research article numerical simulation of unsteady...
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Research ArticleNumerical Simulation of Unsteady-State Flow inDual Porous Coalbed Methane Horizontal Wells withComplex Boundary Conditions
Cheng-yong Li1 Jun Zhou1 Xiang-yi Yi2 Yi Luo1 and Ping-zhi Gong3
1Chengdu University of Technology Chengdu Sichuan 610059 China2State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Chengdu Sichuan 610059 China3CNOOC China Limited Tianjin Branch Tanggu 300452 China
Correspondence should be addressed to Jun Zhou zhoucdut2012126com
Received 20 November 2014 Revised 26 January 2015 Accepted 27 March 2015
Academic Editor Charles M Drain
Copyright copy 2015 Cheng-yong Li et alThis 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
The bottom-hole pressure response which can reflect the gas flow characteristics is important to study A mathematical modelfor description of gas from porous coalbed methane (CBM) reservoirs with complex boundary conditions flowing into horizontalwells has been developed Meanwhile basic solution of boundary elements has been acquired by combination of Lord Kelvin pointsource solution the integral of Bessel function and Poisson superimpose formula for CBMhorizontal wells with complex boundaryconditions Using this model type curves of dimensionless pressure and pressure derivative are obtained and flow characteristicsof horizontal wells in complex boundary reservoirs and relevant factors are accordingly analyzed
1 Introduction
Coalbed methane (CBM) is a kind of green and clean energyThe development and utilization of coalbed methane couldnot only relieve the tense situation of conventional oil andgas in supply but also reduce the atmospheric environmentpollution
Different from conventional gas reservoirs the migrationmechanism of gas in coal is more complex and diverse [1]Coal is dual porous media reservoir matrix is main storagespace of CBM adsorption and fractures are main transportroutes of CBM diffusion-seepage Analyzing bottom-holepressure helps to figure out CBM production status Ertekinand Sung [2] established a nonequilibrium adsorption non-steady seepage flow model in dual porosity media withHenryrsquos law Applying Fickrsquos law Anbarci and Ertekin [3]suggested a single-phase CBM seepage flow mathematicalmodel considering pseudosteady and unsteady diffusion phe-nomenon Clarkson and Bustin [4 5] put forward a new dou-ble diffusion model which assumes that adsorption occurs
only in micropores and conforms to the law of nonlinearadsorption The macropores accumulate the free gas or pro-vide channels of gas migration between micropores andfractures Reeve [6] proposed a new gas-water two-phasetriple-porosity dual-seepage flow mathematical model andthis model can increase the third matrix pore system how-ever the double-permeability model is very complex anddifficult to describe and calculate Tong et al [7ndash9] introducedpermeability modulus considering the deformation of coaland developed a pseudosteady diffusion nonequilibriumadsorption nonsteady seepage mathematical model Hu et al[10] established a gas-water two-phase percolation mathe-matical model of well test interpretation for CBM reservoirsand its correctness has been verified by simulation of CBMseepage flow characteristics Clarkson et al [11] introducedpseudopressure function to analyze production of gas andwater flow performance using the numerical simulationmethod Aminian and Ameri [12] set up a function to predictgas production based on storage and transport mechanismsin CBM reservoirs Recently Cai and Yu [13] introduced
Hindawi Publishing CorporationJournal of ChemistryVolume 2015 Article ID 173975 11 pageshttpdxdoiorg1011552015173975
2 Journal of Chemistry
the fractal theory to study the enhancing recovery mecha-nism in natural gas-saturated porous media by spontaneousimbibition effect
In order to improve the production a large number ofhorizontal wells have been used in the CBM reservoirs Sungand Ertekin [14ndash16] established a two-dimensional two-phasemultiwell gas-water flowmodel and thismodel has theability to simulatemultiple horizontal wells Engler andRajtar[17ndash19] established a mathematical model of single-phasegas flow in the horizontal wells and the analytical solutionsare given for the horizontal well pressure drop and pressurerecovery Wang et al [20] established a mathematical modelin which anisotropy formation heterogeneity permeabilitystress sensitivity and influence of wellbore pressure drop ondirectional pinnate horizontal wells in CBM reservoirs areconsidered Nie et al [21] deduced CBMflow equations basedon Langmuir adsorption inmatrix andDarcy flow in fractureand analyzed the transient transport characteristics of gasfrom CBM reservoirs to horizontal wells
