Karniadakis equation whichaccounts for different flow regimes of the continuumflow slipflow transition flow and free-molecule flow the relationshipbetween flow rate and pressure gradient can be described as
V119892
= minus
1198700
120583
(1 + 120572Kn) (1 + 4Kn1 minus 119887Kn
)
119889119901
119889119909
(3)
Combining (2) and (3) permeability correction factor canbe written as
119870 = 1198700
120585
120585 = (1 + 120572Kn) (1 + 4Kn1 minus 119887Kn
)
(4)
We can see from (4) that the apparent permeabilityis close to the value of absolute permeability when Knapproaches 0 When Knudsen number gets bigger which
Journal of Chemistry 3
2E + 0
1E + 01E minus 3 1E minus 2 1E minus 1
Perm
eabi
lity
adju
stmen
t fac
tor
Knudsen number
120585 = 1 + 4Kn120585 = 1 + 4Kn + 512151205872Kn14
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2 + 2048151205872Kn24
(a) Slippage factor 119887 = 01
2E + 0
1E + 0Perm
eabi
lity
adju
stmen
t fac
tor
1E minus 3 1E minus 2 1E minus 1
Knudsen number
120585 = 1 + 4Kn120585 = 1 + 4Kn + 512151205872Kn14
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2 + 2048151205872Kn24
(b) Slippage factor 119887 = 5
Figure 2 Relationship between permeability correction factor and Knudsen number
means that the flow in microtube is no longer Darcy flow theapparent permeability should be corrected
The rarefaction coefficient is defined by Beskok andKarniadakis (1999) as
120572 =
128
151205872
tanminus1 (4Kn04) (5)
To simplify Beskok-Karniadakis equation [4 9 13] per-meability correction factor can be rewritten as
120585 = 1 + 120572Kn + 4Kn1 minus 119887Kn
+
4120572Kn2
1 minus 119887Kn (6)
The first two terms of (6) represent the first order termto no-slip flow in Beskok-Karniadakis equation When Kn lt01 the following approximations are valid [4 9]
120572 asymp
128
151205872
[4Kn04 minus 1
3
(4Kn04)3
] (7)
Kn1 minus 119887Kn
asymp Kn (1 + 119887Kn + 1198872Kn2) (8)
When Kn gt 01 the value of rarefaction coefficient cal-culated by (7) is negative which is very different fromrarefaction coefficient calculated by (5) so we use anotherapproximation instead
120572 asymp
128
151205872
(4Kn04) (9)
By combining (8) and (9) the permeability correctionfactor can be rewritten as
120585 = 1 + 4Kn + 512
151205872
Kn14 + 4119887Kn2 + 2048
151205872
Kn24
+ 41198872Kn3 + 119900 (Kn3)
(10)
Here the influence of high order terms of (10) on per-meability correction factor will be discussed In Figures 2(a)and 2(b) the term of Kn24 has little influence on permeabilitycorrection factor whether the value of slippage factor 119887 is bigor small the effect of term Kn2 on permeability correctionfactor can be neglected when the value of slippage factor issmall but this effect becomes greater when slippage factorgets bigger For slip flow and continuum flow the high order(gt2) correction term of (10) can be neglected and (3) can bederived as
V119892
= minus
1198700
120583
(1 + 4Kn + 512
151205872
Kn14 + 4119887Kn2)119889119901
119889119909
(11)
The definition of mean path is given as [14]
120582 = radic120587119911119877119879
2119872
120583
119901
(12)
Knudsen diffusion coefficient is mainly changing withpore diameter which is defined as [5 12 15]
119863K =2119903
3
(
8119877119879
120587119872
)
05
(13)
Slippage factor 119887 is changing with pressure and porediameter which is given by [16ndash18]
119887 = (
8120587119877119879
119872
)
05
120583
119901avg119903(
2
119891
minus 1) (14)
Knudsen number can be represented by Knudsen diffu-sion coefficient according to (12) and (13)
Kn = 3120587
81199032
120583
119901
119863K (15)
4 Journal of Chemistry
Taking (15) into (11) we can get
V119892
= minus
1198700
120583
[1 + 4
3120587
81199032
120583
119901
119863K +512
151205872
(
3120587
81199032
120583
119901
119863K)14
+ 4119887(
3120587
81199032
120583
119901
119863K)2
]
119889119901
119889119909
(16)
According to Hagen-Poiseuille model relationshipbetween permeability and pore diameter can be written as
1198700
=
1199032
8
(17)
Taking (17) into (16) we can derive
V119892
= minus
1198700
120583
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119889119901
119889119909
(18)
Considering both sides of (18) multiplied by seepage areaand divided by volume factor we can get the volume flow rateat standard condition
119902V =V119860119861119892
=
1198601198700
119861119892
120583119892
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119889119901
119889119909
(19)
where the gas volume factor 119861119892
can be derived according tothe equation of state
119861119892
=
119899119911119877119879119901
119899119877119879sc119901sc=
119901sc119901
119911119879
119879sc (20)
4 Productivity Model of Fractured Well
The natural productivity of shale gas reservoirs is very lowso formation has to be artificially stimulated (hydraulicallyfractured) to render commercial production So this paperestablishes steady seepage model of fractured well for shalegas reservoirs by conformal transformation According to theprinciple of conformal transformation the gas productionand pressure of the boundary remains the same only thelength of line and flow forms change [19 20]
41 Physical Assumptions To make this mathematical modelmore tractable the following simplified assumptions aremade
(1) The reservoir is isotropic and gas flow in shale gasreservoir is planar (steady state)
(2) Flow in nanopores of shale matrix is assumed to benon-Darcy and fluid in shale gas reservoirs is singlephase
Table 2 Conformal transformations of the fractured well
Location ofpoint 119882 plane 119885 plane
A 119906 = 0 V = 0 119909 = 119871119891
119910 = 0
B 119906 = 0 V = 1205872 119909 = 0 119910 = 0
C 119906 = 0 V = 120587 119909 = minus119871119891
119910 = 0
D 119906 = 1199060
V = 0 119909 = 119871119891
ch1199060
119910 = 0
E 119906 = 1199060
V = 1205872 119909 = 0 119910 = 119871119891
sh1199060
F 119906 = 1199060
V = 120587 119909 = minus119871119891
ch1199060
119910 = 0
(3) Fractured vertical well is produced at a constantwellbore pressure and is penetrated fully in shale gasreservoir The height of hydraulic fractures is equal tothe thickness of formation
(4) The fractures are vertical and symmetrically locatedat two sides of wellbore Fracture half-length is 119871
119891
fracture width is negligible when the vertical fracturesare assumed to have infinite conductivity and is 119882
119891
when vertical fractures are assumed to have finiteconductivity
42 Fractured Well with Infinite Conductivity Conformaltransformation function is defined as
119885 = 119871119891
ch119882 (21)
Taking 119885 = 119909 + 119894119910119882 = 119906 + 119894V into (21) we can get
119909 = 119871119891
ch119906 cos V
119910 = 119871119891
sh119906 sin V(22)
We can see from Table 2 that the upper half planeformation in 119885 plane (Figure 3(a)) is transformed to a halfinfinite formationwithwidth of120587 at the right side of fracturedwell in 119882 plane (Figure 3(b)) elliptical constant pressureboundarywith radius of119877
119890
is transformed to a linear onewithlength of 120587 fractured well with length of 2119871
119891
is transformedto a line sink in119882 plane
Equipotential line equation of gas flow in 119885 plane can beobtained through (22)
1199092
1198712
119891
ch2119906+
1199102
1198712
119891
sh2119906= cos2V + sin2V = 1 (23)
The boundary can be viewed as circular when the bound-ary is far from fractured well so
ch1199060
asymp sh1199060
asymp
1
2
1198901199060 (24)
Then the equipotential line equation of boundary can berewritten as
1199092
+ 1199102
= 1198712
119891
(
1
2
1198901199060)
2
= 1198772
119890
(25)
Journal of Chemistry 5
y
xA BO
v
Hydraulic fracture
Wellbore C
6 3
3
22
1 4
5
1
y
Z = LfchW5
4
6
Q
2
O998400 u0 u
(a) Z plane (b) W plane
Figure 3 Sketch map of conformal transformation to fractured well
Solving (25) we can get the relationship between thelength of drainage area 119906
0
in 119882 plane and the radius ofboundary 119877
119890
in 119885 plane
1199060
= ln2119877119890
119871119891
(26)
Combined with (19) production rate of drainage area in119882 plane can be derived
119876V =21205871198700
ℎ119879sc119901sc1198791205831199111199060
[
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(27)
Gas production of fractured well in 119885 plane can beobtained according to (26)
119876V =21205871198700
ℎ119879sc
119901sc119879120583119911 ln (2119877119890119871119891)
sdot [
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(28)
43 Fractured Well with Finite Conductivity Compared withhydraulic fracture with infinite conductivity fractured ver-tical well with finite conductivity is more close to realcondition and the effects of parameters of hydraulic fractureson well performance can be analyzed as well
Considering an element of hydraulic fracture in119882 planein Figure 3(b) we assume that gas only flows in V direction(flows to wellbore) in hydraulic fracture (Figure 4) So flow
u
d
u
Wf
Figure 4 Sketch map of hydraulic fracture element
rate in fracture VV is a constant value in 119906 direction and avariable value in V direction flow rate from reservoir to theelement V
119906
equals a constant valueThe mass balance equation of gas flow in hydraulic frac-
ture can be obtained by principle of mass conservation
minus (VV1003816100381610038161003816V+ΔV minus VV
1003816100381610038161003816V) sdot
1
2
119882119891
ℎ + V119906
sdot ΔVℎ = 0 (29)
Considering both sides of (29) divided by ΔV the partialdifference equation can be derivedwhenΔV takes an infinites-imal
119889VV119889V
sdot
1
2
119882119891
= V119906
(30)
Pseudopressure function of gas flow in hydraulic fractureis defined as
119898(119901) = 2int
119901
119901119890
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119901
120583119911
119889119901
(31)
Equation (30) can be
1205972
119898
120597V2minus
1198700
(12) 119896119891
119882119891
1
ln (2119877119890
119871119891
)
119898
= minus
1198700
(12) 119896119891
119882119891
1
ln (2119877119890
119871119891
)
119898119890
(32)
6 Journal of Chemistry
The boundary conditions of hydraulic fracture are givenas follows
119889119898
119889V= 0 V = 0
119898 = 119898119908
V =120587
2
(33)
Combining with (33) we can get the solution of (32)
119898(119901) = 1198881
119890120582V+ 1198882
119890minus120582V
+ 119898119890
(34)
where the parameters 1198881
1198882
and 120582 are defined as follows
120582 = radic
21198700
119896119891
119882119891
1
ln 2119877119890
119871119891
1198881
= 1198882
=
119898119908
minus 119898119890
119890(1205872)120582
+ 119890minus(1205872)120582
(35)
The gas production of fractured vertical well can be writ-ten as
119876119891
= minus
119896119891
119882119891
ℎ119879sc
119901sc119879
119889119898
119889V
10038161003816100381610038161003816100381610038161003816V=1205872
=
119896119891
119882119891
ℎ119879sc
119901sc119879120582 (119898119890
minus 119898119908
)
119890(1205872)120582
minus 119890minus(1205872)120582
119890(1205872)120582
+ 119890minus(1205872)120582
(36)
Combined with (31) we can get the productivity formulaof fractured vertical well with finite conductivity
119876119891
=
2119896119891
119882119891
ℎ120582119879sc
119901sc119879120583119911tanh 120587120582
2
sdot [
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(37)
where the first term in the bracket is the gas productioncalculated by Darcy formula which is defined as 119876
119863
5 Results and Discussion
According to the productivity formula of fractured well withfinite conductivity the effects of permeability correctionfactor slippage factor Knudsen diffusion coefficient matrixpermeability fracture half-length and fracture conductivityon productivity of fractured well are analyzed by combiningthe data from a single well of shale gas reservoirs in ChinaThe data for production simulation are presented in Table 3
51 Effect of Permeability Correction Factor on ProductivityFigure 5 presents the effect of permeability correction factoron productivity of fractured well and we can see fromthis figure that permeability correction factor has a great
Table 3 Data of shale gas reservoir for production simulation
Parameter Value UnitsPermeability 119870 0005 mDPorosity 120601 007 mdashFormation temperature 119879 36615 KSurface temperature 119879sc 293 KFormation pressure 119901
119890
28 MPaFormation thickness ℎ 305 mPressure relief radius 119877
119890
400 mGas viscosity 120583 0014 mPasdotsGas compressibility factor 119911 089 mdashRadius of wellbore 119903
119908
01 m
120585 = 1 + 4Kn + 512151205872Kn14
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
Field data
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
120585 = 1 (calculated by Darcy formula)120585 = 1 + 4Kn
Figure 5 Effect of permeability correction factor 120585 on productivity
influence on productivity Production of fractured well isvery small when our model only considers Darcy flow innanomicroscale pores (120585 = 1) but gas production increasesa lot when matrix apparent permeability is corrected bypermeability correction factor 120585 Field data shows that gasproduction of a fractured vertical well is 12 times 104m3 whenproduction pressure drop keeps 7MPa and fracture half-length is 200m [4] The calculation results of multiscalenon-Darcy seepage model are much more close to thefield data compared with the production rate calculated byDarcy formula andproduction rate calculated by productivityformula only considering first order term of permeabilitycorrection factor
52 Effects of Formation and Fracture Parameters on Produc-tivity The effect of matrix permeability 119870 on productivityof fractured well is illustrated in Figure 6 Gas productionof fractured well increases with the matrix permeabilityand gas production increases to 171 times 232 times and287 times when matrix permeability increases to 2 times 3times and 4 times respectively Figure 7 reflects productivity
Journal of Chemistry 7
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
K = 0005mDK = 0010mD
K = 0015mDK = 0020mD
Gas production rate (times104 m3d)
Figure 6 Effect of matrix permeability 119870 on productivity
0
5
10
15
20
25
30
0 05 1 15 2 25 3
Well
bore
pre
ssur
e (M
Pa)
Gas production rate (times104 m3d)
Lf = 50mLf = 150m
Lf = 250mLf = 350m
Figure 7 Effect of fracture half-length 119871119891
on productivity
of fractured well affected by fracture half-length Gas pro-duction increases to 154 times 206 times and 265 timeswhen fracture half-length increases to 3 times 4 times and5 times respectively We can find that the increase of gasproduction rate affected by matrix permeability is still morethan production rate affected by fracture half-length even ifthe increase ratio of fracture half-length is bigger thanmatrixpermeability which means that longer fracture half-lengthonly can improve local seepage ability of formation adjacentto hydraulic fractures but improvement of flow capacityin the whole formation can increase the gas productionsubstantially
Effect of fracture conductivity on productivity of frac-tured well has been analyzed under different fracture pene-tration ratio (119871
119891
119877119890
) We can see from Figure 8 that gas pro-duction increases with the fracture conductivity and fracturepenetration ratio But gas production increase becomes slowwhen the fracture conductivity increases to a certain valueand this value becomes bigger when fracture penetrationratio increases For examples this value will be 2 120583m2sdotcmwhen fracture penetration ratio is 01 but this value willincrease to 4120583m2sdotcm when fracture penetration ratio is 03
0
05
1
15
2
25
3
0 1 2 3 4 5 6
Gas
pro
duct
ion
rate
(times10
4m
3d
)
LfRe = 01
LfRe = 03LfRe = 05LfRe = 07
kf middot Wf (120583m2middotcm)
Figure 8 Effect of fracture conductivity 119896119891
sdot 119882119891
on productivity
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 1 2 3 4 5
Gas production rate (times104 m3d)
b = 0b = 1
b = 2
b = 3
Figure 9 Effect of slippage factor 119887 on productivity
This chart can help to optimize fracture conductivity underdifferent fracture penetration ratio
53 Effects of Slippage and Knudsen Diffusion on ProductivityThe relationship between wellbore pressure of fractured wellwith finite conductivity and gas production under differentslippage factor 119887 has been shown in Figure 9 Slippage effecthas negligible influence on productivity of fractured wellwhen wellbore pressure is high but slippage effect begins toaffect gas production when wellbore pressure is lower than15MPa and productivity of the fractured well increases withthe slippage factor So decreasing the wellbore pressure offractured well properly can improve gas production whenexploiting shale gas reservoirs We can see from Figure 10that gas production rate increases with Knudsen diffusioncoefficient which has a greater impact on gas production thanslippage effect
As we can see in Figure 11 the productivity of fracturedwell derived in this paper (119876
119891
) has been compared withproduction calculated by Darcy formula (119876
119863
) under differ-ent matrix permeability and wellbore pressure Figure 11(a)shows that bigger matrix permeability and a lower wellbore
8 Journal of Chemistry
Table 4 Comparison of production at turning point and production at minimum permeability under different wellbore pressure (the gasproduction rate unit times104 m3d)
119875119908
MPa 1 3 5 7 9 11 13 15 17 19 21 23 25 27119876119870min 0505 0333 0250 0197 0158 0127 0103 0082 0065 0050 0037 0025 0014 0005
119876TP 0315 0238 0193 0160 0134 0112 0093 0076 0061 0048 0035 0024 0014 0004119876119870min119876TP 1601 1398 1296 1228 1178 1138 1109 1082 1064 1046 1034 1024 1014 1010
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
DK = 1 times 10minus6 m2sDK = 3 times 10minus6 m2s
DK = 5 times 10minus6 m2sDK = 7 times 10minus6 m2s
Figure 10 Effect of Knudsen diffusion coefficient119863K on productiv-ity
pressure lead to a higher productivity of fractured well andgas production 119876
119891
is much bigger than Darcy productionrate 119876
119863
The effects of Knudsen diffusion and slip flow onproductivity of fractured well can be observed by taking theratio of119876
119891
and119876119863
in Figure 11(b)The effects on gas produc-tion increase whenmatrix permeability becomes smaller andwellbore pressure declines which become most significantunder the smallest matrix permeability (1 times 10minus5mD) and thelowest wellbore pressure (1MPa) in Figure 11(b)
Figure 12 represents the relationship of gas productionrate119876
119891
and matrix permeability119870 at different wellbore pres-sure For conventional reservoirs production rate decreaseswith formation permeability for shale gas reservoirs gasproduction rate decreases with matrix permeability first butthen increases with permeability owing to the effects ofKnudsen diffusion and slip flow after a certain permeabilityvalue which is called turning point in this paperThe value ofturning point decreases when wellbore pressure rises whichis because Knudsen diffusion and slippage effect are morelikely to occur in formation with smaller permeability andlower pressure so a smaller turning point would be requiredwhen pressure rises
Table 4 shows the ratio of gas production rate at min-imum permeability 119876
119870min in Figure 12 and production atturning point under different wellbore pressures 119876TP Theratios decrease when wellbore pressure increases whichmeans that Knudsen diffusion and slippage effects have agreater influence on productivity of fractured well under alower wellbore pressure But the gas production rate119876
119891
(119870 =1 times 10minus5mD 119875
119908
= 1MPa) is only 16 times the production
119876119891
at turning point (Table 4) even though the production119876119891
is almost 150 times the Darcy production rate 119876119863
in Figure 11(b) Knudsen diffusion and slippage effects canimprove the productivity at extremely low permeabilitysignificantly compared with gas production calculated byDarcy formula (without considering effects of diffusion andslip flow) but the improvement of productivity is still smallcompared with higher permeability when Knudsen diffusionand slippage effects are considered in both situations
6 Conclusions