Generally the theories of calculating reservoir and bot-tom-hole pressures which can reflect the gas flow characteris-tics are mostly based on homogeneous reservoirs and regulargeometry such as infinite boundary or circular boundaryOuter boundary conditions of a reservoir have also beensimplified it is generally regarded as simple situations asconstant pressure or closed boundary However influencedby characteristics of geological structures the true reservoirsusually have complex and diversiform boundaries In thiscase the conventional flow theories and solving methodscould do nothing to calculate reservoir or bottom-hole pres-sures with mixed boundary conditions
The boundary element method (BEM) is a numericalcomputational method of solving linear partial differentialequations developed after the better-known finite elementmethod (FEM) and finite difference method (FDM) TheBEM could be able to reduce dimension and save computermemory and running time Couplingwith the BEM [22] bot-tom-hole pressures and complicated flow characteristicsunder the condition of irregularly shaped area with mixedboundary could be calculated and analyzed Numbere andTiab [23] developed a streamline simulationwith the BEM forhomogeneous or partly homogeneous reservoirs with irreg-ular boundary to make the simulation match physical modelbetter Kikani and Horne [24] employed the BEM to analyzethe transient pressure response in reservoirs with arbitraryboundary and developed two formulas namely convolutionformula and Laplace domain space formula to solve tran-sient fluid flow through porous medium in homogeneousreservoirs Hou et al [25] used the BEM to simulate flowlinemap in homogeneous reservoirs with irregular boundaryChaiyo et al [26] used the BEM to solve free boundary sat-urated seepage problem Rafiezadeh and Ataie-Ashtiani [27]studied the flow mechanism in anisotropic media by three-dimensional boundary elements
In this paper a mathematical model is developed todescribe gas flow in horizontal wells in CBM reservoirs basedon the theory of fluid flow through porousThe type curves of
pressure derivative characteristics of gas flow with complexexternal boundary and relevant affecting factors are ana-lyzed
2 Physical Model of Gas Flow
Coal reservoir is a dual porous medium composed of matrixand fracture The matrix is the main reservoir space ofcoalbed methane (Figure 1(a)) and the fracture is the flowchannel of fluid (Figure 1(b)) The average pore size of CBMreservoirs is much smaller than those of conventional reser-voirs Pore size can be divided into three categories [28]macropore (aperture gt 20 nm) mesopore (2 nm lt aperture lt20 nm) andmicropore (aperture lt 2 nm)The small porosityof coal leads to the great specific surface areaHencemethanecould be strongly adsorbed resulting in the fact that thecontent of CBM is far more than its pore volume
According to the reservoir dual pore structure of coal aphysical model is set up for gas flow The migration of gasin coal is shown in Figure 2 In the production of CBM for-mation pressure keeps declining When formation pressuredrops below the critical desorption pressure CBM starts todesorb from the coal matrix surface Meanwhile the originalstate of equilibrium is broken causing the flow of gas in frac-ture This process is very similar to spontaneous imbibitionin fractured porous media [29]
To facilitate the derivation it is assumed that the length ofa horizontal well is119871 the center location of the horizontal wellis (119909119908 119910119908 119911119908) and the thickness of a CBM reservoir is ℎ In
this paper gas flow into horizontal wells with closed bound-ary (Figure 3(a)) constant pressure boundary (Figure 3(b))and mixed boundary (Figure 3(c)) is considered
The fundamental assumptions are as follows
(1) CBMdiffuses directly frommatrix to fracture and theprocess of diffusion is unsteady
(2) Gas flow in fracture is radial laminar flow in agree-ment with Darcyrsquos law
(3) Only single-phase gas flow exists in coal
(4) The effects of gravity and capillary force are negligible
(5) The effects of temperature are negligible
(6) CBM isothermal adsorption process is in line withthe Langmuir isotherm adsorption law and the initialstate conforms to the isothermal adsorption curve
(7) Radius of the gas well is regarded infinitesimal andgas well production is constant
3 Mathematical Model
31 Gas Diffusion Model in Matrix In combination with themass conservation equation with the second Fickrsquos diffusionlaw the change of gas concentration with time is given by
120601
120597 (119888)
120597119905
= nabla (1198631015840
nabla119888) (1)
Journal of Chemistry 3
(a) (b)
Figure 1 Pore characteristics of coal ((a) micropore (b) fracture)
Figure 2 Desorption and diffusion in the CBM reservoirs
L
(a)
L
(b)
L
(c)
Figure 3 Horizontal wells with different boundaries ((a) closed boundary (b) constant pressure boundary (c) mixed boundary)
The planar radial flow equation is as follows
120597119888
120597119905
=
119863
119903119868
119894
120597
120597119903119894
(119903119868
119894
120597119888
120597119903119894
) (2)
The dimensionless equation of gas diffusion in matrix is
1
119903119894119863
2
120597
120597119903119894119863
(119903119894119863
2120597119888119863
120597119903119894119863
) = 1205821
120597119888119863
120597119905119863
(3)
where 119903119894119863
is the dimensionless radius defined by 119903119894119863= 119903119894119877
119888119863is the dimensionless diffusion concentration defined by
119888119863= 119888 minus 119888
119894 119905119863is the dimensionless time defined by 119905
119863=
36119896119905120579119903119908
2 1205821is interporosity flow coefficient defined by
1205821= 36119896119877
2
120579119863119903119908
2 120579 is comprehensive storage coefficientdefined by 120579 = 120601
119891119888119905120583+6119901
119904119888119879119902119863119879119904119888120595119894 119902119863is the dimensionless
production defined by 119902119863= 1842times10
minus3
119902119904119888119901119904119888119879119896ℎ119879
119904119888120595119894 119888119894is
the initial concentration of gas kgm3 119896 is the permeability120583m2 and 120595
119894is the pseudopressure
32 Gas Flow Model in Matrix and Fracture In the processof diffusion of CBM the concentration often changes There-fore unsteady diffusion equation which accords with actualsituation is used
Free gas concentration is as follows
1198881= 1205881120601 =
119872119901120601
119877119879119885
(4)
Concentration of the adsorbed gas is
1198882= 119881119871
119875
119875119871+ 119875
(5)
4 Journal of Chemistry
Gas concentration in coal is
119888 =
119872119901120601
119877119879119885
+ 119881119871
119875
119875119871+ 119875
(6)
Volume of gas desorption from coal is
119902119889= minus
120597119872119889
120597119905
= minus120588119904119888
120597119881119889
120597119905
(7)
where119872119889is the gas mass under standard condition kgm3
119881119889is the volume of gas adsorption in coal under standard
condition m3m3 120588119904119888is the density of gas under standard
condition kgm3Density of gas is defined as
120588119904119888=
119872119901119904119888
119911119877119879119904119888
(8)
Spherical matrix has the following relationship
120597119881119889
120597119905
= minus
119860
119881
119869 = minus
119860
119881
119863
120597119888
120597119903119894
=
3119863
119877
120597119888
120597119903119894
(9)
where119881 is the volume of coalmatrixm3119860 is the surface areaof coal matrix m2 119863 is the mass diffusion coefficient m2s119869 is the diffusion flux gm2s
Combining (7) (8) and (9) the volume of gas desorptionis given by
119902119889=
119872119901119904119888
119877119879119904119888
3119863
119877
120597119888
120597119903119894
(10)
Based on the basic principle of material balance the gov-erning equations of gas flow in fracture system are given by
1
119903
120597
120597119903
(119903120588
119896
120583119892
120597119901
120597119903
) + 119902119889= 120588120601119862
119905
120597119901
120597119905
(11)
Equation (11) is simplified into the following equation
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596
120597120595119863
120597119905119863
(12)
where 120596 is the fracture storage ratio given by 120596 = (120601119888119905120583)120579 120582
is the diffusion coefficient given by 120582 = 36119896120591120579119903119908
2 120595119863is the
dimensionless pseudopressure given by 120595119863= (120595119894minus 120595)120595
119894119902119863