In this paper a multiscale comprehensive mathematicalmodel had been established which can simulate differentflow regimes including continuum flow slip flow transitionflow and free-molecule flow We deduced the productivityformula for fractured well with limited conductivity for shalegas reservoirs and the influencing parameters were analyzedthoroughly The following conclusions can be drawn
(1) Flow regimes in shale gas reservoir were analyzedby calculating Knudsen number under different pres-sure and different pore throat diameter Gas flow inshale gas reservoirs with nanomicroscale pores is acombination of continuum flow slip flow transitionflow and free-molecule flow which cannot simply berepresented by Darcy formula anymore
(2) A new non-Darcy equation was developed based onBeskok-Karniadakis equation which is applicable forall flow regimes and effects of high order terms ofBK equation on permeability correction factor wereanalyzed under different Knudsen number Perme-ability correction factor is increasingly deviating froma Darcy type as Knudsen number increases
(3) Productivity formula of fractured well satisfying vari-able mass flowing in fractures was obtained andthe simulated results showed that production ratecalculated by non-Darcy seepage model with higherorder terms of permeability correction factor is moreclose to the field production data compared withthe production rate calculated by Darcy formula andproduction rate calculated by productivity formulaonly considering first order term of permeabilitycorrection factor
(4) Effects of slippage and diffusion which are changingwith pressure and pore sizes in nanoscale pores wereconsidered in simulation Simulation results showedthat slippage effect affects gas production of fracturedwell only when wellbore pressure less than 15MPa
Journal of Chemistry 9
010
20300
05
1
15
2
Matrix permeability (mD)Wellbore pressure (MPa)
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
QD
Qf
times104
Prod
uctio
n (m
3d
)
(a) Comparison between 119876119863
and 119876119891
(at different permeability andwellbore pressure)
010
20300
50
100
150
Matrix permeability (mD)Wellbore pressure (MPa)
Prod
uctio
n co
mpa
rison
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
(b) The value of 119876119891
119876
119863
at different matrix permeability and wellborepressure
Figure 11 Effects of Knudsen diffusion coefficient119863K and slip factor 119887 on productivity
0
04
08
12
16
2
Turning point
Effects of slip flow andKnudsen diffusion
Matrix permeability (mD)1E minus 5 1E minus 4 1E minus 3 1E minus 2
Pw = 1Pw = 3Pw = 5Pw = 7Pw = 9Pw = 11
(MPa
)
Pw = 13
Pw = 15
Pw = 17
Pw = 19
Pw = 21
Pw = 23
Pw = 25
Pw = 27
Gas
pro
duct
ion
rate
(times10
4m
3d
)
Figure 12 The relationship between gas production and matrix permeability at different wellbore pressure
and the effects of slippage and diffusion have a greaterinfluence on gas production of reservoirs with smallermatrix permeability especially when permeability isat nano-Darcy scale Matrix permeability fracturehalf length fracture penetration ratio and fractureconductivity can improve gas production at differentdegrees as well
Nomenclature
Latin
119887 Slippage coefficient119861119892
Gas volume factor119863K Knudsen diffusion coefficient m2s119891 Fraction of molecules striking pore wall which are
diffusely reflected
ℎ Thickness of shale gas reservoir m119870 Apparent permeabilityKn Knudsen number119896119891
Permeability of hydraulic fracture1198700
Absolute permeability119871119891
Fracture half length119872 Molecular mass kgmol119898 Pseudopressure119898119890
Initial pseudopressure119898119908
Pseudopressure of fractured well119901 Pressure Pa119901119890
Formation pressure Pa119901119908
Wellbore pressure Pa119901avg Average pressure Pa119901sc Pressure at standard condition Pa119876119863
Volume flow rate calculated by Darcy formula m3s
10 Journal of Chemistry
119876119891
Volume flow rate of fractured well with finiteconductivity m3s
119876TP Volume flow rate of fractured well at turning pointunder different matrix permeability 104m3d
119876V Volume flow rate of fractured well with infiniteconductivity m3s
119902V Volume flow rate m3s119877 Universal gas constant 8314 JKmol119877119890
Radius of boundary m119903 Pore radius119879 Temperature at formation condition K119879sc Temperature at standard condition K119906 Coordinate in119882 plane1199060
Distance between boundary and line sink in119882 planeV Coordinate in119882 planeV119892
Gas flow rate msV119906
Flow rate in 119906 direction of119882 planeVV Flow rate in V direction of119882 plane119882 119882 plane119882119891
Fracture width119909 Coordinate in 119885 plane119884 Coordinate in 119885 plane119885 119885 plane119911 Gas compressibility factor
Greek
120572 Rarefaction coefficient120585 Permeability correction factor120582 Gas molecule mean free path m120583 Gas viscosity Pasdots
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportof the National Program on Key Basic Research Project (973Program) (Grant no 2013CB228002) through the effectivedevelopment research of the Southern Marine Shale GasReservoirs in China
References
[1] M Elgmati Shale Gas Rock Characterization and 3D SubmicronPore Network Reconstruction Missouri University of Scienceand Technology Rolla Mo USA 2011
[2] R G Loucks R M Reed S C Ruppel and D M JarvieldquoMorphology genesis and distribution of nanometer-scalepores in siliceousmudstones of themississippian barnett shalerdquoJournal of Sedimentary Research vol 79 no 12 pp 848ndash8612009
[3] C Zou R Zhu B Bai et al ldquoFirst discovery of nano-pore throatin oil and gas reservoir in China and its scientific valuerdquo ActaPetrologica Sinica vol 27 no 6 pp 1857ndash1864 2011
[4] J DengW Zhu and QMa ldquoA new seepage model for shale gasreservoir and productivity analysis of fractured wellrdquo Fuel vol124 pp 232ndash240 2014
[5] F Javadpour D Fisher and M Unsworth ldquoNanoscale gasflow in shale gas sedimentsrdquo Journal of Canadian PetroleumTechnology vol 46 no 10 pp 55ndash61 2007
[6] F P Wang and R M Reed ldquoPore networks and fluid flow in gasshalesrdquo in Proceedings of the SPE Annual Technical Conferenceand Exhibition (ATCE rsquo09) SPE 124253 pp 1550ndash1557 NewOrleans La USA October 2009
[7] T Huang X Guo and F Chen ldquoModeling transient flowbehavior of a multiscale triple porosity model for shale gasreservoirsrdquo Journal of Natural Gas Science and Engineering vol23 pp 33ndash46 2015
[8] T Huang X Guo and F Chen ldquoModeling transient pressurebehavior of a fractured well for shale gas reservoirs based onthe properties of nanoporesrdquo Journal of Natural Gas Science andEngineering vol 23 pp 387ndash398 2015
[9] G G Michel R F Sigal F Civan and D Devegowda ldquoPara-metric investigation of shale gas production considering nano-scale pore size distribution formation factor and non-darcyflow mechanismsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE 147438 Denver Colo USAOctober-November 2011
[10] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Transactions ofthe ASME Journal of Fluids Engineering vol 118 no 3 pp 448ndash456 1996
[11] A Beskok and G E Karniadakis ldquoA model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[12] S Roy R Raju H F Chuang B A Cruden andMMeyyappanldquoModeling gas flow through microchannels and nanoporesrdquoJournal of Applied Physics vol 93 no 8 pp 4870ndash4879 2003
[13] F Civan ldquoEffective correlation of apparent gas permeability intight porous mediardquo Transport in Porous Media vol 82 no 2pp 375ndash384 2010
[14] E A Guggenheim Elements of the Kinetic Theory of GasesPergamon Press Oxford UK 1960
[15] M Knudsen ldquoThe law of the molecular flow and viscosity ofgases moving through tubesrdquo Annals of Physics vol 28 no 1pp 75ndash130 1909
[16] G P Brown A Dinardo G K Cheng and T K SherwoodldquoThe flow of gases in pipes at low pressuresrdquo Journal of AppliedPhysics vol 17 no 10 pp 802ndash813 1946
[17] G J I Igwe ldquoGas transport mechanism and slippage phe-nomenon in porous mediardquo Tech Rep SPE-16479-MS Societyof Petroleum Engineers 1987
[18] F Javadpour ldquoNanopores and apparent permeability of gasflow in mudrocks (shales and siltstone)rdquo Journal of CanadianPetroleum Technology vol 48 no 8 pp 16ndash21 2009
[19] T JiangW Shan and Y Yang ldquoCalculation of stable productioncapability of vertically fractured wellrdquo Petroleum Explorationand Development vol 28 no 2 p 61 2001
[20] Y Wang T Jiang and B Zeng ldquoProductivity performances ofhydraulically fractured gas wellrdquoActa Petrolei Sinica vol 24 no4 pp 65ndash68 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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CatalystsJournal of
Journal of Chemistry 3
2E + 0
1E + 01E minus 3 1E minus 2 1E minus 1
Perm
eabi
lity
adju
stmen
t fac
tor
Knudsen number
120585 = 1 + 4Kn120585 = 1 + 4Kn + 512151205872Kn14
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2 + 2048151205872Kn24
(a) Slippage factor 119887 = 01
2E + 0
1E + 0Perm
eabi
lity
adju
stmen
t fac
tor
1E minus 3 1E minus 2 1E minus 1
Knudsen number
120585 = 1 + 4Kn120585 = 1 + 4Kn + 512151205872Kn14
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2 + 2048151205872Kn24
(b) Slippage factor 119887 = 5
Figure 2 Relationship between permeability correction factor and Knudsen number
means that the flow in microtube is no longer Darcy flow theapparent permeability should be corrected
The rarefaction coefficient is defined by Beskok andKarniadakis (1999) as
120572 =
128
151205872
tanminus1 (4Kn04) (5)
To simplify Beskok-Karniadakis equation [4 9 13] per-meability correction factor can be rewritten as
120585 = 1 + 120572Kn + 4Kn1 minus 119887Kn
+
4120572Kn2
1 minus 119887Kn (6)
The first two terms of (6) represent the first order termto no-slip flow in Beskok-Karniadakis equation When Kn lt01 the following approximations are valid [4 9]
120572 asymp
128
151205872
[4Kn04 minus 1
3
(4Kn04)3
] (7)
Kn1 minus 119887Kn
asymp Kn (1 + 119887Kn + 1198872Kn2) (8)
When Kn gt 01 the value of rarefaction coefficient cal-culated by (7) is negative which is very different fromrarefaction coefficient calculated by (5) so we use anotherapproximation instead
120572 asymp
128
151205872
(4Kn04) (9)
By combining (8) and (9) the permeability correctionfactor can be rewritten as
120585 = 1 + 4Kn + 512
151205872
Kn14 + 4119887Kn2 + 2048
151205872
Kn24
+ 41198872Kn3 + 119900 (Kn3)
(10)
Here the influence of high order terms of (10) on per-meability correction factor will be discussed In Figures 2(a)and 2(b) the term of Kn24 has little influence on permeabilitycorrection factor whether the value of slippage factor 119887 is bigor small the effect of term Kn2 on permeability correctionfactor can be neglected when the value of slippage factor issmall but this effect becomes greater when slippage factorgets bigger For slip flow and continuum flow the high order(gt2) correction term of (10) can be neglected and (3) can bederived as
V119892
= minus
1198700
120583
(1 + 4Kn + 512
151205872
Kn14 + 4119887Kn2)119889119901
119889119909
(11)
The definition of mean path is given as [14]
120582 = radic120587119911119877119879
2119872
120583
119901
(12)
Knudsen diffusion coefficient is mainly changing withpore diameter which is defined as [5 12 15]
119863K =2119903
3
(
8119877119879
120587119872
)
05
(13)
Slippage factor 119887 is changing with pressure and porediameter which is given by [16ndash18]
119887 = (
8120587119877119879
119872
)
05
120583
119901avg119903(
2
119891
minus 1) (14)
Knudsen number can be represented by Knudsen diffu-sion coefficient according to (12) and (13)
Kn = 3120587
81199032
120583
119901
119863K (15)
4 Journal of Chemistry
Taking (15) into (11) we can get
V119892
= minus
1198700
120583
[1 + 4
3120587
81199032
120583
119901
119863K +512
151205872
(
3120587
81199032
120583
119901
119863K)14
+ 4119887(
3120587
81199032
120583
119901
119863K)2
]
119889119901
119889119909
(16)
According to Hagen-Poiseuille model relationshipbetween permeability and pore diameter can be written as
1198700
=
1199032
8
(17)
Taking (17) into (16) we can derive
V119892
= minus
1198700
120583
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119889119901
119889119909
(18)
Considering both sides of (18) multiplied by seepage areaand divided by volume factor we can get the volume flow rateat standard condition
119902V =V119860119861119892
=
1198601198700
119861119892
120583119892
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119889119901
119889119909
(19)
where the gas volume factor 119861119892
can be derived according tothe equation of state
119861119892
=
119899119911119877119879119901
119899119877119879sc119901sc=
119901sc119901
119911119879
119879sc (20)
4 Productivity Model of Fractured Well
The natural productivity of shale gas reservoirs is very lowso formation has to be artificially stimulated (hydraulicallyfractured) to render commercial production So this paperestablishes steady seepage model of fractured well for shalegas reservoirs by conformal transformation According to theprinciple of conformal transformation the gas productionand pressure of the boundary remains the same only thelength of line and flow forms change [19 20]
41 Physical Assumptions To make this mathematical modelmore tractable the following simplified assumptions aremade
(1) The reservoir is isotropic and gas flow in shale gasreservoir is planar (steady state)
(2) Flow in nanopores of shale matrix is assumed to benon-Darcy and fluid in shale gas reservoirs is singlephase
Table 2 Conformal transformations of the fractured well
Location ofpoint 119882 plane 119885 plane
A 119906 = 0 V = 0 119909 = 119871119891
119910 = 0
B 119906 = 0 V = 1205872 119909 = 0 119910 = 0
C 119906 = 0 V = 120587 119909 = minus119871119891
119910 = 0
D 119906 = 1199060
V = 0 119909 = 119871119891
ch1199060
119910 = 0
E 119906 = 1199060
V = 1205872 119909 = 0 119910 = 119871119891
sh1199060
F 119906 = 1199060
V = 120587 119909 = minus119871119891
ch1199060
119910 = 0
(3) Fractured vertical well is produced at a constantwellbore pressure and is penetrated fully in shale gasreservoir The height of hydraulic fractures is equal tothe thickness of formation
(4) The fractures are vertical and symmetrically locatedat two sides of wellbore Fracture half-length is 119871
119891
fracture width is negligible when the vertical fracturesare assumed to have infinite conductivity and is 119882
119891
when vertical fractures are assumed to have finiteconductivity
42 Fractured Well with Infinite Conductivity Conformaltransformation function is defined as
119885 = 119871119891
ch119882 (21)
Taking 119885 = 119909 + 119894119910119882 = 119906 + 119894V into (21) we can get
119909 = 119871119891
ch119906 cos V
119910 = 119871119891
sh119906 sin V(22)
We can see from Table 2 that the upper half planeformation in 119885 plane (Figure 3(a)) is transformed to a halfinfinite formationwithwidth of120587 at the right side of fracturedwell in 119882 plane (Figure 3(b)) elliptical constant pressureboundarywith radius of119877
119890
is transformed to a linear onewithlength of 120587 fractured well with length of 2119871
119891
is transformedto a line sink in119882 plane
Equipotential line equation of gas flow in 119885 plane can beobtained through (22)
1199092
1198712
119891
ch2119906+
1199102
1198712
119891
sh2119906= cos2V + sin2V = 1 (23)
The boundary can be viewed as circular when the bound-ary is far from fractured well so
ch1199060
asymp sh1199060
asymp
1
2
1198901199060 (24)
Then the equipotential line equation of boundary can berewritten as
1199092
+ 1199102
= 1198712
119891
(
1
2
1198901199060)
2
= 1198772
119890
(25)
Journal of Chemistry 5
y
xA BO
v
Hydraulic fracture
Wellbore C
6 3
3
22
1 4
5
1
y
Z = LfchW5
4
6
Q
2
O998400 u0 u
(a) Z plane (b) W plane
Figure 3 Sketch map of conformal transformation to fractured well
Solving (25) we can get the relationship between thelength of drainage area 119906
0
in 119882 plane and the radius ofboundary 119877
119890
in 119885 plane
1199060
= ln2119877119890
119871119891
(26)
Combined with (19) production rate of drainage area in119882 plane can be derived
119876V =21205871198700
ℎ119879sc119901sc1198791205831199111199060
[
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(27)
Gas production of fractured well in 119885 plane can beobtained according to (26)
119876V =21205871198700
ℎ119879sc
119901sc119879120583119911 ln (2119877119890119871119891)
sdot [
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(28)
43 Fractured Well with Finite Conductivity Compared withhydraulic fracture with infinite conductivity fractured ver-tical well with finite conductivity is more close to realcondition and the effects of parameters of hydraulic fractureson well performance can be analyzed as well
Considering an element of hydraulic fracture in119882 planein Figure 3(b) we assume that gas only flows in V direction(flows to wellbore) in hydraulic fracture (Figure 4) So flow
u
d
u
Wf
Figure 4 Sketch map of hydraulic fracture element
rate in fracture VV is a constant value in 119906 direction and avariable value in V direction flow rate from reservoir to theelement V
119906
equals a constant valueThe mass balance equation of gas flow in hydraulic frac-
ture can be obtained by principle of mass conservation
minus (VV1003816100381610038161003816V+ΔV minus VV
1003816100381610038161003816V) sdot
1
2
119882119891
ℎ + V119906
sdot ΔVℎ = 0 (29)
Considering both sides of (29) divided by ΔV the partialdifference equation can be derivedwhenΔV takes an infinites-imal
119889VV119889V
sdot
1
2
119882119891
= V119906
(30)
Pseudopressure function of gas flow in hydraulic fractureis defined as
119898(119901) = 2int
119901
119901119890
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119901
120583119911
119889119901
(31)
Equation (30) can be
1205972
119898
120597V2minus
1198700
(12) 119896119891
119882119891
1
ln (2119877119890
119871119891
)
119898
= minus
1198700
(12) 119896119891
119882119891
1
ln (2119877119890
119871119891
)
119898119890
(32)
6 Journal of Chemistry
The boundary conditions of hydraulic fracture are givenas follows
119889119898
119889V= 0 V = 0
119898 = 119898119908
V =120587
2
(33)
Combining with (33) we can get the solution of (32)
119898(119901) = 1198881
119890120582V+ 1198882
119890minus120582V
+ 119898119890
(34)
where the parameters 1198881
1198882
and 120582 are defined as follows
120582 = radic
21198700
119896119891
119882119891
1
ln 2119877119890
119871119891
1198881
= 1198882
=
119898119908
minus 119898119890
119890(1205872)120582
+ 119890minus(1205872)120582
(35)
The gas production of fractured vertical well can be writ-ten as
119876119891
= minus
119896119891
119882119891
ℎ119879sc
119901sc119879
119889119898
119889V
10038161003816100381610038161003816100381610038161003816V=1205872
=
119896119891
119882119891
ℎ119879sc
119901sc119879120582 (119898119890
minus 119898119908
)
119890(1205872)120582
minus 119890minus(1205872)120582
119890(1205872)120582
+ 119890minus(1205872)120582
(36)
Combined with (31) we can get the productivity formulaof fractured vertical well with finite conductivity
119876119891
=
2119896119891
119882119891
ℎ120582119879sc
119901sc119879120583119911tanh 120587120582
2
sdot [
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(37)
where the first term in the bracket is the gas productioncalculated by Darcy formula which is defined as 119876
119863
5 Results and Discussion
According to the productivity formula of fractured well withfinite conductivity the effects of permeability correctionfactor slippage factor Knudsen diffusion coefficient matrixpermeability fracture half-length and fracture conductivityon productivity of fractured well are analyzed by combiningthe data from a single well of shale gas reservoirs in ChinaThe data for production simulation are presented in Table 3
51 Effect of Permeability Correction Factor on ProductivityFigure 5 presents the effect of permeability correction factoron productivity of fractured well and we can see fromthis figure that permeability correction factor has a great
Table 3 Data of shale gas reservoir for production simulation
Parameter Value UnitsPermeability 119870 0005 mDPorosity 120601 007 mdashFormation temperature 119879 36615 KSurface temperature 119879sc 293 KFormation pressure 119901