120591 is the adsorption time given by 120591 = 1198772119863
33 Mathematical Model Solution Diffusion and governingequations of gas flow in fracture are combined as follows
1
119903119894119863
2
120597
120597119903119894119863
(119903119894119863
2120597119888119863
120597119903119894119863
) = 120582
120597119888119863
120597119905119863
(diffusion in matrix)
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596
120597120595119863
120597119905119863
(flow in fracture)
(13)
Dimensionless initial and boundary conditions are givenas follows
The initial condition is
120595119863(119903119863 119905119863= 0) = 0 (14)
The boundary condition is
120597120595119863
120597119903119863
(119903119863= 1 119905119863) = minus1 (15)
The infinite outer boundary condition is
120595119863(119903119863997888rarr infin 119905
119863) = 0 (16)
The constant pressure boundary condition is
120595119863(119903119890119863 119905119863) = 0 (17)
The closed boundary condition is
120597120595119863
120597119903119863
(119903119890119863 119905119863) = 0 (18)
When119872 = 119903119894119863119888119863 (13) can be transformed into
1205972
119872
120597119903119894119863
2= 120582
120597119872
120597119905119863
(19)
The Laplace transform of (19) is
1205972
119872
120597119903119894119863
2= 120582119906119872 (20)
The general solution of (20) is
119872 = 119860 sinh (radic120582119906119903119894119863) + 119861 cosh (radic120582119906119903
119894119863) (21)
where sinh(119909) = (119890119909 minus 119890minus119909)2 and cosh(119909) = (119890119909 + 119890minus119909)2The coefficients 119860 and 119861 could be obtained by the initial
and boundary conditions
119861 = 0 119860 =
119872119886
sinh (radic120582119906) (22)
Hence
119872 = 119872119886
sinh (radic120582119906119903119894119863)
sinh (radic120582119906) (23)
Therefore the diffusion equation is transformed into
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= 119888119863(radic120582119906 cothradic120582119906 minus 1) (24)
Combining the dimensionless definition 119888119863and Lang-
muir isothermal adsorption equation 119881 = 119881119871119875119898(119875119871+ 119875119898)
(24) is transformed into
120597119888119863
120597119903119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= 119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] (radic120582119906 cothradic120582119906 minus 1)
(25)
Journal of Chemistry 5
where 119871 is Laplace transform
119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] = 120573119875119863 (26)
So
120597119888119863
120597119903119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= minus120573119875119863(radic120582119906 cothradic120582119906 minus 1) (27)
Laplace transform of (20) is
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596119906120595119863 (28)
The initial conditions are
120595119863
1003816100381610038161003816119906rarrinfin
= 0 (29)
The inner boundary conditions are
119903119863
120597120595119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
= minus
1
119906
(30)
The outer boundary conditions are
120595119863
1003816100381610038161003816119903119863rarrinfin
= 0 (infinite)
120595119863
1003816100381610038161003816119903119863=119903119890119863
= 0 (constant pressure)
120597120595119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=119903119890119863
= 0 (closed)
(31)
The solution of diffusion equation (27) is
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= 119888119863(radic120582119906 cothradic120582119906 minus 1) (32)
Substituting the dimensionless definition 119888119863and Lang-
muir isothermal adsorption equation 119881 = 119881119871119875119898(119875119871+ 119875119898)
in (32) it can be written as follows
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= 119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] (radic120582119906 cothradic120582119906 minus 1)
(33)
Substituting 120595119863= (120595119894minus 120595)120595
119894119902119863in (33) it can be exp-
ressed as follows119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
= minus
120595119871119881119871120595119894119902119863
(120595119871+ 120595) (120595
119871+ 120595119894) (120595 + 120595
119894)
120595119863 (34)
Defining 120573 = 120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894)
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= minus120573120595119863(radic120582119906 cothradic120582119906 minus 1) (35)
Substituting (35) in (28)