119890
28 MPaFormation thickness ℎ 305 mPressure relief radius 119877
119890
400 mGas viscosity 120583 0014 mPasdotsGas compressibility factor 119911 089 mdashRadius of wellbore 119903
119908
01 m
120585 = 1 + 4Kn + 512151205872Kn14
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
Field data
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
120585 = 1 (calculated by Darcy formula)120585 = 1 + 4Kn
Figure 5 Effect of permeability correction factor 120585 on productivity
influence on productivity Production of fractured well isvery small when our model only considers Darcy flow innanomicroscale pores (120585 = 1) but gas production increasesa lot when matrix apparent permeability is corrected bypermeability correction factor 120585 Field data shows that gasproduction of a fractured vertical well is 12 times 104m3 whenproduction pressure drop keeps 7MPa and fracture half-length is 200m [4] The calculation results of multiscalenon-Darcy seepage model are much more close to thefield data compared with the production rate calculated byDarcy formula andproduction rate calculated by productivityformula only considering first order term of permeabilitycorrection factor
52 Effects of Formation and Fracture Parameters on Produc-tivity The effect of matrix permeability 119870 on productivityof fractured well is illustrated in Figure 6 Gas productionof fractured well increases with the matrix permeabilityand gas production increases to 171 times 232 times and287 times when matrix permeability increases to 2 times 3times and 4 times respectively Figure 7 reflects productivity
Journal of Chemistry 7
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
K = 0005mDK = 0010mD
K = 0015mDK = 0020mD
Gas production rate (times104 m3d)
Figure 6 Effect of matrix permeability 119870 on productivity
0
5
10
15
20
25
30
0 05 1 15 2 25 3
Well
bore
pre
ssur
e (M
Pa)
Gas production rate (times104 m3d)
Lf = 50mLf = 150m
Lf = 250mLf = 350m
Figure 7 Effect of fracture half-length 119871119891
on productivity
of fractured well affected by fracture half-length Gas pro-duction increases to 154 times 206 times and 265 timeswhen fracture half-length increases to 3 times 4 times and5 times respectively We can find that the increase of gasproduction rate affected by matrix permeability is still morethan production rate affected by fracture half-length even ifthe increase ratio of fracture half-length is bigger thanmatrixpermeability which means that longer fracture half-lengthonly can improve local seepage ability of formation adjacentto hydraulic fractures but improvement of flow capacityin the whole formation can increase the gas productionsubstantially
Effect of fracture conductivity on productivity of frac-tured well has been analyzed under different fracture pene-tration ratio (119871
119891
119877119890
) We can see from Figure 8 that gas pro-duction increases with the fracture conductivity and fracturepenetration ratio But gas production increase becomes slowwhen the fracture conductivity increases to a certain valueand this value becomes bigger when fracture penetrationratio increases For examples this value will be 2 120583m2sdotcmwhen fracture penetration ratio is 01 but this value willincrease to 4120583m2sdotcm when fracture penetration ratio is 03
0
05
1
15
2
25
3
0 1 2 3 4 5 6
Gas
pro
duct
ion
rate
(times10
4m
3d
)
LfRe = 01
LfRe = 03LfRe = 05LfRe = 07
kf middot Wf (120583m2middotcm)
Figure 8 Effect of fracture conductivity 119896119891
sdot 119882119891
on productivity
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 1 2 3 4 5
Gas production rate (times104 m3d)
b = 0b = 1
b = 2
b = 3
Figure 9 Effect of slippage factor 119887 on productivity
This chart can help to optimize fracture conductivity underdifferent fracture penetration ratio
53 Effects of Slippage and Knudsen Diffusion on ProductivityThe relationship between wellbore pressure of fractured wellwith finite conductivity and gas production under differentslippage factor 119887 has been shown in Figure 9 Slippage effecthas negligible influence on productivity of fractured wellwhen wellbore pressure is high but slippage effect begins toaffect gas production when wellbore pressure is lower than15MPa and productivity of the fractured well increases withthe slippage factor So decreasing the wellbore pressure offractured well properly can improve gas production whenexploiting shale gas reservoirs We can see from Figure 10that gas production rate increases with Knudsen diffusioncoefficient which has a greater impact on gas production thanslippage effect
As we can see in Figure 11 the productivity of fracturedwell derived in this paper (119876
119891
) has been compared withproduction calculated by Darcy formula (119876
119863
) under differ-ent matrix permeability and wellbore pressure Figure 11(a)shows that bigger matrix permeability and a lower wellbore
8 Journal of Chemistry
Table 4 Comparison of production at turning point and production at minimum permeability under different wellbore pressure (the gasproduction rate unit times104 m3d)
119875119908
MPa 1 3 5 7 9 11 13 15 17 19 21 23 25 27119876119870min 0505 0333 0250 0197 0158 0127 0103 0082 0065 0050 0037 0025 0014 0005
119876TP 0315 0238 0193 0160 0134 0112 0093 0076 0061 0048 0035 0024 0014 0004119876119870min119876TP 1601 1398 1296 1228 1178 1138 1109 1082 1064 1046 1034 1024 1014 1010
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
DK = 1 times 10minus6 m2sDK = 3 times 10minus6 m2s
DK = 5 times 10minus6 m2sDK = 7 times 10minus6 m2s
Figure 10 Effect of Knudsen diffusion coefficient119863K on productiv-ity
pressure lead to a higher productivity of fractured well andgas production 119876
119891
is much bigger than Darcy productionrate 119876
119863
The effects of Knudsen diffusion and slip flow onproductivity of fractured well can be observed by taking theratio of119876
119891
and119876119863
in Figure 11(b)The effects on gas produc-tion increase whenmatrix permeability becomes smaller andwellbore pressure declines which become most significantunder the smallest matrix permeability (1 times 10minus5mD) and thelowest wellbore pressure (1MPa) in Figure 11(b)
Figure 12 represents the relationship of gas productionrate119876
119891
and matrix permeability119870 at different wellbore pres-sure For conventional reservoirs production rate decreaseswith formation permeability for shale gas reservoirs gasproduction rate decreases with matrix permeability first butthen increases with permeability owing to the effects ofKnudsen diffusion and slip flow after a certain permeabilityvalue which is called turning point in this paperThe value ofturning point decreases when wellbore pressure rises whichis because Knudsen diffusion and slippage effect are morelikely to occur in formation with smaller permeability andlower pressure so a smaller turning point would be requiredwhen pressure rises
Table 4 shows the ratio of gas production rate at min-imum permeability 119876
119870min in Figure 12 and production atturning point under different wellbore pressures 119876TP Theratios decrease when wellbore pressure increases whichmeans that Knudsen diffusion and slippage effects have agreater influence on productivity of fractured well under alower wellbore pressure But the gas production rate119876
119891
(119870 =1 times 10minus5mD 119875
119908
= 1MPa) is only 16 times the production
119876119891
at turning point (Table 4) even though the production119876119891
is almost 150 times the Darcy production rate 119876119863
in Figure 11(b) Knudsen diffusion and slippage effects canimprove the productivity at extremely low permeabilitysignificantly compared with gas production calculated byDarcy formula (without considering effects of diffusion andslip flow) but the improvement of productivity is still smallcompared with higher permeability when Knudsen diffusionand slippage effects are considered in both situations
6 Conclusions
In this paper a multiscale comprehensive mathematicalmodel had been established which can simulate differentflow regimes including continuum flow slip flow transitionflow and free-molecule flow We deduced the productivityformula for fractured well with limited conductivity for shalegas reservoirs and the influencing parameters were analyzedthoroughly The following conclusions can be drawn
(1) Flow regimes in shale gas reservoir were analyzedby calculating Knudsen number under different pres-sure and different pore throat diameter Gas flow inshale gas reservoirs with nanomicroscale pores is acombination of continuum flow slip flow transitionflow and free-molecule flow which cannot simply berepresented by Darcy formula anymore
(2) A new non-Darcy equation was developed based onBeskok-Karniadakis equation which is applicable forall flow regimes and effects of high order terms ofBK equation on permeability correction factor wereanalyzed under different Knudsen number Perme-ability correction factor is increasingly deviating froma Darcy type as Knudsen number increases
(3) Productivity formula of fractured well satisfying vari-able mass flowing in fractures was obtained andthe simulated results showed that production ratecalculated by non-Darcy seepage model with higherorder terms of permeability correction factor is moreclose to the field production data compared withthe production rate calculated by Darcy formula andproduction rate calculated by productivity formulaonly considering first order term of permeabilitycorrection factor
(4) Effects of slippage and diffusion which are changingwith pressure and pore sizes in nanoscale pores wereconsidered in simulation Simulation results showedthat slippage effect affects gas production of fracturedwell only when wellbore pressure less than 15MPa
Journal of Chemistry 9
010
20300
05
1
15
2
Matrix permeability (mD)Wellbore pressure (MPa)
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
QD
Qf
times104
Prod
uctio
n (m
3d
)
(a) Comparison between 119876119863
and 119876119891
(at different permeability andwellbore pressure)
010
20300
50
100
150
Matrix permeability (mD)Wellbore pressure (MPa)
Prod
uctio
n co
mpa
rison
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
(b) The value of 119876119891
119876
119863
at different matrix permeability and wellborepressure
Figure 11 Effects of Knudsen diffusion coefficient119863K and slip factor 119887 on productivity
0
04
08
12
16
2
Turning point
Effects of slip flow andKnudsen diffusion
Matrix permeability (mD)1E minus 5 1E minus 4 1E minus 3 1E minus 2
Pw = 1Pw = 3Pw = 5Pw = 7Pw = 9Pw = 11
(MPa
)
Pw = 13
Pw = 15
Pw = 17
Pw = 19
Pw = 21
Pw = 23
Pw = 25
Pw = 27
Gas
pro
duct
ion
rate
(times10
4m
3d
)
Figure 12 The relationship between gas production and matrix permeability at different wellbore pressure
and the effects of slippage and diffusion have a greaterinfluence on gas production of reservoirs with smallermatrix permeability especially when permeability isat nano-Darcy scale Matrix permeability fracturehalf length fracture penetration ratio and fractureconductivity can improve gas production at differentdegrees as well
Nomenclature
Latin
119887 Slippage coefficient119861119892
Gas volume factor119863K Knudsen diffusion coefficient m2s119891 Fraction of molecules striking pore wall which are
diffusely reflected
ℎ Thickness of shale gas reservoir m119870 Apparent permeabilityKn Knudsen number119896119891
Permeability of hydraulic fracture1198700
Absolute permeability119871119891
Fracture half length119872 Molecular mass kgmol119898 Pseudopressure119898119890
Initial pseudopressure119898119908
Pseudopressure of fractured well119901 Pressure Pa119901119890
Formation pressure Pa119901119908
Wellbore pressure Pa119901avg Average pressure Pa119901sc Pressure at standard condition Pa119876119863
Volume flow rate calculated by Darcy formula m3s
10 Journal of Chemistry
119876119891
Volume flow rate of fractured well with finiteconductivity m3s
119876TP Volume flow rate of fractured well at turning pointunder different matrix permeability 104m3d
119876V Volume flow rate of fractured well with infiniteconductivity m3s
119902V Volume flow rate m3s119877 Universal gas constant 8314 JKmol119877119890
Radius of boundary m119903 Pore radius119879 Temperature at formation condition K119879sc Temperature at standard condition K119906 Coordinate in119882 plane1199060
Distance between boundary and line sink in119882 planeV Coordinate in119882 planeV119892
Gas flow rate msV119906
Flow rate in 119906 direction of119882 planeVV Flow rate in V direction of119882 plane119882 119882 plane119882119891
Fracture width119909 Coordinate in 119885 plane119884 Coordinate in 119885 plane119885 119885 plane119911 Gas compressibility factor
Greek
120572 Rarefaction coefficient120585 Permeability correction factor120582 Gas molecule mean free path m120583 Gas viscosity Pasdots
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportof the National Program on Key Basic Research Project (973Program) (Grant no 2013CB228002) through the effectivedevelopment research of the Southern Marine Shale GasReservoirs in China
References
[1] M Elgmati Shale Gas Rock Characterization and 3D SubmicronPore Network Reconstruction Missouri University of Scienceand Technology Rolla Mo USA 2011
[2] R G Loucks R M Reed S C Ruppel and D M JarvieldquoMorphology genesis and distribution of nanometer-scalepores in siliceousmudstones of themississippian barnett shalerdquoJournal of Sedimentary Research vol 79 no 12 pp 848ndash8612009
[3] C Zou R Zhu B Bai et al ldquoFirst discovery of nano-pore throatin oil and gas reservoir in China and its scientific valuerdquo ActaPetrologica Sinica vol 27 no 6 pp 1857ndash1864 2011
[4] J DengW Zhu and QMa ldquoA new seepage model for shale gasreservoir and productivity analysis of fractured wellrdquo Fuel vol124 pp 232ndash240 2014
[5] F Javadpour D Fisher and M Unsworth ldquoNanoscale gasflow in shale gas sedimentsrdquo Journal of Canadian PetroleumTechnology vol 46 no 10 pp 55ndash61 2007
[6] F P Wang and R M Reed ldquoPore networks and fluid flow in gasshalesrdquo in Proceedings of the SPE Annual Technical Conferenceand Exhibition (ATCE rsquo09) SPE 124253 pp 1550ndash1557 NewOrleans La USA October 2009
[7] T Huang X Guo and F Chen ldquoModeling transient flowbehavior of a multiscale triple porosity model for shale gasreservoirsrdquo Journal of Natural Gas Science and Engineering vol23 pp 33ndash46 2015
[8] T Huang X Guo and F Chen ldquoModeling transient pressurebehavior of a fractured well for shale gas reservoirs based onthe properties of nanoporesrdquo Journal of Natural Gas Science andEngineering vol 23 pp 387ndash398 2015
[9] G G Michel R F Sigal F Civan and D Devegowda ldquoPara-metric investigation of shale gas production considering nano-scale pore size distribution formation factor and non-darcyflow mechanismsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE 147438 Denver Colo USAOctober-November 2011
[10] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Transactions ofthe ASME Journal of Fluids Engineering vol 118 no 3 pp 448ndash456 1996
[11] A Beskok and G E Karniadakis ldquoA model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[12] S Roy R Raju H F Chuang B A Cruden andMMeyyappanldquoModeling gas flow through microchannels and nanoporesrdquoJournal of Applied Physics vol 93 no 8 pp 4870ndash4879 2003
[13] F Civan ldquoEffective correlation of apparent gas permeability intight porous mediardquo Transport in Porous Media vol 82 no 2pp 375ndash384 2010
[14] E A Guggenheim Elements of the Kinetic Theory of GasesPergamon Press Oxford UK 1960
[15] M Knudsen ldquoThe law of the molecular flow and viscosity ofgases moving through tubesrdquo Annals of Physics vol 28 no 1pp 75ndash130 1909
[16] G P Brown A Dinardo G K Cheng and T K SherwoodldquoThe flow of gases in pipes at low pressuresrdquo Journal of AppliedPhysics vol 17 no 10 pp 802ndash813 1946
[17] G J I Igwe ldquoGas transport mechanism and slippage phe-nomenon in porous mediardquo Tech Rep SPE-16479-MS Societyof Petroleum Engineers 1987
[18] F Javadpour ldquoNanopores and apparent permeability of gasflow in mudrocks (shales and siltstone)rdquo Journal of CanadianPetroleum Technology vol 48 no 8 pp 16ndash21 2009
[19] T JiangW Shan and Y Yang ldquoCalculation of stable productioncapability of vertically fractured wellrdquo Petroleum Explorationand Development vol 28 no 2 p 61 2001
[20] Y Wang T Jiang and B Zeng ldquoProductivity performances ofhydraulically fractured gas wellrdquoActa Petrolei Sinica vol 24 no4 pp 65ndash68 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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CatalystsJournal of
4 Journal of Chemistry
Taking (15) into (11) we can get
V119892
= minus
1198700
120583
[1 + 4
3120587
81199032
120583
119901
119863K +512
151205872
(
3120587
81199032
120583
119901
119863K)14
+ 4119887(
3120587
81199032
120583
119901
119863K)2
]
119889119901
119889119909
(16)
According to Hagen-Poiseuille model relationshipbetween permeability and pore diameter can be written as
1198700
=
1199032
8
(17)
Taking (17) into (16) we can derive
V119892
= minus
1198700
120583
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119889119901
119889119909
(18)
Considering both sides of (18) multiplied by seepage areaand divided by volume factor we can get the volume flow rateat standard condition
119902V =V119860119861119892
=
1198601198700
119861119892
120583119892
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119889119901
119889119909
(19)
where the gas volume factor 119861119892
can be derived according tothe equation of state
119861119892
=
119899119911119877119879119901
119899119877119879sc119901sc=
119901sc119901
119911119879
119879sc (20)
4 Productivity Model of Fractured Well
The natural productivity of shale gas reservoirs is very lowso formation has to be artificially stimulated (hydraulicallyfractured) to render commercial production So this paperestablishes steady seepage model of fractured well for shalegas reservoirs by conformal transformation According to theprinciple of conformal transformation the gas productionand pressure of the boundary remains the same only thelength of line and flow forms change [19 20]
41 Physical Assumptions To make this mathematical modelmore tractable the following simplified assumptions aremade
(1) The reservoir is isotropic and gas flow in shale gasreservoir is planar (steady state)
(2) Flow in nanopores of shale matrix is assumed to benon-Darcy and fluid in shale gas reservoirs is singlephase
Table 2 Conformal transformations of the fractured well
Location ofpoint 119882 plane 119885 plane
A 119906 = 0 V = 0 119909 = 119871119891
119910 = 0
B 119906 = 0 V = 1205872 119909 = 0 119910 = 0
C 119906 = 0 V = 120587 119909 = minus119871119891
119910 = 0
D 119906 = 1199060
V = 0 119909 = 119871119891
ch1199060
119910 = 0
E 119906 = 1199060
V = 1205872 119909 = 0 119910 = 119871119891
sh1199060
F 119906 = 1199060
V = 120587 119909 = minus119871119891
ch1199060
119910 = 0
(3) Fractured vertical well is produced at a constantwellbore pressure and is penetrated fully in shale gasreservoir The height of hydraulic fractures is equal tothe thickness of formation
(4) The fractures are vertical and symmetrically locatedat two sides of wellbore Fracture half-length is 119871
119891
fracture width is negligible when the vertical fracturesare assumed to have infinite conductivity and is 119882
119891
when vertical fractures are assumed to have finiteconductivity
42 Fractured Well with Infinite Conductivity Conformaltransformation function is defined as
119885 = 119871119891
ch119882 (21)
Taking 119885 = 119909 + 119894119910119882 = 119906 + 119894V into (21) we can get
119909 = 119871119891
ch119906 cos V
119910 = 119871119891
sh119906 sin V(22)
We can see from Table 2 that the upper half planeformation in 119885 plane (Figure 3(a)) is transformed to a halfinfinite formationwithwidth of120587 at the right side of fracturedwell in 119882 plane (Figure 3(b)) elliptical constant pressureboundarywith radius of119877
119890
is transformed to a linear onewithlength of 120587 fractured well with length of 2119871
119891
is transformedto a line sink in119882 plane
Equipotential line equation of gas flow in 119885 plane can beobtained through (22)
1199092
1198712
119891
ch2119906+
1199102
1198712
119891
sh2119906= cos2V + sin2V = 1 (23)
The boundary can be viewed as circular when the bound-ary is far from fractured well so
ch1199060
asymp sh1199060
asymp
1
2
1198901199060 (24)
Then the equipotential line equation of boundary can berewritten as
1199092
+ 1199102
= 1198712
119891
(
1
2
1198901199060)
2
= 1198772
119890
(25)
Journal of Chemistry 5
y
xA BO
v
Hydraulic fracture
Wellbore C
6 3
3
22
1 4
5
1
y
Z = LfchW5
4
6
Q
2
O998400 u0 u
(a) Z plane (b) W plane
Figure 3 Sketch map of conformal transformation to fractured well
Solving (25) we can get the relationship between thelength of drainage area 119906
0
in 119882 plane and the radius ofboundary 119877
119890
in 119885 plane
1199060
= ln2119877119890
119871119891
(26)
Combined with (19) production rate of drainage area in119882 plane can be derived
119876V =21205871198700
ℎ119879sc119901sc1198791205831199111199060
[
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(27)
Gas production of fractured well in 119885 plane can beobtained according to (26)
119876V =21205871198700
ℎ119879sc
119901sc119879120583119911 ln (2119877119890119871119891)
sdot [
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(28)
43 Fractured Well with Finite Conductivity Compared withhydraulic fracture with infinite conductivity fractured ver-tical well with finite conductivity is more close to realcondition and the effects of parameters of hydraulic fractureson well performance can be analyzed as well
Considering an element of hydraulic fracture in119882 planein Figure 3(b) we assume that gas only flows in V direction(flows to wellbore) in hydraulic fracture (Figure 4) So flow
u
d
u
Wf
Figure 4 Sketch map of hydraulic fracture element
rate in fracture VV is a constant value in 119906 direction and avariable value in V direction flow rate from reservoir to theelement V
119906
equals a constant valueThe mass balance equation of gas flow in hydraulic frac-
ture can be obtained by principle of mass conservation
minus (VV1003816100381610038161003816V+ΔV minus VV
1003816100381610038161003816V) sdot
1
2
119882119891
ℎ + V119906
sdot ΔVℎ = 0 (29)
Considering both sides of (29) divided by ΔV the partialdifference equation can be derivedwhenΔV takes an infinites-imal
119889VV119889V
sdot
1
2
119882119891
= V119906
(30)
Pseudopressure function of gas flow in hydraulic fractureis defined as
119898(119901) = 2int
119901
119901119890
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119901
120583119911
119889119901
(31)
Equation (30) can be
1205972
119898
120597V2minus
1198700
(12) 119896119891
119882119891
1
ln (2119877119890
119871119891
)
119898
= minus
1198700
(12) 119896119891
119882119891
1
ln (2119877119890
119871119891
)
119898119890
(32)
6 Journal of Chemistry
The boundary conditions of hydraulic fracture are givenas follows
119889119898
119889V= 0 V = 0
119898 = 119898119908
V =120587
2
(33)
Combining with (33) we can get the solution of (32)
119898(119901) = 1198881
119890120582V+ 1198882
119890minus120582V
+ 119898119890
(34)
where the parameters 1198881
1198882
and 120582 are defined as follows
120582 = radic
21198700
119896119891
119882119891
1
ln 2119877119890
119871119891
1198881
= 1198882
=
119898119908
minus 119898119890
119890(1205872)120582
+ 119890minus(1205872)120582
(35)
The gas production of fractured vertical well can be writ-ten as
119876119891
= minus
119896119891
119882119891
ℎ119879sc
119901sc119879
119889119898
119889V
10038161003816100381610038161003816100381610038161003816V=1205872
=
119896119891
119882119891
ℎ119879sc
119901sc119879120582 (119898119890
minus 119898119908
)
119890(1205872)120582
minus 119890minus(1205872)120582
119890(1205872)120582
+ 119890minus(1205872)120582
(36)
Combined with (31) we can get the productivity formulaof fractured vertical well with finite conductivity
119876119891
=
2119896119891
119882119891
ℎ120582119879sc
119901sc119879120583119911tanh 120587120582
2
sdot [
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(37)
where the first term in the bracket is the gas productioncalculated by Darcy formula which is defined as 119876
119863
5 Results and Discussion
According to the productivity formula of fractured well withfinite conductivity the effects of permeability correctionfactor slippage factor Knudsen diffusion coefficient matrixpermeability fracture half-length and fracture conductivityon productivity of fractured well are analyzed by combiningthe data from a single well of shale gas reservoirs in ChinaThe data for production simulation are presented in Table 3
51 Effect of Permeability Correction Factor on ProductivityFigure 5 presents the effect of permeability correction factoron productivity of fractured well and we can see fromthis figure that permeability correction factor has a great
Table 3 Data of shale gas reservoir for production simulation
Parameter Value UnitsPermeability 119870 0005 mDPorosity 120601 007 mdashFormation temperature 119879 36615 KSurface temperature 119879sc 293 KFormation pressure 119901
119890
28 MPaFormation thickness ℎ 305 mPressure relief radius 119877
119890
400 mGas viscosity 120583 0014 mPasdotsGas compressibility factor 119911 089 mdashRadius of wellbore 119903
119908
01 m
120585 = 1 + 4Kn + 512151205872Kn14
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
Field data
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
120585 = 1 (calculated by Darcy formula)120585 = 1 + 4Kn
Figure 5 Effect of permeability correction factor 120585 on productivity
influence on productivity Production of fractured well isvery small when our model only considers Darcy flow innanomicroscale pores (120585 = 1) but gas production increasesa lot when matrix apparent permeability is corrected bypermeability correction factor 120585 Field data shows that gasproduction of a fractured vertical well is 12 times 104m3 whenproduction pressure drop keeps 7MPa and fracture half-length is 200m [4] The calculation results of multiscalenon-Darcy seepage model are much more close to thefield data compared with the production rate calculated byDarcy formula andproduction rate calculated by productivityformula only considering first order term of permeabilitycorrection factor
52 Effects of Formation and Fracture Parameters on Produc-tivity The effect of matrix permeability 119870 on productivityof fractured well is illustrated in Figure 6 Gas productionof fractured well increases with the matrix permeabilityand gas production increases to 171 times 232 times and287 times when matrix permeability increases to 2 times 3times and 4 times respectively Figure 7 reflects productivity
Journal of Chemistry 7
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
K = 0005mDK = 0010mD
K = 0015mDK = 0020mD
Gas production rate (times104 m3d)
Figure 6 Effect of matrix permeability 119870 on productivity
0
5
10
15
20
25
30
0 05 1 15 2 25 3
Well
bore
pre
ssur
e (M
Pa)
Gas production rate (times104 m3d)
Lf = 50mLf = 150m
Lf = 250mLf = 350m
Figure 7 Effect of fracture half-length 119871119891
on productivity
of fractured well affected by fracture half-length Gas pro-duction increases to 154 times 206 times and 265 timeswhen fracture half-length increases to 3 times 4 times and5 times respectively We can find that the increase of gasproduction rate affected by matrix permeability is still morethan production rate affected by fracture half-length even ifthe increase ratio of fracture half-length is bigger thanmatrixpermeability which means that longer fracture half-lengthonly can improve local seepage ability of formation adjacentto hydraulic fractures but improvement of flow capacityin the whole formation can increase the gas productionsubstantially
Effect of fracture conductivity on productivity of frac-tured well has been analyzed under different fracture pene-tration ratio (119871
119891
119877119890
) We can see from Figure 8 that gas pro-duction increases with the fracture conductivity and fracturepenetration ratio But gas production increase becomes slowwhen the fracture conductivity increases to a certain valueand this value becomes bigger when fracture penetrationratio increases For examples this value will be 2 120583m2sdotcmwhen fracture penetration ratio is 01 but this value willincrease to 4120583m2sdotcm when fracture penetration ratio is 03
0
05
1
15
2
25
3
0 1 2 3 4 5 6
Gas
pro
duct
ion
rate
(times10
4m
3d
)
LfRe = 01
LfRe = 03LfRe = 05LfRe = 07
kf middot Wf (120583m2middotcm)
Figure 8 Effect of fracture conductivity 119896119891
sdot 119882119891
on productivity
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 1 2 3 4 5
Gas production rate (times104 m3d)
b = 0b = 1
b = 2
b = 3
Figure 9 Effect of slippage factor 119887 on productivity
This chart can help to optimize fracture conductivity underdifferent fracture penetration ratio
53 Effects of Slippage and Knudsen Diffusion on ProductivityThe relationship between wellbore pressure of fractured wellwith finite conductivity and gas production under differentslippage factor 119887 has been shown in Figure 9 Slippage effecthas negligible influence on productivity of fractured wellwhen wellbore pressure is high but slippage effect begins toaffect gas production when wellbore pressure is lower than15MPa and productivity of the fractured well increases withthe slippage factor So decreasing the wellbore pressure offractured well properly can improve gas production whenexploiting shale gas reservoirs We can see from Figure 10that gas production rate increases with Knudsen diffusioncoefficient which has a greater impact on gas production thanslippage effect
As we can see in Figure 11 the productivity of fracturedwell derived in this paper (119876
119891
) has been compared withproduction calculated by Darcy formula (119876
119863
) under differ-ent matrix permeability and wellbore pressure Figure 11(a)shows that bigger matrix permeability and a lower wellbore
8 Journal of Chemistry
Table 4 Comparison of production at turning point and production at minimum permeability under different wellbore pressure (the gasproduction rate unit times104 m3d)
119875119908
MPa 1 3 5 7 9 11 13 15 17 19 21 23 25 27119876119870min 0505 0333 0250 0197 0158 0127 0103 0082 0065 0050 0037 0025 0014 0005
119876TP 0315 0238 0193 0160 0134 0112 0093 0076 0061 0048 0035 0024 0014 0004119876119870min119876TP 1601 1398 1296 1228 1178 1138 1109 1082 1064 1046 1034 1024 1014 1010
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
DK = 1 times 10minus6 m2sDK = 3 times 10minus6 m2s
DK = 5 times 10minus6 m2sDK = 7 times 10minus6 m2s
Figure 10 Effect of Knudsen diffusion coefficient119863K on productiv-ity
pressure lead to a higher productivity of fractured well andgas production 119876
119891
is much bigger than Darcy productionrate 119876
119863
The effects of Knudsen diffusion and slip flow onproductivity of fractured well can be observed by taking theratio of119876
119891
and119876119863
in Figure 11(b)The effects on gas produc-tion increase whenmatrix permeability becomes smaller andwellbore pressure declines which become most significantunder the smallest matrix permeability (1 times 10minus5mD) and thelowest wellbore pressure (1MPa) in Figure 11(b)
Figure 12 represents the relationship of gas productionrate119876
119891
and matrix permeability119870 at different wellbore pres-sure For conventional reservoirs production rate decreaseswith formation permeability for shale gas reservoirs gasproduction rate decreases with matrix permeability first butthen increases with permeability owing to the effects ofKnudsen diffusion and slip flow after a certain permeabilityvalue which is called turning point in this paperThe value ofturning point decreases when wellbore pressure rises whichis because Knudsen diffusion and slippage effect are morelikely to occur in formation with smaller permeability andlower pressure so a smaller turning point would be requiredwhen pressure rises
Table 4 shows the ratio of gas production rate at min-imum permeability 119876
119870min in Figure 12 and production atturning point under different wellbore pressures 119876TP Theratios decrease when wellbore pressure increases whichmeans that Knudsen diffusion and slippage effects have agreater influence on productivity of fractured well under alower wellbore pressure But the gas production rate119876
119891
(119870 =1 times 10minus5mD 119875
119908
= 1MPa) is only 16 times the production
119876119891
at turning point (Table 4) even though the production119876119891
is almost 150 times the Darcy production rate 119876119863
in Figure 11(b) Knudsen diffusion and slippage effects canimprove the productivity at extremely low permeabilitysignificantly compared with gas production calculated byDarcy formula (without considering effects of diffusion andslip flow) but the improvement of productivity is still smallcompared with higher permeability when Knudsen diffusionand slippage effects are considered in both situations
6 Conclusions
In this paper a multiscale comprehensive mathematicalmodel had been established which can simulate differentflow regimes including continuum flow slip flow transitionflow and free-molecule flow We deduced the productivityformula for fractured well with limited conductivity for shalegas reservoirs and the influencing parameters were analyzedthoroughly The following conclusions can be drawn
(1) Flow regimes in shale gas reservoir were analyzedby calculating Knudsen number under different pres-sure and different pore throat diameter Gas flow inshale gas reservoirs with nanomicroscale pores is acombination of continuum flow slip flow transitionflow and free-molecule flow which cannot simply berepresented by Darcy formula anymore
(2) A new non-Darcy equation was developed based onBeskok-Karniadakis equation which is applicable forall flow regimes and effects of high order terms ofBK equation on permeability correction factor wereanalyzed under different Knudsen number Perme-ability correction factor is increasingly deviating froma Darcy type as Knudsen number increases
(3) Productivity formula of fractured well satisfying vari-able mass flowing in fractures was obtained andthe simulated results showed that production ratecalculated by non-Darcy seepage model with higherorder terms of permeability correction factor is moreclose to the field production data compared withthe production rate calculated by Darcy formula andproduction rate calculated by productivity formulaonly considering first order term of permeabilitycorrection factor
(4) Effects of slippage and diffusion which are changingwith pressure and pore sizes in nanoscale pores wereconsidered in simulation Simulation results showedthat slippage effect affects gas production of fracturedwell only when wellbore pressure less than 15MPa
Journal of Chemistry 9
010
20300
05
1
15
2
Matrix permeability (mD)Wellbore pressure (MPa)
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
QD
Qf
times104
Prod
uctio
n (m
3d
)
(a) Comparison between 119876119863
and 119876119891
(at different permeability andwellbore pressure)
010
20300
50
100
150
Matrix permeability (mD)Wellbore pressure (MPa)
Prod
uctio
n co
mpa
rison
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
(b) The value of 119876119891
119876
119863
at different matrix permeability and wellborepressure
Figure 11 Effects of Knudsen diffusion coefficient119863K and slip factor 119887 on productivity
0
04
08
12
16
2
Turning point
Effects of slip flow andKnudsen diffusion
Matrix permeability (mD)1E minus 5 1E minus 4 1E minus 3 1E minus 2
Pw = 1Pw = 3Pw = 5Pw = 7Pw = 9Pw = 11
(MPa
)
Pw = 13
Pw = 15
Pw = 17
Pw = 19
Pw = 21
Pw = 23
Pw = 25
Pw = 27
Gas
pro
duct
ion
rate
(times10
4m
3d
)
Figure 12 The relationship between gas production and matrix permeability at different wellbore pressure
and the effects of slippage and diffusion have a greaterinfluence on gas production of reservoirs with smallermatrix permeability especially when permeability isat nano-Darcy scale Matrix permeability fracturehalf length fracture penetration ratio and fractureconductivity can improve gas production at differentdegrees as well
Nomenclature
Latin
119887 Slippage coefficient119861119892
Gas volume factor119863K Knudsen diffusion coefficient m2s119891 Fraction of molecules striking pore wall which are
diffusely reflected
ℎ Thickness of shale gas reservoir m119870 Apparent permeabilityKn Knudsen number119896119891
Permeability of hydraulic fracture1198700
Absolute permeability119871119891
Fracture half length119872 Molecular mass kgmol119898 Pseudopressure119898119890
Initial pseudopressure119898119908
Pseudopressure of fractured well119901 Pressure Pa119901119890
Formation pressure Pa119901119908
Wellbore pressure Pa119901avg Average pressure Pa119901sc Pressure at standard condition Pa119876119863
Volume flow rate calculated by Darcy formula m3s
10 Journal of Chemistry
119876119891
Volume flow rate of fractured well with finiteconductivity m3s
119876TP Volume flow rate of fractured well at turning pointunder different matrix permeability 104m3d
119876V Volume flow rate of fractured well with infiniteconductivity m3s
119902V Volume flow rate m3s119877 Universal gas constant 8314 JKmol119877119890
Radius of boundary m119903 Pore radius119879 Temperature at formation condition K119879sc Temperature at standard condition K119906 Coordinate in119882 plane1199060
Distance between boundary and line sink in119882 planeV Coordinate in119882 planeV119892
Gas flow rate msV119906
Flow rate in 119906 direction of119882 planeVV Flow rate in V direction of119882 plane119882 119882 plane119882119891
Fracture width119909 Coordinate in 119885 plane119884 Coordinate in 119885 plane119885 119885 plane119911 Gas compressibility factor
Greek
120572 Rarefaction coefficient120585 Permeability correction factor120582 Gas molecule mean free path m120583 Gas viscosity Pasdots
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportof the National Program on Key Basic Research Project (973Program) (Grant no 2013CB228002) through the effectivedevelopment research of the Southern Marine Shale GasReservoirs in China
References
[1] M Elgmati Shale Gas Rock Characterization and 3D SubmicronPore Network Reconstruction Missouri University of Scienceand Technology Rolla Mo USA 2011
[2] R G Loucks R M Reed S C Ruppel and D M JarvieldquoMorphology genesis and distribution of nanometer-scalepores in siliceousmudstones of themississippian barnett shalerdquoJournal of Sedimentary Research vol 79 no 12 pp 848ndash8612009
[3] C Zou R Zhu B Bai et al ldquoFirst discovery of nano-pore throatin oil and gas reservoir in China and its scientific valuerdquo ActaPetrologica Sinica vol 27 no 6 pp 1857ndash1864 2011
[4] J DengW Zhu and QMa ldquoA new seepage model for shale gasreservoir and productivity analysis of fractured wellrdquo Fuel vol124 pp 232ndash240 2014
[5] F Javadpour D Fisher and M Unsworth ldquoNanoscale gasflow in shale gas sedimentsrdquo Journal of Canadian PetroleumTechnology vol 46 no 10 pp 55ndash61 2007
[6] F P Wang and R M Reed ldquoPore networks and fluid flow in gasshalesrdquo in Proceedings of the SPE Annual Technical Conferenceand Exhibition (ATCE rsquo09) SPE 124253 pp 1550ndash1557 NewOrleans La USA October 2009
[7] T Huang X Guo and F Chen ldquoModeling transient flowbehavior of a multiscale triple porosity model for shale gasreservoirsrdquo Journal of Natural Gas Science and Engineering vol23 pp 33ndash46 2015
[8] T Huang X Guo and F Chen ldquoModeling transient pressurebehavior of a fractured well for shale gas reservoirs based onthe properties of nanoporesrdquo Journal of Natural Gas Science andEngineering vol 23 pp 387ndash398 2015
[9] G G Michel R F Sigal F Civan and D Devegowda ldquoPara-metric investigation of shale gas production considering nano-scale pore size distribution formation factor and non-darcyflow mechanismsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE 147438 Denver Colo USAOctober-November 2011
[10] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Transactions ofthe ASME Journal of Fluids Engineering vol 118 no 3 pp 448ndash456 1996