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) = 119891 (119906) 120595119863
119891 (119906) = 120596119906 +
1 minus 120596
120582
120573 (radic120582119906 cothradic120582119906 minus 1)
(36)
4 Equation of Boundary Condition
41 Boundary Integral Equation Applying the theory ofboundary element previous equation can be solved from theintegral transformation of the governing equation based onthe expression of the fundamental solutions and differentialequation of fluid flow through porous medium
int
Ω
[120595119863(119875 119906) nabla
2
119866 (119875119876 119906) minus 119866 (119875 119876 119906) nabla2
120595119863(119875 119906)
+ 120575 (119875 119876) 120595119863(119875 119906) minus
1
119906
119873119908
sum
119894=1
119902119863119894120575 (119909119863minus 119909119863119894 119910119863minus 119910119863119894)
sdot 119866 (119875 119876 119906)] 119889Ω = 0
(37)
where 119866(119875119876 119906) is the fundamental solution of horizontalwells in complex boundary reservoirs
According to the properties of 120575 function and the secondorder of Green formula it can be simplified to the boundaryintegral equation
120595119863(119876119896 119906) = int
Γ
[119866 (119875119876119896 119906)
120597120595119863(1198751015840
119906)
120597119899
minus120595119863(119875 119906)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889Γ (1198751015840
)
+
1
119906
119873119908
sum
119894=1
119902119863119894119866 (119875119876
119894 119906)
(38)
The boundary Γ is divided into119873119887cells which are located
at the end point and are taken as the nodes of boundaryelements Cell properties are assumed as linear distributionMeanwhile the boundary sections near nodes are assumed asarcs with nodes as their centers the resulting boundary integ-ral equation is
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
int
Γ119894
[119866 (1198751015840
119876119896 119906)
120597120595119863(1198751015840
119906)
120597119899
minus120595119863(1198751015840
119906)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889Γ119894(1198751015840
)
+
1
119906
119873119908
sum
119894=1
119902119863119894119866 (119875119876
119894 119906)
(39)
6 Journal of Chemistry
where 120579119896represents interior angles between any two adjacent
boundary elements Consider
120579119896=
1 the point in domain 120579119894= 2120587
05 the point at smooth boundary 120579119894= 120587
120579119894
2120587
the point at smoothless boundary
(40)
Using the linear interpolation in boundary element theboundary integral formula is deformed as follows
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
119897119894
2
int
1
minus1
[119866 (1198751015840
119876119896 119906) (120593
1(120585)
120597120595119863119894
120597119899
+ 1205932(120585)
120597120595119863119894+1
120597119899
)
minus (1205931(120585) 120595119863119894+ 1205932(120585) 120595119863119894+1
)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889120585
+
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876119894 119906)
(41)
where 1206011(120585) = (1 minus 120585)2 and 120601
2(120585) = (1 + 120585)2 are linear
interpolation formula 119897119894= radic(119909
119894+1minus 119909119894)2
+ (119910119894+1minus 119910119894)2 minus1 lt
120585 lt 1 and Γ119894is the length of linearity cell
42 Fundamental Solution of Boundary Element IntegralEquation It is crucial to find its fundamental solution whenthe horizontal flow problem in complex reservoirs is resolvedusing the boundary element method According to the prop-erties of the boundary element and the mathematic equationgoverning pressure transmission in porous medium thefundamental solution must satisfy the modified Helmholtzoperator The equation is given by
1
119903
120597
120597119903
(119903
120597119866
120597119903
) minus 119891 (119906)119866 = minus2120587120575 (1198721198631198721015840
119863) (42)
With Lord Kelvin point source solution the fundamentalsolution of (42) can be derived as follows
120574 =
exp (minus120588119863radic119891 (119906))
4120587120588119863
(43)
With mirror image method of fluid mechanics in porousmedium the transient point source fundamental solution ofclosed boundary is
120574 =
1
4120587
+infin
sum
minusinfin
exp (minusradic119891 (119906)radic1198772119863+ (119885119863+ 1198851015840
119863minus 2119899119885
119890119863)2
)
radic1198772
119863+ (119885119863+ 1198851015840
119863minus 2119899119885
119890119863)2
+