[11] A Beskok and G E Karniadakis ldquoA model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[12] S Roy R Raju H F Chuang B A Cruden andMMeyyappanldquoModeling gas flow through microchannels and nanoporesrdquoJournal of Applied Physics vol 93 no 8 pp 4870ndash4879 2003
[13] F Civan ldquoEffective correlation of apparent gas permeability intight porous mediardquo Transport in Porous Media vol 82 no 2pp 375ndash384 2010
[14] E A Guggenheim Elements of the Kinetic Theory of GasesPergamon Press Oxford UK 1960
[15] M Knudsen ldquoThe law of the molecular flow and viscosity ofgases moving through tubesrdquo Annals of Physics vol 28 no 1pp 75ndash130 1909
[16] G P Brown A Dinardo G K Cheng and T K SherwoodldquoThe flow of gases in pipes at low pressuresrdquo Journal of AppliedPhysics vol 17 no 10 pp 802ndash813 1946
[17] G J I Igwe ldquoGas transport mechanism and slippage phe-nomenon in porous mediardquo Tech Rep SPE-16479-MS Societyof Petroleum Engineers 1987
[18] F Javadpour ldquoNanopores and apparent permeability of gasflow in mudrocks (shales and siltstone)rdquo Journal of CanadianPetroleum Technology vol 48 no 8 pp 16ndash21 2009
[19] T JiangW Shan and Y Yang ldquoCalculation of stable productioncapability of vertically fractured wellrdquo Petroleum Explorationand Development vol 28 no 2 p 61 2001
[20] Y Wang T Jiang and B Zeng ldquoProductivity performances ofhydraulically fractured gas wellrdquoActa Petrolei Sinica vol 24 no4 pp 65ndash68 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
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Carbohydrate Chemistry
International Journal of
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Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
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Analytical Methods in Chemistry
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Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
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Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Journal of Chemistry 5
y
xA BO
v
Hydraulic fracture
Wellbore C
6 3
3
22
1 4
5
1
y
Z = LfchW5
4
6
Q
2
O998400 u0 u
(a) Z plane (b) W plane
Figure 3 Sketch map of conformal transformation to fractured well
Solving (25) we can get the relationship between thelength of drainage area 119906
0
in 119882 plane and the radius ofboundary 119877
119890
in 119885 plane
1199060
= ln2119877119890
119871119891
(26)
Combined with (19) production rate of drainage area in119882 plane can be derived
119876V =21205871198700
ℎ119879sc119901sc1198791205831199111199060
[
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(27)
Gas production of fractured well in 119885 plane can beobtained according to (26)
119876V =21205871198700
ℎ119879sc
119901sc119879120583119911 ln (2119877119890119871119891)
sdot [
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(28)
43 Fractured Well with Finite Conductivity Compared withhydraulic fracture with infinite conductivity fractured ver-tical well with finite conductivity is more close to realcondition and the effects of parameters of hydraulic fractureson well performance can be analyzed as well
Considering an element of hydraulic fracture in119882 planein Figure 3(b) we assume that gas only flows in V direction(flows to wellbore) in hydraulic fracture (Figure 4) So flow
u
d
u
Wf
Figure 4 Sketch map of hydraulic fracture element
rate in fracture VV is a constant value in 119906 direction and avariable value in V direction flow rate from reservoir to theelement V
119906
equals a constant valueThe mass balance equation of gas flow in hydraulic frac-
ture can be obtained by principle of mass conservation
minus (VV1003816100381610038161003816V+ΔV minus VV
1003816100381610038161003816V) sdot
1
2
119882119891
ℎ + V119906
sdot ΔVℎ = 0 (29)
Considering both sides of (29) divided by ΔV the partialdifference equation can be derivedwhenΔV takes an infinites-imal
119889VV119889V
sdot
1
2
119882119891
= V119906
(30)
Pseudopressure function of gas flow in hydraulic fractureis defined as
119898(119901) = 2int
119901
119901119890
[1 +
3120587
161198700
120583
119901
119863K +512
151205872
(
3120587
641198700
120583
119901
119863K)14
+ 4119887(
3120587
641198700
120583
119901
119863K)2
]
119901
120583119911
119889119901
(31)
Equation (30) can be
1205972
119898
120597V2minus
1198700
(12) 119896119891
119882119891
1
ln (2119877119890
119871119891
)
119898
= minus
1198700
(12) 119896119891
119882119891
1
ln (2119877119890
119871119891
)
119898119890
(32)
6 Journal of Chemistry
The boundary conditions of hydraulic fracture are givenas follows
119889119898
119889V= 0 V = 0
119898 = 119898119908
V =120587
2
(33)
Combining with (33) we can get the solution of (32)
119898(119901) = 1198881
119890120582V+ 1198882
119890minus120582V
+ 119898119890
(34)
where the parameters 1198881
1198882
and 120582 are defined as follows
120582 = radic
21198700
119896119891
119882119891
1
ln 2119877119890
119871119891
1198881
= 1198882
=
119898119908
minus 119898119890
119890(1205872)120582
+ 119890minus(1205872)120582
(35)
The gas production of fractured vertical well can be writ-ten as
119876119891
= minus
119896119891
119882119891
ℎ119879sc
119901sc119879
119889119898
119889V
10038161003816100381610038161003816100381610038161003816V=1205872
=
119896119891
119882119891
ℎ119879sc
119901sc119879120582 (119898119890
minus 119898119908
)
119890(1205872)120582
minus 119890minus(1205872)120582
119890(1205872)120582
+ 119890minus(1205872)120582
(36)
Combined with (31) we can get the productivity formulaof fractured vertical well with finite conductivity
119876119891
=
2119896119891
119882119891
ℎ120582119879sc
119901sc119879120583119911tanh 120587120582
2
sdot [
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(37)
where the first term in the bracket is the gas productioncalculated by Darcy formula which is defined as 119876
119863
5 Results and Discussion
According to the productivity formula of fractured well withfinite conductivity the effects of permeability correctionfactor slippage factor Knudsen diffusion coefficient matrixpermeability fracture half-length and fracture conductivityon productivity of fractured well are analyzed by combiningthe data from a single well of shale gas reservoirs in ChinaThe data for production simulation are presented in Table 3
51 Effect of Permeability Correction Factor on ProductivityFigure 5 presents the effect of permeability correction factoron productivity of fractured well and we can see fromthis figure that permeability correction factor has a great
Table 3 Data of shale gas reservoir for production simulation
Parameter Value UnitsPermeability 119870 0005 mDPorosity 120601 007 mdashFormation temperature 119879 36615 KSurface temperature 119879sc 293 KFormation pressure 119901
119890
28 MPaFormation thickness ℎ 305 mPressure relief radius 119877
119890
400 mGas viscosity 120583 0014 mPasdotsGas compressibility factor 119911 089 mdashRadius of wellbore 119903
119908
01 m
120585 = 1 + 4Kn + 512151205872Kn14
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
Field data
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
120585 = 1 (calculated by Darcy formula)120585 = 1 + 4Kn
Figure 5 Effect of permeability correction factor 120585 on productivity
influence on productivity Production of fractured well isvery small when our model only considers Darcy flow innanomicroscale pores (120585 = 1) but gas production increasesa lot when matrix apparent permeability is corrected bypermeability correction factor 120585 Field data shows that gasproduction of a fractured vertical well is 12 times 104m3 whenproduction pressure drop keeps 7MPa and fracture half-length is 200m [4] The calculation results of multiscalenon-Darcy seepage model are much more close to thefield data compared with the production rate calculated byDarcy formula andproduction rate calculated by productivityformula only considering first order term of permeabilitycorrection factor
52 Effects of Formation and Fracture Parameters on Produc-tivity The effect of matrix permeability 119870 on productivityof fractured well is illustrated in Figure 6 Gas productionof fractured well increases with the matrix permeabilityand gas production increases to 171 times 232 times and287 times when matrix permeability increases to 2 times 3times and 4 times respectively Figure 7 reflects productivity
Journal of Chemistry 7
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
K = 0005mDK = 0010mD
K = 0015mDK = 0020mD
Gas production rate (times104 m3d)
Figure 6 Effect of matrix permeability 119870 on productivity
0
5
10
15
20
25
30
0 05 1 15 2 25 3
Well
bore
pre
ssur
e (M
Pa)
Gas production rate (times104 m3d)
Lf = 50mLf = 150m
Lf = 250mLf = 350m
Figure 7 Effect of fracture half-length 119871119891
on productivity
of fractured well affected by fracture half-length Gas pro-duction increases to 154 times 206 times and 265 timeswhen fracture half-length increases to 3 times 4 times and5 times respectively We can find that the increase of gasproduction rate affected by matrix permeability is still morethan production rate affected by fracture half-length even ifthe increase ratio of fracture half-length is bigger thanmatrixpermeability which means that longer fracture half-lengthonly can improve local seepage ability of formation adjacentto hydraulic fractures but improvement of flow capacityin the whole formation can increase the gas productionsubstantially
Effect of fracture conductivity on productivity of frac-tured well has been analyzed under different fracture pene-tration ratio (119871
119891
119877119890
) We can see from Figure 8 that gas pro-duction increases with the fracture conductivity and fracturepenetration ratio But gas production increase becomes slowwhen the fracture conductivity increases to a certain valueand this value becomes bigger when fracture penetrationratio increases For examples this value will be 2 120583m2sdotcmwhen fracture penetration ratio is 01 but this value willincrease to 4120583m2sdotcm when fracture penetration ratio is 03
0
05
1
15
2
25
3
0 1 2 3 4 5 6
Gas
pro
duct
ion
rate
(times10
4m
3d
)
LfRe = 01
LfRe = 03LfRe = 05LfRe = 07
kf middot Wf (120583m2middotcm)
Figure 8 Effect of fracture conductivity 119896119891
sdot 119882119891
on productivity
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 1 2 3 4 5
Gas production rate (times104 m3d)
b = 0b = 1
b = 2
b = 3
Figure 9 Effect of slippage factor 119887 on productivity
This chart can help to optimize fracture conductivity underdifferent fracture penetration ratio
53 Effects of Slippage and Knudsen Diffusion on ProductivityThe relationship between wellbore pressure of fractured wellwith finite conductivity and gas production under differentslippage factor 119887 has been shown in Figure 9 Slippage effecthas negligible influence on productivity of fractured wellwhen wellbore pressure is high but slippage effect begins toaffect gas production when wellbore pressure is lower than15MPa and productivity of the fractured well increases withthe slippage factor So decreasing the wellbore pressure offractured well properly can improve gas production whenexploiting shale gas reservoirs We can see from Figure 10that gas production rate increases with Knudsen diffusioncoefficient which has a greater impact on gas production thanslippage effect
As we can see in Figure 11 the productivity of fracturedwell derived in this paper (119876
119891
) has been compared withproduction calculated by Darcy formula (119876
119863
) under differ-ent matrix permeability and wellbore pressure Figure 11(a)shows that bigger matrix permeability and a lower wellbore
8 Journal of Chemistry
Table 4 Comparison of production at turning point and production at minimum permeability under different wellbore pressure (the gasproduction rate unit times104 m3d)
119875119908
MPa 1 3 5 7 9 11 13 15 17 19 21 23 25 27119876119870min 0505 0333 0250 0197 0158 0127 0103 0082 0065 0050 0037 0025 0014 0005
119876TP 0315 0238 0193 0160 0134 0112 0093 0076 0061 0048 0035 0024 0014 0004119876119870min119876TP 1601 1398 1296 1228 1178 1138 1109 1082 1064 1046 1034 1024 1014 1010
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
DK = 1 times 10minus6 m2sDK = 3 times 10minus6 m2s
DK = 5 times 10minus6 m2sDK = 7 times 10minus6 m2s
Figure 10 Effect of Knudsen diffusion coefficient119863K on productiv-ity
pressure lead to a higher productivity of fractured well andgas production 119876
119891
is much bigger than Darcy productionrate 119876
119863
The effects of Knudsen diffusion and slip flow onproductivity of fractured well can be observed by taking theratio of119876
119891
and119876119863
in Figure 11(b)The effects on gas produc-tion increase whenmatrix permeability becomes smaller andwellbore pressure declines which become most significantunder the smallest matrix permeability (1 times 10minus5mD) and thelowest wellbore pressure (1MPa) in Figure 11(b)
Figure 12 represents the relationship of gas productionrate119876
119891
and matrix permeability119870 at different wellbore pres-sure For conventional reservoirs production rate decreaseswith formation permeability for shale gas reservoirs gasproduction rate decreases with matrix permeability first butthen increases with permeability owing to the effects ofKnudsen diffusion and slip flow after a certain permeabilityvalue which is called turning point in this paperThe value ofturning point decreases when wellbore pressure rises whichis because Knudsen diffusion and slippage effect are morelikely to occur in formation with smaller permeability andlower pressure so a smaller turning point would be requiredwhen pressure rises
Table 4 shows the ratio of gas production rate at min-imum permeability 119876
119870min in Figure 12 and production atturning point under different wellbore pressures 119876TP Theratios decrease when wellbore pressure increases whichmeans that Knudsen diffusion and slippage effects have agreater influence on productivity of fractured well under alower wellbore pressure But the gas production rate119876
119891
(119870 =1 times 10minus5mD 119875
119908
= 1MPa) is only 16 times the production
119876119891
at turning point (Table 4) even though the production119876119891
is almost 150 times the Darcy production rate 119876119863
in Figure 11(b) Knudsen diffusion and slippage effects canimprove the productivity at extremely low permeabilitysignificantly compared with gas production calculated byDarcy formula (without considering effects of diffusion andslip flow) but the improvement of productivity is still smallcompared with higher permeability when Knudsen diffusionand slippage effects are considered in both situations
6 Conclusions
In this paper a multiscale comprehensive mathematicalmodel had been established which can simulate differentflow regimes including continuum flow slip flow transitionflow and free-molecule flow We deduced the productivityformula for fractured well with limited conductivity for shalegas reservoirs and the influencing parameters were analyzedthoroughly The following conclusions can be drawn
(1) Flow regimes in shale gas reservoir were analyzedby calculating Knudsen number under different pres-sure and different pore throat diameter Gas flow inshale gas reservoirs with nanomicroscale pores is acombination of continuum flow slip flow transitionflow and free-molecule flow which cannot simply berepresented by Darcy formula anymore
(2) A new non-Darcy equation was developed based onBeskok-Karniadakis equation which is applicable forall flow regimes and effects of high order terms ofBK equation on permeability correction factor wereanalyzed under different Knudsen number Perme-ability correction factor is increasingly deviating froma Darcy type as Knudsen number increases
(3) Productivity formula of fractured well satisfying vari-able mass flowing in fractures was obtained andthe simulated results showed that production ratecalculated by non-Darcy seepage model with higherorder terms of permeability correction factor is moreclose to the field production data compared withthe production rate calculated by Darcy formula andproduction rate calculated by productivity formulaonly considering first order term of permeabilitycorrection factor
(4) Effects of slippage and diffusion which are changingwith pressure and pore sizes in nanoscale pores wereconsidered in simulation Simulation results showedthat slippage effect affects gas production of fracturedwell only when wellbore pressure less than 15MPa
Journal of Chemistry 9
010
20300
05
1
15
2
Matrix permeability (mD)Wellbore pressure (MPa)
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
QD
Qf
times104
Prod
uctio
n (m
3d
)
(a) Comparison between 119876119863
and 119876119891
(at different permeability andwellbore pressure)
010
20300
50
100
150
Matrix permeability (mD)Wellbore pressure (MPa)
Prod
uctio
n co
mpa
rison
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
(b) The value of 119876119891
119876
119863
at different matrix permeability and wellborepressure
Figure 11 Effects of Knudsen diffusion coefficient119863K and slip factor 119887 on productivity
0
04
08
12
16
2
Turning point
Effects of slip flow andKnudsen diffusion
Matrix permeability (mD)1E minus 5 1E minus 4 1E minus 3 1E minus 2
Pw = 1Pw = 3Pw = 5Pw = 7Pw = 9Pw = 11
(MPa
)
Pw = 13
Pw = 15
Pw = 17
Pw = 19
Pw = 21
Pw = 23
Pw = 25
Pw = 27
Gas
pro
duct
ion
rate
(times10
4m
3d
)
Figure 12 The relationship between gas production and matrix permeability at different wellbore pressure
and the effects of slippage and diffusion have a greaterinfluence on gas production of reservoirs with smallermatrix permeability especially when permeability isat nano-Darcy scale Matrix permeability fracturehalf length fracture penetration ratio and fractureconductivity can improve gas production at differentdegrees as well
Nomenclature
Latin
119887 Slippage coefficient119861119892
Gas volume factor119863K Knudsen diffusion coefficient m2s119891 Fraction of molecules striking pore wall which are
diffusely reflected
ℎ Thickness of shale gas reservoir m119870 Apparent permeabilityKn Knudsen number119896119891
Permeability of hydraulic fracture1198700
Absolute permeability119871119891
Fracture half length119872 Molecular mass kgmol119898 Pseudopressure119898119890
Initial pseudopressure119898119908
Pseudopressure of fractured well119901 Pressure Pa119901119890
Formation pressure Pa119901119908
Wellbore pressure Pa119901avg Average pressure Pa119901sc Pressure at standard condition Pa119876119863
Volume flow rate calculated by Darcy formula m3s
10 Journal of Chemistry
119876119891
Volume flow rate of fractured well with finiteconductivity m3s
119876TP Volume flow rate of fractured well at turning pointunder different matrix permeability 104m3d
119876V Volume flow rate of fractured well with infiniteconductivity m3s
119902V Volume flow rate m3s119877 Universal gas constant 8314 JKmol119877119890
Radius of boundary m119903 Pore radius119879 Temperature at formation condition K119879sc Temperature at standard condition K119906 Coordinate in119882 plane1199060
Distance between boundary and line sink in119882 planeV Coordinate in119882 planeV119892
Gas flow rate msV119906
Flow rate in 119906 direction of119882 planeVV Flow rate in V direction of119882 plane119882 119882 plane119882119891
Fracture width119909 Coordinate in 119885 plane119884 Coordinate in 119885 plane119885 119885 plane119911 Gas compressibility factor
Greek
120572 Rarefaction coefficient120585 Permeability correction factor120582 Gas molecule mean free path m120583 Gas viscosity Pasdots
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportof the National Program on Key Basic Research Project (973Program) (Grant no 2013CB228002) through the effectivedevelopment research of the Southern Marine Shale GasReservoirs in China
References
[1] M Elgmati Shale Gas Rock Characterization and 3D SubmicronPore Network Reconstruction Missouri University of Scienceand Technology Rolla Mo USA 2011
[2] R G Loucks R M Reed S C Ruppel and D M JarvieldquoMorphology genesis and distribution of nanometer-scalepores in siliceousmudstones of themississippian barnett shalerdquoJournal of Sedimentary Research vol 79 no 12 pp 848ndash8612009