exp (minusradic119891 (119906)radic1198772119863+(119885119863minus1198851015840
119863minus2119899119885
119890119863)2
)
radic1198772
119863+(119885119863minus 1198851015840
119863minus2119899119885
119890119863)2
(44)
With Poisson superposition formula (44) can be simpli-fied and the transient point source fundamental solution ofsealed boundary at 119885 = 0 and 119885 = 119885119890 is
120574 =
1
2120587119885119890119863
[1198700(119877119863radic119891 (119906))
+ 2
119899=infin
sum
119899=1
1198700(119877119863radic119891 (119906) +
1198992
1205872
119885119890119863
2)
sdot cos(119899120587 119885119863119885119890119863
) cos(1198991205871198851015840
119863
119885119890119863
)]
(45)
The fundamental boundary element solution of horizon-tal wells in a reservoir with closed top and bottom boundariesis
119866(1198751015840
119876 119906)
=
1
2
int
1
minus1
1198700(119877119863radic119891 (119906)) 119889120572
+
119899=infin
sum
119899=1
cos (119899120587119911119863) cos (119899120587119911
119908119863)
sdot int
1
minus1
1198700(radic(119909
119863minus 120572)2
+ 119910119863
2radic119891 (119906) +
1198992
1205872
119885119890119863
2)119889120572
120597119866 (1198751015840
119876 119906)
120597119899
= minus
1
2
int
1
minus1
radic119891 (119906)1198701((119903119863minus 120572)radic119891 (119906))
120597119903119863
120597119899
119889120572
minus
119899=infin
sum
119899=1
cos (119899120587119911119863)
sdot cos (119899120587119911119908119863) int
1
minus1
radic119891 (119906) +
1198992
1205872
119885119890119863
2
sdot 1198701((119903119863minus 120572)
sdotradic119891 (119906) +
1198992
1205872
119885119890119863
2)
120597119903119863
120597119899
119889120572
(46)
where120597119903119863
120597119899
= plusmn
100381610038161003816100381610038161003816100381610038161003816
((119909120585minus 119909) (119910
119894minus 119910119894+1) minus (119910
120585minus 119910) (119909
119894minus 119909119894+1))
sdot (radic(119909119894minus 119909119894+1)2
+ (119910119894minus 119910119894+1)2
)
minus1100381610038161003816100381610038161003816100381610038161003816
sdot (radic(119909120585minus 119909119894)2
+ (119910120585minus 119910119894)2
)
minus1
(47)
Journal of Chemistry 7
If 119899 and 1198751015840 are in the same direction then 120597119903119863120597119899 is posi-
tive otherwise it is negative
43 Solving of Integral Equation The integral boundary equa-tion (37) can be simplified as
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(48)
where
1198671015840
1=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206011(120585) 119889120585
1198671015840
2=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206012(120585) 119889120585
1198671015840
3=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206011(120585) 119889120585
1198671015840
4=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206012(120585) 119889120585
(49)
The unknown variable in the integral boundary equationis 120597120595119863119894120597119899 or 120595
119863119894 The number of the nodes at the boundary
is 119873119887 so 119873
119887equation with form of (48) can be established
When the boundary properties are known there are just 119873119887
unknown variables So we can solve the set of equationswhose matrix expression is
[[[[[[[
[
11986711
11986712
sdot sdot sdot 1198671119873119887
11986721
11986722
sdot sdot sdot 1198672119873119887
1198671198731198871
1198671198731198872
sdot sdot sdot 119867119873119887119873119887
]]]]]]]
]
[[[[[[
[
1199091
1199092
119909119873119887
]]]]]]
]
=
[[[[[[
[
1198651
1198652
119865119873119887
]]]]]]
]
(50)
where119909119894is 120597120595119863119894120597119899 or120595
119863119894and119865119894is (1119906)sum119873119908
119894=1119902119863119894119866(1198751015840
119876 119906)Once the unknown variables are acquired we can solve
119875119863
of arbitrary point in the research domain using theboundary integral equation (50)
120595119863(119876 119906) =
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894
+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(51)
5 Validation of the Model
In order to validate the gas flowmodel proposed in this paperwe test the adsorption and desorption data of coal Then wecompare the boundary element model with those publishedworks
001 01 1 10 100
1E16
1E17
120595D
120595998400D
TD
120595D
and120595998400 D
Figure 4 Double logarithmic curve of pressure drop of test data
51 Validation of the Gas FlowModel in CBM Reservoirs Theadsorption and desorption data of coal are tested in the labo-ratory the pressure continues to drop as the coal continuesto adsorb methane We deal with the experimental data inlog-log coordinates as shown in Figure 4 It can be seen fromthe diagram that the results of the numerical simulation arein good match with the experimental data and are consistentwith the gas flow law in CBM reservoirs
52 Validation of Gas Flow in a Horizontal Well with ComplexBoundary A pressure buildup test example of a horizontalgas well in tight gas reservoir has been presented byHan [30]the production and pressure data fitting results show that thehorizontal gas well has a closed boundary
Figure 5 is the comparison of pressure curves betweenhorizontal wells in CBM and conventional gas reservoirs andthe values of parameters are fromHanrsquos paper [30] Accordingto the analysis of flow characteristics there is an additionalradial flowwhich reflects gas flow in fracture After this stagethe pressure transmits from fracture to matrix and gas startsto desorbTheflowcharacteristics of this stagewould be influ-enced by desorption velocity and desorption quantity Thelast stage of pressure derivative curves in different reservoirsis in good match which reflects the type of boundary
6 Analysis of Flow Characteristics andField Application
61 Flow Regime Recognition Type curves for horizontalwells in a CBM reservoir with complex boundary calculatedby the BEM in this paper are shown in Figure 6 The figureshows that flow characteristics can be divided into sevenflow periods (1) wellbore storage period influenced by earlywellbore storage effect the curves of pressure and pressurederivative are straight lines with the same slope in this period(2) the first transition flow section which reflects the degreeof pollution near the bottom and is mainly affected by skin
8 Journal of Chemistry
001 0
1 1 10 100
1000
1000
0
1000
00
01
1
10
100
1Eminus4
1Eminus3
120595D of conventional gas reservoir horizontal well120595998400D of conventional gas reservoir horizontal well
120595D of CBM horizontal well120595998400D of CBM horizontal well
TD
120595D
and120595998400 D
Figure 5 Comparison of typical curves between horizontal wells inCBM and conventional gas reservoirs
001 0
1 1 10 100
1000
1000
0
1000
00
001
01
1
10
100
Closed boundaryConstant pressure boundaryMixed boundary
1Eminus6
1Eminus5
1Eminus4
1Eminus3
I II III IV V VI VII
TD
120595D
and120595998400 D
Figure 6 Typical curve of CBM horizontal well with complexboundary conditions (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
factor 119878 (3) the first radial flow section which is perpendic-ular to the horizontal wellbore the pressure derivative curveis shown as a horizontal line which reflects the early radialflow perpendicular to the horizontal wellbore (4) the secondtransition section which is the transition stage of radial flowfrom wellbore to fracture (5) the second radial flow stagewhich reflects the free gas flow in fracture this period isaffected by the flow of free gas and adsorbed gas diffusioncoefficient 120582 and Langmuir adsorption parameter 120573 (6) thethird radial flow section which reflects radial flow of whole
001 1 100 10000 1000000
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
120582 = 1
120582 = 100
120582 = 10000
Figure 7 Influence of diffusion coefficient 120582 on bottom-holepressure with mixed boundary (120596 = 001 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
system after pressure balance and pressure derivative curveappears as a flat behavior (7) boundary response sectionThepressure and pressure derivative curves with closed boundarywould be upward after pressure transmitting to the boundaryThe pressure derivative curve with constant pressure bound-ary would be in decline after pressure transmitting to theboundary Compared with the constant pressure boundarythe falling range of pressure derivative curve with mixedboundary is less
62 Parameter Sensitivity Analysis Figure 7 shows the influ-ence of diffusion coefficient 120582 on the bottom-hole pressurewith mixed boundary There is relevance between diffusioncoefficient 120582 = 120572119896120591120579119871
2 and adsorption time 120591 = 1198772