[3] C Zou R Zhu B Bai et al ldquoFirst discovery of nano-pore throatin oil and gas reservoir in China and its scientific valuerdquo ActaPetrologica Sinica vol 27 no 6 pp 1857ndash1864 2011
[4] J DengW Zhu and QMa ldquoA new seepage model for shale gasreservoir and productivity analysis of fractured wellrdquo Fuel vol124 pp 232ndash240 2014
[5] F Javadpour D Fisher and M Unsworth ldquoNanoscale gasflow in shale gas sedimentsrdquo Journal of Canadian PetroleumTechnology vol 46 no 10 pp 55ndash61 2007
[6] F P Wang and R M Reed ldquoPore networks and fluid flow in gasshalesrdquo in Proceedings of the SPE Annual Technical Conferenceand Exhibition (ATCE rsquo09) SPE 124253 pp 1550ndash1557 NewOrleans La USA October 2009
[7] T Huang X Guo and F Chen ldquoModeling transient flowbehavior of a multiscale triple porosity model for shale gasreservoirsrdquo Journal of Natural Gas Science and Engineering vol23 pp 33ndash46 2015
[8] T Huang X Guo and F Chen ldquoModeling transient pressurebehavior of a fractured well for shale gas reservoirs based onthe properties of nanoporesrdquo Journal of Natural Gas Science andEngineering vol 23 pp 387ndash398 2015
[9] G G Michel R F Sigal F Civan and D Devegowda ldquoPara-metric investigation of shale gas production considering nano-scale pore size distribution formation factor and non-darcyflow mechanismsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE 147438 Denver Colo USAOctober-November 2011
[10] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Transactions ofthe ASME Journal of Fluids Engineering vol 118 no 3 pp 448ndash456 1996
[11] A Beskok and G E Karniadakis ldquoA model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[12] S Roy R Raju H F Chuang B A Cruden andMMeyyappanldquoModeling gas flow through microchannels and nanoporesrdquoJournal of Applied Physics vol 93 no 8 pp 4870ndash4879 2003
[13] F Civan ldquoEffective correlation of apparent gas permeability intight porous mediardquo Transport in Porous Media vol 82 no 2pp 375ndash384 2010
[14] E A Guggenheim Elements of the Kinetic Theory of GasesPergamon Press Oxford UK 1960
[15] M Knudsen ldquoThe law of the molecular flow and viscosity ofgases moving through tubesrdquo Annals of Physics vol 28 no 1pp 75ndash130 1909
[16] G P Brown A Dinardo G K Cheng and T K SherwoodldquoThe flow of gases in pipes at low pressuresrdquo Journal of AppliedPhysics vol 17 no 10 pp 802ndash813 1946
[17] G J I Igwe ldquoGas transport mechanism and slippage phe-nomenon in porous mediardquo Tech Rep SPE-16479-MS Societyof Petroleum Engineers 1987
[18] F Javadpour ldquoNanopores and apparent permeability of gasflow in mudrocks (shales and siltstone)rdquo Journal of CanadianPetroleum Technology vol 48 no 8 pp 16ndash21 2009
[19] T JiangW Shan and Y Yang ldquoCalculation of stable productioncapability of vertically fractured wellrdquo Petroleum Explorationand Development vol 28 no 2 p 61 2001
[20] Y Wang T Jiang and B Zeng ldquoProductivity performances ofhydraulically fractured gas wellrdquoActa Petrolei Sinica vol 24 no4 pp 65ndash68 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
6 Journal of Chemistry
The boundary conditions of hydraulic fracture are givenas follows
119889119898
119889V= 0 V = 0
119898 = 119898119908
V =120587
2
(33)
Combining with (33) we can get the solution of (32)
119898(119901) = 1198881
119890120582V+ 1198882
119890minus120582V
+ 119898119890
(34)
where the parameters 1198881
1198882
and 120582 are defined as follows
120582 = radic
21198700
119896119891
119882119891
1
ln 2119877119890
119871119891
1198881
= 1198882
=
119898119908
minus 119898119890
119890(1205872)120582
+ 119890minus(1205872)120582
(35)
The gas production of fractured vertical well can be writ-ten as
119876119891
= minus
119896119891
119882119891
ℎ119879sc
119901sc119879
119889119898
119889V
10038161003816100381610038161003816100381610038161003816V=1205872
=
119896119891
119882119891
ℎ119879sc
119901sc119879120582 (119898119890
minus 119898119908
)
119890(1205872)120582
minus 119890minus(1205872)120582
119890(1205872)120582
+ 119890minus(1205872)120582
(36)
Combined with (31) we can get the productivity formulaof fractured vertical well with finite conductivity
119876119891
=
2119896119891
119882119891
ℎ120582119879sc
119901sc119879120583119911tanh 120587120582
2
sdot [
1199012
119890
minus 1199012
119908
2
+
3120587120583119863K161198700
(119901119890
minus 119901119908
)
+
512
91205872
(
3120587120583119863K641198700
)
14
(11990106
119890
minus 11990106
119908
)
+ 4119887 (
3120587120583119863K641198700
)
2
ln119901119890
119901119908
]
(37)
where the first term in the bracket is the gas productioncalculated by Darcy formula which is defined as 119876
119863
5 Results and Discussion
According to the productivity formula of fractured well withfinite conductivity the effects of permeability correctionfactor slippage factor Knudsen diffusion coefficient matrixpermeability fracture half-length and fracture conductivityon productivity of fractured well are analyzed by combiningthe data from a single well of shale gas reservoirs in ChinaThe data for production simulation are presented in Table 3
51 Effect of Permeability Correction Factor on ProductivityFigure 5 presents the effect of permeability correction factoron productivity of fractured well and we can see fromthis figure that permeability correction factor has a great
Table 3 Data of shale gas reservoir for production simulation
Parameter Value UnitsPermeability 119870 0005 mDPorosity 120601 007 mdashFormation temperature 119879 36615 KSurface temperature 119879sc 293 KFormation pressure 119901
119890
28 MPaFormation thickness ℎ 305 mPressure relief radius 119877
119890
400 mGas viscosity 120583 0014 mPasdotsGas compressibility factor 119911 089 mdashRadius of wellbore 119903
119908
01 m
120585 = 1 + 4Kn + 512151205872Kn14
120585 = 1 + 4Kn + 512151205872Kn14 + 4bKn2
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
Field data
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
120585 = 1 (calculated by Darcy formula)120585 = 1 + 4Kn
Figure 5 Effect of permeability correction factor 120585 on productivity
influence on productivity Production of fractured well isvery small when our model only considers Darcy flow innanomicroscale pores (120585 = 1) but gas production increasesa lot when matrix apparent permeability is corrected bypermeability correction factor 120585 Field data shows that gasproduction of a fractured vertical well is 12 times 104m3 whenproduction pressure drop keeps 7MPa and fracture half-length is 200m [4] The calculation results of multiscalenon-Darcy seepage model are much more close to thefield data compared with the production rate calculated byDarcy formula andproduction rate calculated by productivityformula only considering first order term of permeabilitycorrection factor
52 Effects of Formation and Fracture Parameters on Produc-tivity The effect of matrix permeability 119870 on productivityof fractured well is illustrated in Figure 6 Gas productionof fractured well increases with the matrix permeabilityand gas production increases to 171 times 232 times and287 times when matrix permeability increases to 2 times 3times and 4 times respectively Figure 7 reflects productivity
Journal of Chemistry 7
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
K = 0005mDK = 0010mD
K = 0015mDK = 0020mD
Gas production rate (times104 m3d)
Figure 6 Effect of matrix permeability 119870 on productivity
0
5
10
15
20
25
30
0 05 1 15 2 25 3
Well
bore
pre
ssur
e (M
Pa)
Gas production rate (times104 m3d)
Lf = 50mLf = 150m
Lf = 250mLf = 350m
Figure 7 Effect of fracture half-length 119871119891
on productivity
of fractured well affected by fracture half-length Gas pro-duction increases to 154 times 206 times and 265 timeswhen fracture half-length increases to 3 times 4 times and5 times respectively We can find that the increase of gasproduction rate affected by matrix permeability is still morethan production rate affected by fracture half-length even ifthe increase ratio of fracture half-length is bigger thanmatrixpermeability which means that longer fracture half-lengthonly can improve local seepage ability of formation adjacentto hydraulic fractures but improvement of flow capacityin the whole formation can increase the gas productionsubstantially
Effect of fracture conductivity on productivity of frac-tured well has been analyzed under different fracture pene-tration ratio (119871
119891
119877119890
) We can see from Figure 8 that gas pro-duction increases with the fracture conductivity and fracturepenetration ratio But gas production increase becomes slowwhen the fracture conductivity increases to a certain valueand this value becomes bigger when fracture penetrationratio increases For examples this value will be 2 120583m2sdotcmwhen fracture penetration ratio is 01 but this value willincrease to 4120583m2sdotcm when fracture penetration ratio is 03
0
05
1
15
2
25
3
0 1 2 3 4 5 6
Gas
pro
duct
ion
rate
(times10
4m
3d
)
LfRe = 01
LfRe = 03LfRe = 05LfRe = 07
kf middot Wf (120583m2middotcm)
Figure 8 Effect of fracture conductivity 119896119891
sdot 119882119891
on productivity
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 1 2 3 4 5
Gas production rate (times104 m3d)
b = 0b = 1
b = 2
b = 3
Figure 9 Effect of slippage factor 119887 on productivity
This chart can help to optimize fracture conductivity underdifferent fracture penetration ratio
53 Effects of Slippage and Knudsen Diffusion on ProductivityThe relationship between wellbore pressure of fractured wellwith finite conductivity and gas production under differentslippage factor 119887 has been shown in Figure 9 Slippage effecthas negligible influence on productivity of fractured wellwhen wellbore pressure is high but slippage effect begins toaffect gas production when wellbore pressure is lower than15MPa and productivity of the fractured well increases withthe slippage factor So decreasing the wellbore pressure offractured well properly can improve gas production whenexploiting shale gas reservoirs We can see from Figure 10that gas production rate increases with Knudsen diffusioncoefficient which has a greater impact on gas production thanslippage effect
As we can see in Figure 11 the productivity of fracturedwell derived in this paper (119876
119891
) has been compared withproduction calculated by Darcy formula (119876
119863
) under differ-ent matrix permeability and wellbore pressure Figure 11(a)shows that bigger matrix permeability and a lower wellbore
8 Journal of Chemistry
Table 4 Comparison of production at turning point and production at minimum permeability under different wellbore pressure (the gasproduction rate unit times104 m3d)
119875119908
MPa 1 3 5 7 9 11 13 15 17 19 21 23 25 27119876119870min 0505 0333 0250 0197 0158 0127 0103 0082 0065 0050 0037 0025 0014 0005
119876TP 0315 0238 0193 0160 0134 0112 0093 0076 0061 0048 0035 0024 0014 0004119876119870min119876TP 1601 1398 1296 1228 1178 1138 1109 1082 1064 1046 1034 1024 1014 1010
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
DK = 1 times 10minus6 m2sDK = 3 times 10minus6 m2s
DK = 5 times 10minus6 m2sDK = 7 times 10minus6 m2s
Figure 10 Effect of Knudsen diffusion coefficient119863K on productiv-ity
pressure lead to a higher productivity of fractured well andgas production 119876
119891
is much bigger than Darcy productionrate 119876
119863
The effects of Knudsen diffusion and slip flow onproductivity of fractured well can be observed by taking theratio of119876
119891
and119876119863
in Figure 11(b)The effects on gas produc-tion increase whenmatrix permeability becomes smaller andwellbore pressure declines which become most significantunder the smallest matrix permeability (1 times 10minus5mD) and thelowest wellbore pressure (1MPa) in Figure 11(b)
Figure 12 represents the relationship of gas productionrate119876
119891
and matrix permeability119870 at different wellbore pres-sure For conventional reservoirs production rate decreaseswith formation permeability for shale gas reservoirs gasproduction rate decreases with matrix permeability first butthen increases with permeability owing to the effects ofKnudsen diffusion and slip flow after a certain permeabilityvalue which is called turning point in this paperThe value ofturning point decreases when wellbore pressure rises whichis because Knudsen diffusion and slippage effect are morelikely to occur in formation with smaller permeability andlower pressure so a smaller turning point would be requiredwhen pressure rises
Table 4 shows the ratio of gas production rate at min-imum permeability 119876
119870min in Figure 12 and production atturning point under different wellbore pressures 119876TP Theratios decrease when wellbore pressure increases whichmeans that Knudsen diffusion and slippage effects have agreater influence on productivity of fractured well under alower wellbore pressure But the gas production rate119876
119891
(119870 =1 times 10minus5mD 119875
119908
= 1MPa) is only 16 times the production
119876119891
at turning point (Table 4) even though the production119876119891
is almost 150 times the Darcy production rate 119876119863
in Figure 11(b) Knudsen diffusion and slippage effects canimprove the productivity at extremely low permeabilitysignificantly compared with gas production calculated byDarcy formula (without considering effects of diffusion andslip flow) but the improvement of productivity is still smallcompared with higher permeability when Knudsen diffusionand slippage effects are considered in both situations
6 Conclusions
In this paper a multiscale comprehensive mathematicalmodel had been established which can simulate differentflow regimes including continuum flow slip flow transitionflow and free-molecule flow We deduced the productivityformula for fractured well with limited conductivity for shalegas reservoirs and the influencing parameters were analyzedthoroughly The following conclusions can be drawn
(1) Flow regimes in shale gas reservoir were analyzedby calculating Knudsen number under different pres-sure and different pore throat diameter Gas flow inshale gas reservoirs with nanomicroscale pores is acombination of continuum flow slip flow transitionflow and free-molecule flow which cannot simply berepresented by Darcy formula anymore
(2) A new non-Darcy equation was developed based onBeskok-Karniadakis equation which is applicable forall flow regimes and effects of high order terms ofBK equation on permeability correction factor wereanalyzed under different Knudsen number Perme-ability correction factor is increasingly deviating froma Darcy type as Knudsen number increases
(3) Productivity formula of fractured well satisfying vari-able mass flowing in fractures was obtained andthe simulated results showed that production ratecalculated by non-Darcy seepage model with higherorder terms of permeability correction factor is moreclose to the field production data compared withthe production rate calculated by Darcy formula andproduction rate calculated by productivity formulaonly considering first order term of permeabilitycorrection factor
(4) Effects of slippage and diffusion which are changingwith pressure and pore sizes in nanoscale pores wereconsidered in simulation Simulation results showedthat slippage effect affects gas production of fracturedwell only when wellbore pressure less than 15MPa
Journal of Chemistry 9
010
20300
05
1
15
2
Matrix permeability (mD)Wellbore pressure (MPa)
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
QD
Qf
times104
Prod
uctio
n (m
3d
)
(a) Comparison between 119876119863
and 119876119891
(at different permeability andwellbore pressure)
010
20300
50
100
150
Matrix permeability (mD)Wellbore pressure (MPa)
Prod
uctio
n co
mpa
rison
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
(b) The value of 119876119891
119876
119863
at different matrix permeability and wellborepressure
Figure 11 Effects of Knudsen diffusion coefficient119863K and slip factor 119887 on productivity
0
04
08
12
16
2
Turning point
Effects of slip flow andKnudsen diffusion
Matrix permeability (mD)1E minus 5 1E minus 4 1E minus 3 1E minus 2
Pw = 1Pw = 3Pw = 5Pw = 7Pw = 9Pw = 11
(MPa
)
Pw = 13
Pw = 15
Pw = 17
Pw = 19
Pw = 21
Pw = 23
Pw = 25
Pw = 27
Gas
pro
duct
ion
rate
(times10
4m
3d
)
Figure 12 The relationship between gas production and matrix permeability at different wellbore pressure
and the effects of slippage and diffusion have a greaterinfluence on gas production of reservoirs with smallermatrix permeability especially when permeability isat nano-Darcy scale Matrix permeability fracturehalf length fracture penetration ratio and fractureconductivity can improve gas production at differentdegrees as well
Nomenclature
Latin
119887 Slippage coefficient119861119892
Gas volume factor119863K Knudsen diffusion coefficient m2s119891 Fraction of molecules striking pore wall which are
diffusely reflected
ℎ Thickness of shale gas reservoir m119870 Apparent permeabilityKn Knudsen number119896119891
Permeability of hydraulic fracture1198700
Absolute permeability119871119891
Fracture half length119872 Molecular mass kgmol119898 Pseudopressure119898119890
Initial pseudopressure119898119908
Pseudopressure of fractured well119901 Pressure Pa119901119890
Formation pressure Pa119901119908
Wellbore pressure Pa119901avg Average pressure Pa119901sc Pressure at standard condition Pa119876119863
Volume flow rate calculated by Darcy formula m3s
10 Journal of Chemistry
119876119891
Volume flow rate of fractured well with finiteconductivity m3s
119876TP Volume flow rate of fractured well at turning pointunder different matrix permeability 104m3d
119876V Volume flow rate of fractured well with infiniteconductivity m3s
119902V Volume flow rate m3s119877 Universal gas constant 8314 JKmol119877119890
Radius of boundary m119903 Pore radius119879 Temperature at formation condition K119879sc Temperature at standard condition K119906 Coordinate in119882 plane1199060
Distance between boundary and line sink in119882 planeV Coordinate in119882 planeV119892
Gas flow rate msV119906
Flow rate in 119906 direction of119882 planeVV Flow rate in V direction of119882 plane119882 119882 plane119882119891
Fracture width119909 Coordinate in 119885 plane119884 Coordinate in 119885 plane119885 119885 plane119911 Gas compressibility factor
Greek
120572 Rarefaction coefficient120585 Permeability correction factor120582 Gas molecule mean free path m120583 Gas viscosity Pasdots
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportof the National Program on Key Basic Research Project (973Program) (Grant no 2013CB228002) through the effectivedevelopment research of the Southern Marine Shale GasReservoirs in China
References
[1] M Elgmati Shale Gas Rock Characterization and 3D SubmicronPore Network Reconstruction Missouri University of Scienceand Technology Rolla Mo USA 2011
[2] R G Loucks R M Reed S C Ruppel and D M JarvieldquoMorphology genesis and distribution of nanometer-scalepores in siliceousmudstones of themississippian barnett shalerdquoJournal of Sedimentary Research vol 79 no 12 pp 848ndash8612009
[3] C Zou R Zhu B Bai et al ldquoFirst discovery of nano-pore throatin oil and gas reservoir in China and its scientific valuerdquo ActaPetrologica Sinica vol 27 no 6 pp 1857ndash1864 2011
[4] J DengW Zhu and QMa ldquoA new seepage model for shale gasreservoir and productivity analysis of fractured wellrdquo Fuel vol124 pp 232ndash240 2014
[5] F Javadpour D Fisher and M Unsworth ldquoNanoscale gasflow in shale gas sedimentsrdquo Journal of Canadian PetroleumTechnology vol 46 no 10 pp 55ndash61 2007
[6] F P Wang and R M Reed ldquoPore networks and fluid flow in gasshalesrdquo in Proceedings of the SPE Annual Technical Conferenceand Exhibition (ATCE rsquo09) SPE 124253 pp 1550ndash1557 NewOrleans La USA October 2009
[7] T Huang X Guo and F Chen ldquoModeling transient flowbehavior of a multiscale triple porosity model for shale gasreservoirsrdquo Journal of Natural Gas Science and Engineering vol23 pp 33ndash46 2015
[8] T Huang X Guo and F Chen ldquoModeling transient pressurebehavior of a fractured well for shale gas reservoirs based onthe properties of nanoporesrdquo Journal of Natural Gas Science andEngineering vol 23 pp 387ndash398 2015
[9] G G Michel R F Sigal F Civan and D Devegowda ldquoPara-metric investigation of shale gas production considering nano-scale pore size distribution formation factor and non-darcyflow mechanismsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE 147438 Denver Colo USAOctober-November 2011
[10] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Transactions ofthe ASME Journal of Fluids Engineering vol 118 no 3 pp 448ndash456 1996
[11] A Beskok and G E Karniadakis ldquoA model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[12] S Roy R Raju H F Chuang B A Cruden andMMeyyappanldquoModeling gas flow through microchannels and nanoporesrdquoJournal of Applied Physics vol 93 no 8 pp 4870ndash4879 2003
[13] F Civan ldquoEffective correlation of apparent gas permeability intight porous mediardquo Transport in Porous Media vol 82 no 2pp 375ndash384 2010
[14] E A Guggenheim Elements of the Kinetic Theory of GasesPergamon Press Oxford UK 1960
[15] M Knudsen ldquoThe law of the molecular flow and viscosity ofgases moving through tubesrdquo Annals of Physics vol 28 no 1pp 75ndash130 1909
[16] G P Brown A Dinardo G K Cheng and T K SherwoodldquoThe flow of gases in pipes at low pressuresrdquo Journal of AppliedPhysics vol 17 no 10 pp 802ndash813 1946
[17] G J I Igwe ldquoGas transport mechanism and slippage phe-nomenon in porous mediardquo Tech Rep SPE-16479-MS Societyof Petroleum Engineers 1987
[18] F Javadpour ldquoNanopores and apparent permeability of gasflow in mudrocks (shales and siltstone)rdquo Journal of CanadianPetroleum Technology vol 48 no 8 pp 16ndash21 2009
[19] T JiangW Shan and Y Yang ldquoCalculation of stable productioncapability of vertically fractured wellrdquo Petroleum Explorationand Development vol 28 no 2 p 61 2001
[20] Y Wang T Jiang and B Zeng ldquoProductivity performances ofhydraulically fractured gas wellrdquoActa Petrolei Sinica vol 24 no4 pp 65ndash68 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Journal of Chemistry 7
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
K = 0005mDK = 0010mD
K = 0015mDK = 0020mD
Gas production rate (times104 m3d)
Figure 6 Effect of matrix permeability 119870 on productivity
0
5
10
15
20
25
30
0 05 1 15 2 25 3
Well
bore
pre
ssur
e (M
Pa)
Gas production rate (times104 m3d)
Lf = 50mLf = 150m
Lf = 250mLf = 350m
Figure 7 Effect of fracture half-length 119871119891
on productivity
of fractured well affected by fracture half-length Gas pro-duction increases to 154 times 206 times and 265 timeswhen fracture half-length increases to 3 times 4 times and5 times respectively We can find that the increase of gasproduction rate affected by matrix permeability is still morethan production rate affected by fracture half-length even ifthe increase ratio of fracture half-length is bigger thanmatrixpermeability which means that longer fracture half-lengthonly can improve local seepage ability of formation adjacentto hydraulic fractures but improvement of flow capacityin the whole formation can increase the gas productionsubstantially
Effect of fracture conductivity on productivity of frac-tured well has been analyzed under different fracture pene-tration ratio (119871
119891
119877119890
) We can see from Figure 8 that gas pro-duction increases with the fracture conductivity and fracturepenetration ratio But gas production increase becomes slowwhen the fracture conductivity increases to a certain valueand this value becomes bigger when fracture penetrationratio increases For examples this value will be 2 120583m2sdotcmwhen fracture penetration ratio is 01 but this value willincrease to 4120583m2sdotcm when fracture penetration ratio is 03
0
05
1
15
2
25
3
0 1 2 3 4 5 6
Gas
pro
duct
ion
rate
(times10
4m
3d
)
LfRe = 01
LfRe = 03LfRe = 05LfRe = 07
kf middot Wf (120583m2middotcm)
Figure 8 Effect of fracture conductivity 119896119891
sdot 119882119891
on productivity
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 1 2 3 4 5
Gas production rate (times104 m3d)
b = 0b = 1
b = 2
b = 3
Figure 9 Effect of slippage factor 119887 on productivity
This chart can help to optimize fracture conductivity underdifferent fracture penetration ratio
53 Effects of Slippage and Knudsen Diffusion on ProductivityThe relationship between wellbore pressure of fractured wellwith finite conductivity and gas production under differentslippage factor 119887 has been shown in Figure 9 Slippage effecthas negligible influence on productivity of fractured wellwhen wellbore pressure is high but slippage effect begins toaffect gas production when wellbore pressure is lower than15MPa and productivity of the fractured well increases withthe slippage factor So decreasing the wellbore pressure offractured well properly can improve gas production whenexploiting shale gas reservoirs We can see from Figure 10that gas production rate increases with Knudsen diffusioncoefficient which has a greater impact on gas production thanslippage effect
As we can see in Figure 11 the productivity of fracturedwell derived in this paper (119876
119891
) has been compared withproduction calculated by Darcy formula (119876
119863
) under differ-ent matrix permeability and wellbore pressure Figure 11(a)shows that bigger matrix permeability and a lower wellbore
8 Journal of Chemistry
Table 4 Comparison of production at turning point and production at minimum permeability under different wellbore pressure (the gasproduction rate unit times104 m3d)
119875119908
MPa 1 3 5 7 9 11 13 15 17 19 21 23 25 27119876119870min 0505 0333 0250 0197 0158 0127 0103 0082 0065 0050 0037 0025 0014 0005
119876TP 0315 0238 0193 0160 0134 0112 0093 0076 0061 0048 0035 0024 0014 0004119876119870min119876TP 1601 1398 1296 1228 1178 1138 1109 1082 1064 1046 1034 1024 1014 1010
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
DK = 1 times 10minus6 m2sDK = 3 times 10minus6 m2s
DK = 5 times 10minus6 m2sDK = 7 times 10minus6 m2s
Figure 10 Effect of Knudsen diffusion coefficient119863K on productiv-ity
pressure lead to a higher productivity of fractured well andgas production 119876
119891
is much bigger than Darcy productionrate 119876
119863
The effects of Knudsen diffusion and slip flow onproductivity of fractured well can be observed by taking theratio of119876
119891
and119876119863
in Figure 11(b)The effects on gas produc-tion increase whenmatrix permeability becomes smaller andwellbore pressure declines which become most significantunder the smallest matrix permeability (1 times 10minus5mD) and thelowest wellbore pressure (1MPa) in Figure 11(b)
Figure 12 represents the relationship of gas productionrate119876
119891
and matrix permeability119870 at different wellbore pres-sure For conventional reservoirs production rate decreaseswith formation permeability for shale gas reservoirs gasproduction rate decreases with matrix permeability first butthen increases with permeability owing to the effects ofKnudsen diffusion and slip flow after a certain permeabilityvalue which is called turning point in this paperThe value ofturning point decreases when wellbore pressure rises whichis because Knudsen diffusion and slippage effect are morelikely to occur in formation with smaller permeability andlower pressure so a smaller turning point would be requiredwhen pressure rises
Table 4 shows the ratio of gas production rate at min-imum permeability 119876
119870min in Figure 12 and production atturning point under different wellbore pressures 119876TP Theratios decrease when wellbore pressure increases whichmeans that Knudsen diffusion and slippage effects have agreater influence on productivity of fractured well under alower wellbore pressure But the gas production rate119876
119891
(119870 =1 times 10minus5mD 119875
119908
= 1MPa) is only 16 times the production
119876119891
at turning point (Table 4) even though the production119876119891
is almost 150 times the Darcy production rate 119876119863
in Figure 11(b) Knudsen diffusion and slippage effects canimprove the productivity at extremely low permeabilitysignificantly compared with gas production calculated byDarcy formula (without considering effects of diffusion andslip flow) but the improvement of productivity is still smallcompared with higher permeability when Knudsen diffusionand slippage effects are considered in both situations
6 Conclusions
In this paper a multiscale comprehensive mathematicalmodel had been established which can simulate differentflow regimes including continuum flow slip flow transitionflow and free-molecule flow We deduced the productivityformula for fractured well with limited conductivity for shalegas reservoirs and the influencing parameters were analyzedthoroughly The following conclusions can be drawn
(1) Flow regimes in shale gas reservoir were analyzedby calculating Knudsen number under different pres-sure and different pore throat diameter Gas flow inshale gas reservoirs with nanomicroscale pores is acombination of continuum flow slip flow transitionflow and free-molecule flow which cannot simply berepresented by Darcy formula anymore
(2) A new non-Darcy equation was developed based onBeskok-Karniadakis equation which is applicable forall flow regimes and effects of high order terms ofBK equation on permeability correction factor wereanalyzed under different Knudsen number Perme-ability correction factor is increasingly deviating froma Darcy type as Knudsen number increases
(3) Productivity formula of fractured well satisfying vari-able mass flowing in fractures was obtained andthe simulated results showed that production ratecalculated by non-Darcy seepage model with higherorder terms of permeability correction factor is moreclose to the field production data compared withthe production rate calculated by Darcy formula andproduction rate calculated by productivity formulaonly considering first order term of permeabilitycorrection factor
(4) Effects of slippage and diffusion which are changingwith pressure and pore sizes in nanoscale pores wereconsidered in simulation Simulation results showedthat slippage effect affects gas production of fracturedwell only when wellbore pressure less than 15MPa
Journal of Chemistry 9
010
20300
05
1
15
2
Matrix permeability (mD)Wellbore pressure (MPa)
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
QD
Qf
times104
Prod
uctio
n (m
3d
)
(a) Comparison between 119876119863
and 119876119891
(at different permeability andwellbore pressure)
010
20300
50
100
150
Matrix permeability (mD)Wellbore pressure (MPa)
Prod
uctio
n co
mpa
rison
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
(b) The value of 119876119891
119876
119863
at different matrix permeability and wellborepressure
Figure 11 Effects of Knudsen diffusion coefficient119863K and slip factor 119887 on productivity
0
04
08
12
16
2
Turning point
Effects of slip flow andKnudsen diffusion
Matrix permeability (mD)1E minus 5 1E minus 4 1E minus 3 1E minus 2
Pw = 1Pw = 3Pw = 5Pw = 7Pw = 9Pw = 11
(MPa
)
Pw = 13
Pw = 15
Pw = 17
Pw = 19
Pw = 21
Pw = 23
Pw = 25
Pw = 27
Gas
pro
duct
ion
rate
(times10
4m
3d
)
Figure 12 The relationship between gas production and matrix permeability at different wellbore pressure
and the effects of slippage and diffusion have a greaterinfluence on gas production of reservoirs with smallermatrix permeability especially when permeability isat nano-Darcy scale Matrix permeability fracturehalf length fracture penetration ratio and fractureconductivity can improve gas production at differentdegrees as well
Nomenclature
Latin
119887 Slippage coefficient119861119892
Gas volume factor119863K Knudsen diffusion coefficient m2s119891 Fraction of molecules striking pore wall which are
diffusely reflected
ℎ Thickness of shale gas reservoir m119870 Apparent permeabilityKn Knudsen number119896119891
Permeability of hydraulic fracture1198700
Absolute permeability119871119891
Fracture half length119872 Molecular mass kgmol119898 Pseudopressure119898119890
Initial pseudopressure119898119908
Pseudopressure of fractured well119901 Pressure Pa119901119890
Formation pressure Pa119901119908
Wellbore pressure Pa119901avg Average pressure Pa119901sc Pressure at standard condition Pa119876119863
Volume flow rate calculated by Darcy formula m3s
10 Journal of Chemistry
119876119891
Volume flow rate of fractured well with finiteconductivity m3s
119876TP Volume flow rate of fractured well at turning pointunder different matrix permeability 104m3d
119876V Volume flow rate of fractured well with infiniteconductivity m3s
119902V Volume flow rate m3s119877 Universal gas constant 8314 JKmol119877119890
Radius of boundary m119903 Pore radius119879 Temperature at formation condition K119879sc Temperature at standard condition K119906 Coordinate in119882 plane1199060
Distance between boundary and line sink in119882 planeV Coordinate in119882 planeV119892
Gas flow rate msV119906
Flow rate in 119906 direction of119882 planeVV Flow rate in V direction of119882 plane119882 119882 plane119882119891
Fracture width119909 Coordinate in 119885 plane119884 Coordinate in 119885 plane119885 119885 plane119911 Gas compressibility factor
Greek
120572 Rarefaction coefficient120585 Permeability correction factor120582 Gas molecule mean free path m120583 Gas viscosity Pasdots
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportof the National Program on Key Basic Research Project (973Program) (Grant no 2013CB228002) through the effectivedevelopment research of the Southern Marine Shale GasReservoirs in China
References
[1] M Elgmati Shale Gas Rock Characterization and 3D SubmicronPore Network Reconstruction Missouri University of Scienceand Technology Rolla Mo USA 2011
[2] R G Loucks R M Reed S C Ruppel and D M JarvieldquoMorphology genesis and distribution of nanometer-scalepores in siliceousmudstones of themississippian barnett shalerdquoJournal of Sedimentary Research vol 79 no 12 pp 848ndash8612009
[3] C Zou R Zhu B Bai et al ldquoFirst discovery of nano-pore throatin oil and gas reservoir in China and its scientific valuerdquo ActaPetrologica Sinica vol 27 no 6 pp 1857ndash1864 2011
[4] J DengW Zhu and QMa ldquoA new seepage model for shale gasreservoir and productivity analysis of fractured wellrdquo Fuel vol124 pp 232ndash240 2014
[5] F Javadpour D Fisher and M Unsworth ldquoNanoscale gasflow in shale gas sedimentsrdquo Journal of Canadian PetroleumTechnology vol 46 no 10 pp 55ndash61 2007
[6] F P Wang and R M Reed ldquoPore networks and fluid flow in gasshalesrdquo in Proceedings of the SPE Annual Technical Conferenceand Exhibition (ATCE rsquo09) SPE 124253 pp 1550ndash1557 NewOrleans La USA October 2009
[7] T Huang X Guo and F Chen ldquoModeling transient flowbehavior of a multiscale triple porosity model for shale gasreservoirsrdquo Journal of Natural Gas Science and Engineering vol23 pp 33ndash46 2015
[8] T Huang X Guo and F Chen ldquoModeling transient pressurebehavior of a fractured well for shale gas reservoirs based onthe properties of nanoporesrdquo Journal of Natural Gas Science andEngineering vol 23 pp 387ndash398 2015
[9] G G Michel R F Sigal F Civan and D Devegowda ldquoPara-metric investigation of shale gas production considering nano-scale pore size distribution formation factor and non-darcyflow mechanismsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE 147438 Denver Colo USAOctober-November 2011
[10] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Transactions ofthe ASME Journal of Fluids Engineering vol 118 no 3 pp 448ndash456 1996
[11] A Beskok and G E Karniadakis ldquoA model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[12] S Roy R Raju H F Chuang B A Cruden andMMeyyappanldquoModeling gas flow through microchannels and nanoporesrdquoJournal of Applied Physics vol 93 no 8 pp 4870ndash4879 2003
[13] F Civan ldquoEffective correlation of apparent gas permeability intight porous mediardquo Transport in Porous Media vol 82 no 2pp 375ndash384 2010
[14] E A Guggenheim Elements of the Kinetic Theory of GasesPergamon Press Oxford UK 1960
[15] M Knudsen ldquoThe law of the molecular flow and viscosity ofgases moving through tubesrdquo Annals of Physics vol 28 no 1pp 75ndash130 1909
[16] G P Brown A Dinardo G K Cheng and T K SherwoodldquoThe flow of gases in pipes at low pressuresrdquo Journal of AppliedPhysics vol 17 no 10 pp 802ndash813 1946
[17] G J I Igwe ldquoGas transport mechanism and slippage phe-nomenon in porous mediardquo Tech Rep SPE-16479-MS Societyof Petroleum Engineers 1987
[18] F Javadpour ldquoNanopores and apparent permeability of gasflow in mudrocks (shales and siltstone)rdquo Journal of CanadianPetroleum Technology vol 48 no 8 pp 16ndash21 2009
[19] T JiangW Shan and Y Yang ldquoCalculation of stable productioncapability of vertically fractured wellrdquo Petroleum Explorationand Development vol 28 no 2 p 61 2001
[20] Y Wang T Jiang and B Zeng ldquoProductivity performances ofhydraulically fractured gas wellrdquoActa Petrolei Sinica vol 24 no4 pp 65ndash68 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
8 Journal of Chemistry
Table 4 Comparison of production at turning point and production at minimum permeability under different wellbore pressure (the gasproduction rate unit times104 m3d)
119875119908
MPa 1 3 5 7 9 11 13 15 17 19 21 23 25 27119876119870min 0505 0333 0250 0197 0158 0127 0103 0082 0065 0050 0037 0025 0014 0005
119876TP 0315 0238 0193 0160 0134 0112 0093 0076 0061 0048 0035 0024 0014 0004119876119870min119876TP 1601 1398 1296 1228 1178 1138 1109 1082 1064 1046 1034 1024 1014 1010
0
5
10
15
20
25
30
Well
bore
pre
ssur
e (M
Pa)
0 05 1 15 2 25 3 35 4
Gas production rate (times104 m3d)
DK = 1 times 10minus6 m2sDK = 3 times 10minus6 m2s
DK = 5 times 10minus6 m2sDK = 7 times 10minus6 m2s
Figure 10 Effect of Knudsen diffusion coefficient119863K on productiv-ity
pressure lead to a higher productivity of fractured well andgas production 119876
119891
is much bigger than Darcy productionrate 119876
119863
The effects of Knudsen diffusion and slip flow onproductivity of fractured well can be observed by taking theratio of119876
119891
and119876119863
in Figure 11(b)The effects on gas produc-tion increase whenmatrix permeability becomes smaller andwellbore pressure declines which become most significantunder the smallest matrix permeability (1 times 10minus5mD) and thelowest wellbore pressure (1MPa) in Figure 11(b)
Figure 12 represents the relationship of gas productionrate119876
119891
and matrix permeability119870 at different wellbore pres-sure For conventional reservoirs production rate decreaseswith formation permeability for shale gas reservoirs gasproduction rate decreases with matrix permeability first butthen increases with permeability owing to the effects ofKnudsen diffusion and slip flow after a certain permeabilityvalue which is called turning point in this paperThe value ofturning point decreases when wellbore pressure rises whichis because Knudsen diffusion and slippage effect are morelikely to occur in formation with smaller permeability andlower pressure so a smaller turning point would be requiredwhen pressure rises
Table 4 shows the ratio of gas production rate at min-imum permeability 119876
119870min in Figure 12 and production atturning point under different wellbore pressures 119876TP Theratios decrease when wellbore pressure increases whichmeans that Knudsen diffusion and slippage effects have agreater influence on productivity of fractured well under alower wellbore pressure But the gas production rate119876
119891
(119870 =1 times 10minus5mD 119875
119908
= 1MPa) is only 16 times the production
119876119891
at turning point (Table 4) even though the production119876119891
is almost 150 times the Darcy production rate 119876119863
in Figure 11(b) Knudsen diffusion and slippage effects canimprove the productivity at extremely low permeabilitysignificantly compared with gas production calculated byDarcy formula (without considering effects of diffusion andslip flow) but the improvement of productivity is still smallcompared with higher permeability when Knudsen diffusionand slippage effects are considered in both situations
6 Conclusions
In this paper a multiscale comprehensive mathematicalmodel had been established which can simulate differentflow regimes including continuum flow slip flow transitionflow and free-molecule flow We deduced the productivityformula for fractured well with limited conductivity for shalegas reservoirs and the influencing parameters were analyzedthoroughly The following conclusions can be drawn
(1) Flow regimes in shale gas reservoir were analyzedby calculating Knudsen number under different pres-sure and different pore throat diameter Gas flow inshale gas reservoirs with nanomicroscale pores is acombination of continuum flow slip flow transitionflow and free-molecule flow which cannot simply berepresented by Darcy formula anymore
(2) A new non-Darcy equation was developed based onBeskok-Karniadakis equation which is applicable forall flow regimes and effects of high order terms ofBK equation on permeability correction factor wereanalyzed under different Knudsen number Perme-ability correction factor is increasingly deviating froma Darcy type as Knudsen number increases
(3) Productivity formula of fractured well satisfying vari-able mass flowing in fractures was obtained andthe simulated results showed that production ratecalculated by non-Darcy seepage model with higherorder terms of permeability correction factor is moreclose to the field production data compared withthe production rate calculated by Darcy formula andproduction rate calculated by productivity formulaonly considering first order term of permeabilitycorrection factor
(4) Effects of slippage and diffusion which are changingwith pressure and pore sizes in nanoscale pores wereconsidered in simulation Simulation results showedthat slippage effect affects gas production of fracturedwell only when wellbore pressure less than 15MPa
Journal of Chemistry 9
010
20300
05
1
15
2
Matrix permeability (mD)Wellbore pressure (MPa)
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
QD
Qf
times104
Prod
uctio
n (m
3d
)
(a) Comparison between 119876119863
and 119876119891
(at different permeability andwellbore pressure)
010
20300
50
100
150
Matrix permeability (mD)Wellbore pressure (MPa)
Prod
uctio
n co
mpa
rison
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
(b) The value of 119876119891
119876
119863
at different matrix permeability and wellborepressure
Figure 11 Effects of Knudsen diffusion coefficient119863K and slip factor 119887 on productivity
0
04
08
12
16
2
Turning point
Effects of slip flow andKnudsen diffusion
Matrix permeability (mD)1E minus 5 1E minus 4 1E minus 3 1E minus 2
Pw = 1Pw = 3Pw = 5Pw = 7Pw = 9Pw = 11
(MPa
)
Pw = 13
Pw = 15
Pw = 17
Pw = 19
Pw = 21
Pw = 23
Pw = 25
Pw = 27
Gas
pro
duct
ion
rate
(times10
4m
3d
)
Figure 12 The relationship between gas production and matrix permeability at different wellbore pressure
and the effects of slippage and diffusion have a greaterinfluence on gas production of reservoirs with smallermatrix permeability especially when permeability isat nano-Darcy scale Matrix permeability fracturehalf length fracture penetration ratio and fractureconductivity can improve gas production at differentdegrees as well
Nomenclature
Latin
119887 Slippage coefficient119861119892
Gas volume factor119863K Knudsen diffusion coefficient m2s119891 Fraction of molecules striking pore wall which are
diffusely reflected
ℎ Thickness of shale gas reservoir m119870 Apparent permeabilityKn Knudsen number119896119891
Permeability of hydraulic fracture1198700
Absolute permeability119871119891
Fracture half length119872 Molecular mass kgmol119898 Pseudopressure119898119890
Initial pseudopressure119898119908
Pseudopressure of fractured well119901 Pressure Pa119901119890
Formation pressure Pa119901119908
Wellbore pressure Pa119901avg Average pressure Pa119901sc Pressure at standard condition Pa119876119863
Volume flow rate calculated by Darcy formula m3s
10 Journal of Chemistry
119876119891
Volume flow rate of fractured well with finiteconductivity m3s
119876TP Volume flow rate of fractured well at turning pointunder different matrix permeability 104m3d
119876V Volume flow rate of fractured well with infiniteconductivity m3s
119902V Volume flow rate m3s119877 Universal gas constant 8314 JKmol119877119890
Radius of boundary m119903 Pore radius119879 Temperature at formation condition K119879sc Temperature at standard condition K119906 Coordinate in119882 plane1199060
Distance between boundary and line sink in119882 planeV Coordinate in119882 planeV119892
Gas flow rate msV119906
Flow rate in 119906 direction of119882 planeVV Flow rate in V direction of119882 plane119882 119882 plane119882119891
Fracture width119909 Coordinate in 119885 plane119884 Coordinate in 119885 plane119885 119885 plane119911 Gas compressibility factor
Greek
120572 Rarefaction coefficient120585 Permeability correction factor120582 Gas molecule mean free path m120583 Gas viscosity Pasdots
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportof the National Program on Key Basic Research Project (973Program) (Grant no 2013CB228002) through the effectivedevelopment research of the Southern Marine Shale GasReservoirs in China
References
[1] M Elgmati Shale Gas Rock Characterization and 3D SubmicronPore Network Reconstruction Missouri University of Scienceand Technology Rolla Mo USA 2011
[2] R G Loucks R M Reed S C Ruppel and D M JarvieldquoMorphology genesis and distribution of nanometer-scalepores in siliceousmudstones of themississippian barnett shalerdquoJournal of Sedimentary Research vol 79 no 12 pp 848ndash8612009
[3] C Zou R Zhu B Bai et al ldquoFirst discovery of nano-pore throatin oil and gas reservoir in China and its scientific valuerdquo ActaPetrologica Sinica vol 27 no 6 pp 1857ndash1864 2011
[4] J DengW Zhu and QMa ldquoA new seepage model for shale gasreservoir and productivity analysis of fractured wellrdquo Fuel vol124 pp 232ndash240 2014
[5] F Javadpour D Fisher and M Unsworth ldquoNanoscale gasflow in shale gas sedimentsrdquo Journal of Canadian PetroleumTechnology vol 46 no 10 pp 55ndash61 2007
[6] F P Wang and R M Reed ldquoPore networks and fluid flow in gasshalesrdquo in Proceedings of the SPE Annual Technical Conferenceand Exhibition (ATCE rsquo09) SPE 124253 pp 1550ndash1557 NewOrleans La USA October 2009
[7] T Huang X Guo and F Chen ldquoModeling transient flowbehavior of a multiscale triple porosity model for shale gasreservoirsrdquo Journal of Natural Gas Science and Engineering vol23 pp 33ndash46 2015
[8] T Huang X Guo and F Chen ldquoModeling transient pressurebehavior of a fractured well for shale gas reservoirs based onthe properties of nanoporesrdquo Journal of Natural Gas Science andEngineering vol 23 pp 387ndash398 2015
[9] G G Michel R F Sigal F Civan and D Devegowda ldquoPara-metric investigation of shale gas production considering nano-scale pore size distribution formation factor and non-darcyflow mechanismsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE 147438 Denver Colo USAOctober-November 2011
[10] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Transactions ofthe ASME Journal of Fluids Engineering vol 118 no 3 pp 448ndash456 1996
[11] A Beskok and G E Karniadakis ldquoA model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[12] S Roy R Raju H F Chuang B A Cruden andMMeyyappanldquoModeling gas flow through microchannels and nanoporesrdquoJournal of Applied Physics vol 93 no 8 pp 4870ndash4879 2003
[13] F Civan ldquoEffective correlation of apparent gas permeability intight porous mediardquo Transport in Porous Media vol 82 no 2pp 375ndash384 2010
[14] E A Guggenheim Elements of the Kinetic Theory of GasesPergamon Press Oxford UK 1960
[15] M Knudsen ldquoThe law of the molecular flow and viscosity ofgases moving through tubesrdquo Annals of Physics vol 28 no 1pp 75ndash130 1909
[16] G P Brown A Dinardo G K Cheng and T K SherwoodldquoThe flow of gases in pipes at low pressuresrdquo Journal of AppliedPhysics vol 17 no 10 pp 802ndash813 1946
[17] G J I Igwe ldquoGas transport mechanism and slippage phe-nomenon in porous mediardquo Tech Rep SPE-16479-MS Societyof Petroleum Engineers 1987
[18] F Javadpour ldquoNanopores and apparent permeability of gasflow in mudrocks (shales and siltstone)rdquo Journal of CanadianPetroleum Technology vol 48 no 8 pp 16ndash21 2009
[19] T JiangW Shan and Y Yang ldquoCalculation of stable productioncapability of vertically fractured wellrdquo Petroleum Explorationand Development vol 28 no 2 p 61 2001
[20] Y Wang T Jiang and B Zeng ldquoProductivity performances ofhydraulically fractured gas wellrdquoActa Petrolei Sinica vol 24 no4 pp 65ndash68 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Journal of Chemistry 9
010
20300
05
1
15
2
Matrix permeability (mD)Wellbore pressure (MPa)
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
QD
Qf
times104
Prod
uctio
n (m
3d
)
(a) Comparison between 119876119863
and 119876119891
(at different permeability andwellbore pressure)
010
20300
50
100
150
Matrix permeability (mD)Wellbore pressure (MPa)
Prod
uctio
n co
mpa
rison
Effects of slip flow and Knudsen diffusion
10minus210minus3
10minus410minus5
(b) The value of 119876119891
119876
119863
at different matrix permeability and wellborepressure
Figure 11 Effects of Knudsen diffusion coefficient119863K and slip factor 119887 on productivity
0
04
08
12
16
2
Turning point
Effects of slip flow andKnudsen diffusion
Matrix permeability (mD)1E minus 5 1E minus 4 1E minus 3 1E minus 2
Pw = 1Pw = 3Pw = 5Pw = 7Pw = 9Pw = 11
(MPa
)
Pw = 13
Pw = 15
Pw = 17
Pw = 19
Pw = 21
Pw = 23
Pw = 25
Pw = 27
Gas
pro
duct
ion
rate
(times10
4m
3d
)
Figure 12 The relationship between gas production and matrix permeability at different wellbore pressure
and the effects of slippage and diffusion have a greaterinfluence on gas production of reservoirs with smallermatrix permeability especially when permeability isat nano-Darcy scale Matrix permeability fracturehalf length fracture penetration ratio and fractureconductivity can improve gas production at differentdegrees as well
Nomenclature
Latin
119887 Slippage coefficient119861119892
Gas volume factor119863K Knudsen diffusion coefficient m2s119891 Fraction of molecules striking pore wall which are
diffusely reflected
ℎ Thickness of shale gas reservoir m119870 Apparent permeabilityKn Knudsen number119896119891
Permeability of hydraulic fracture1198700
Absolute permeability119871119891
Fracture half length119872 Molecular mass kgmol119898 Pseudopressure119898119890
Initial pseudopressure119898119908
Pseudopressure of fractured well119901 Pressure Pa119901119890
Formation pressure Pa119901119908
Wellbore pressure Pa119901avg Average pressure Pa119901sc Pressure at standard condition Pa119876119863
Volume flow rate calculated by Darcy formula m3s
10 Journal of Chemistry
119876119891
Volume flow rate of fractured well with finiteconductivity m3s
119876TP Volume flow rate of fractured well at turning pointunder different matrix permeability 104m3d
119876V Volume flow rate of fractured well with infiniteconductivity m3s
119902V Volume flow rate m3s119877 Universal gas constant 8314 JKmol119877119890
Radius of boundary m119903 Pore radius119879 Temperature at formation condition K119879sc Temperature at standard condition K119906 Coordinate in119882 plane1199060
Distance between boundary and line sink in119882 planeV Coordinate in119882 planeV119892
Gas flow rate msV119906
Flow rate in 119906 direction of119882 planeVV Flow rate in V direction of119882 plane119882 119882 plane119882119891
Fracture width119909 Coordinate in 119885 plane119884 Coordinate in 119885 plane119885 119885 plane119911 Gas compressibility factor
Greek
120572 Rarefaction coefficient120585 Permeability correction factor120582 Gas molecule mean free path m120583 Gas viscosity Pasdots
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportof the National Program on Key Basic Research Project (973Program) (Grant no 2013CB228002) through the effectivedevelopment research of the Southern Marine Shale GasReservoirs in China
References
[1] M Elgmati Shale Gas Rock Characterization and 3D SubmicronPore Network Reconstruction Missouri University of Scienceand Technology Rolla Mo USA 2011
[2] R G Loucks R M Reed S C Ruppel and D M JarvieldquoMorphology genesis and distribution of nanometer-scalepores in siliceousmudstones of themississippian barnett shalerdquoJournal of Sedimentary Research vol 79 no 12 pp 848ndash8612009
[3] C Zou R Zhu B Bai et al ldquoFirst discovery of nano-pore throatin oil and gas reservoir in China and its scientific valuerdquo ActaPetrologica Sinica vol 27 no 6 pp 1857ndash1864 2011
[4] J DengW Zhu and QMa ldquoA new seepage model for shale gasreservoir and productivity analysis of fractured wellrdquo Fuel vol124 pp 232ndash240 2014
[5] F Javadpour D Fisher and M Unsworth ldquoNanoscale gasflow in shale gas sedimentsrdquo Journal of Canadian PetroleumTechnology vol 46 no 10 pp 55ndash61 2007
[6] F P Wang and R M Reed ldquoPore networks and fluid flow in gasshalesrdquo in Proceedings of the SPE Annual Technical Conferenceand Exhibition (ATCE rsquo09) SPE 124253 pp 1550ndash1557 NewOrleans La USA October 2009
[7] T Huang X Guo and F Chen ldquoModeling transient flowbehavior of a multiscale triple porosity model for shale gasreservoirsrdquo Journal of Natural Gas Science and Engineering vol23 pp 33ndash46 2015
[8] T Huang X Guo and F Chen ldquoModeling transient pressurebehavior of a fractured well for shale gas reservoirs based onthe properties of nanoporesrdquo Journal of Natural Gas Science andEngineering vol 23 pp 387ndash398 2015
[9] G G Michel R F Sigal F Civan and D Devegowda ldquoPara-metric investigation of shale gas production considering nano-scale pore size distribution formation factor and non-darcyflow mechanismsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE 147438 Denver Colo USAOctober-November 2011
[10] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Transactions ofthe ASME Journal of Fluids Engineering vol 118 no 3 pp 448ndash456 1996
[11] A Beskok and G E Karniadakis ldquoA model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[12] S Roy R Raju H F Chuang B A Cruden andMMeyyappanldquoModeling gas flow through microchannels and nanoporesrdquoJournal of Applied Physics vol 93 no 8 pp 4870ndash4879 2003
[13] F Civan ldquoEffective correlation of apparent gas permeability intight porous mediardquo Transport in Porous Media vol 82 no 2pp 375ndash384 2010
[14] E A Guggenheim Elements of the Kinetic Theory of GasesPergamon Press Oxford UK 1960
[15] M Knudsen ldquoThe law of the molecular flow and viscosity ofgases moving through tubesrdquo Annals of Physics vol 28 no 1pp 75ndash130 1909
[16] G P Brown A Dinardo G K Cheng and T K SherwoodldquoThe flow of gases in pipes at low pressuresrdquo Journal of AppliedPhysics vol 17 no 10 pp 802ndash813 1946
[17] G J I Igwe ldquoGas transport mechanism and slippage phe-nomenon in porous mediardquo Tech Rep SPE-16479-MS Societyof Petroleum Engineers 1987
[18] F Javadpour ldquoNanopores and apparent permeability of gasflow in mudrocks (shales and siltstone)rdquo Journal of CanadianPetroleum Technology vol 48 no 8 pp 16ndash21 2009
[19] T JiangW Shan and Y Yang ldquoCalculation of stable productioncapability of vertically fractured wellrdquo Petroleum Explorationand Development vol 28 no 2 p 61 2001
[20] Y Wang T Jiang and B Zeng ldquoProductivity performances ofhydraulically fractured gas wellrdquoActa Petrolei Sinica vol 24 no4 pp 65ndash68 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
10 Journal of Chemistry
119876119891
Volume flow rate of fractured well with finiteconductivity m3s
119876TP Volume flow rate of fractured well at turning pointunder different matrix permeability 104m3d
119876V Volume flow rate of fractured well with infiniteconductivity m3s
119902V Volume flow rate m3s119877 Universal gas constant 8314 JKmol119877119890
Radius of boundary m119903 Pore radius119879 Temperature at formation condition K119879sc Temperature at standard condition K119906 Coordinate in119882 plane1199060
Distance between boundary and line sink in119882 planeV Coordinate in119882 planeV119892
Gas flow rate msV119906
Flow rate in 119906 direction of119882 planeVV Flow rate in V direction of119882 plane119882 119882 plane119882119891
Fracture width119909 Coordinate in 119885 plane119884 Coordinate in 119885 plane119885 119885 plane119911 Gas compressibility factor
Greek
120572 Rarefaction coefficient120585 Permeability correction factor120582 Gas molecule mean free path m120583 Gas viscosity Pasdots
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the financial supportof the National Program on Key Basic Research Project (973Program) (Grant no 2013CB228002) through the effectivedevelopment research of the Southern Marine Shale GasReservoirs in China
References
[1] M Elgmati Shale Gas Rock Characterization and 3D SubmicronPore Network Reconstruction Missouri University of Scienceand Technology Rolla Mo USA 2011
[2] R G Loucks R M Reed S C Ruppel and D M JarvieldquoMorphology genesis and distribution of nanometer-scalepores in siliceousmudstones of themississippian barnett shalerdquoJournal of Sedimentary Research vol 79 no 12 pp 848ndash8612009
[3] C Zou R Zhu B Bai et al ldquoFirst discovery of nano-pore throatin oil and gas reservoir in China and its scientific valuerdquo ActaPetrologica Sinica vol 27 no 6 pp 1857ndash1864 2011
[4] J DengW Zhu and QMa ldquoA new seepage model for shale gasreservoir and productivity analysis of fractured wellrdquo Fuel vol124 pp 232ndash240 2014
[5] F Javadpour D Fisher and M Unsworth ldquoNanoscale gasflow in shale gas sedimentsrdquo Journal of Canadian PetroleumTechnology vol 46 no 10 pp 55ndash61 2007
[6] F P Wang and R M Reed ldquoPore networks and fluid flow in gasshalesrdquo in Proceedings of the SPE Annual Technical Conferenceand Exhibition (ATCE rsquo09) SPE 124253 pp 1550ndash1557 NewOrleans La USA October 2009
[7] T Huang X Guo and F Chen ldquoModeling transient flowbehavior of a multiscale triple porosity model for shale gasreservoirsrdquo Journal of Natural Gas Science and Engineering vol23 pp 33ndash46 2015
[8] T Huang X Guo and F Chen ldquoModeling transient pressurebehavior of a fractured well for shale gas reservoirs based onthe properties of nanoporesrdquo Journal of Natural Gas Science andEngineering vol 23 pp 387ndash398 2015
[9] G G Michel R F Sigal F Civan and D Devegowda ldquoPara-metric investigation of shale gas production considering nano-scale pore size distribution formation factor and non-darcyflow mechanismsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition SPE 147438 Denver Colo USAOctober-November 2011
[10] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Transactions ofthe ASME Journal of Fluids Engineering vol 118 no 3 pp 448ndash456 1996
[11] A Beskok and G E Karniadakis ldquoA model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[12] S Roy R Raju H F Chuang B A Cruden andMMeyyappanldquoModeling gas flow through microchannels and nanoporesrdquoJournal of Applied Physics vol 93 no 8 pp 4870ndash4879 2003
[13] F Civan ldquoEffective correlation of apparent gas permeability intight porous mediardquo Transport in Porous Media vol 82 no 2pp 375ndash384 2010
[14] E A Guggenheim Elements of the Kinetic Theory of GasesPergamon Press Oxford UK 1960
[15] M Knudsen ldquoThe law of the molecular flow and viscosity ofgases moving through tubesrdquo Annals of Physics vol 28 no 1pp 75ndash130 1909
[16] G P Brown A Dinardo G K Cheng and T K SherwoodldquoThe flow of gases in pipes at low pressuresrdquo Journal of AppliedPhysics vol 17 no 10 pp 802ndash813 1946
[17] G J I Igwe ldquoGas transport mechanism and slippage phe-nomenon in porous mediardquo Tech Rep SPE-16479-MS Societyof Petroleum Engineers 1987
[18] F Javadpour ldquoNanopores and apparent permeability of gasflow in mudrocks (shales and siltstone)rdquo Journal of CanadianPetroleum Technology vol 48 no 8 pp 16ndash21 2009
[19] T JiangW Shan and Y Yang ldquoCalculation of stable productioncapability of vertically fractured wellrdquo Petroleum Explorationand Development vol 28 no 2 p 61 2001
[20] Y Wang T Jiang and B Zeng ldquoProductivity performances ofhydraulically fractured gas wellrdquoActa Petrolei Sinica vol 24 no4 pp 65ndash68 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
![[11] Seepage [Rev2]](https://static.fdocuments.net/doc/165x107/55cf9210550346f57b932a1e/11-seepage-rev2.jpg)