119863The smaller the 120591 the shorter the time of desorption-diffusionand the shorter the time of pressure balance between fractureand matrix As shown in the diagram diffusion coefficient 120582mainly affects the duration of the second radial flow periodand appearance of third radial flow The greater the diffusioncoefficient 120582 the longer the duration of second radial flowperiod and the later the 119905 appearance of third radial flow andvice versa If 120582 is small enough the second radial flow periodwould be covered
Figure 8 shows the influence of Langmuir parameter 120573on the bottom-hole pressure with mixed boundary 120573 =
120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894) is related to Langmuir
adsorption pressure 119901119871and Langmuir adsorption volume119881
119871
The larger the 120573 the greater the adsorption ability of coalAs shown in the graph 120573 only affects the second radial flowsection The greater the 120573 the smaller the pressure derivativevalues of second radial flow period
Figure 9 shows the influence of fracture storage ratio120596 onthe bottom-hole pressure with mixed boundary Its expres-sion 120596 = 120601
119891119888119905120583(120601119891119888119905120583 + 6119901
119904119888119879119902119863119879119904119888120595119894) indicates that the
smaller the 120601119891119888119905 the smaller the 120596 and the more the radial
Journal of Chemistry 9
001 1 100 10000 1000000
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
0011E8
120573 = 01
120573 = 1
120573 = 10
Figure 8 Influence of 120573 on the bottom-hole pressure with mixedboundary (120596 = 001 120582 = 300 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
1E8
120596 = 04
120596 = 01
120596 = 001
Figure 9 Influence of 120596 on the bottom-hole pressure with mixedboundary (120582 = 300 120573 = 50 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
flow in fracture From Figure 8 almost all of the flow periodswould be affected by 120596 except for wellbore storage periodThe greater the 120596 the smaller the hump value of pressurederivative and the longer the duration of first radial flowperiod the greater the pressure derivative value in secondradial flow period
Figure 10 shows the influence of eccentricity of horizontalwell 119885
119908119889on the bottom-hole pressure with mixed boundary
The streamline shape and pressure distribution of a horizon-tal well would be affected by the size of 119885
119908119889 The smaller
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
Zwd = 01
Zwd = 03
Zwd = 05
Figure 10 Influence of119885119908119889
on the bottom-hole pressure withmixedboundary (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25 119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
LD = 1
LD = 25
LD = 50
Figure 11 Influence of 119871119863on the bottom-hole pressure with mixed
boundary (120596 = 001 120582 = 300 120573 = 50119885119908119889= 05119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
the119885119908119889 the larger the pressure derivative value in first radial
flow periodFigure 11 shows the influence of length of horizontal well
119871119863on the bottom-hole pressure with mixed boundary It
shows large influence of 119871119863on the pressure derivative value
in first radial flow period The smaller the 119871119863 the larger the
pressure derivative value in first radial flow period
63 Field Example To demonstrate the application of themodel proposed in this paper a horizontal well ZX-12 in aCBM reservoir is studied The vertical depth and horizontallength of this well are 689m and 536m respectively withwell radius 119903
119908of 01m coal thickness of 45m and initial
10 Journal of Chemistry
10 100100
1000
1000
TD
120595D
and120595998400 D
120595D of CBM horizontal well fitted curve120595998400D CBM horizontal well fitted curve
120595D of CBM horizontal well product curve120595998400D of CBM horizontal well product data curve
Figure 12 Fitted curves of test data from an actual well
pressure of 55MPa According to test results the Langmuirvolume 119881
119871is 3275m3t and the Langmuir pressure 119875
119871is
249MPa Figure 12 is the fitted curves which were obtainedby calculating the actual production and pressure data Asshown in Figure 12 the field data are in good match with thetheoretical results calculated by the new model in this paper
According to the actual field data some of reservoirparameters are fitted as follows the porosity is 004 and thepermeability is 3mD The stage of desorption and boundaryresponse can be clearly seen in Figure 12 Because of thefalling of the pressure derivative curve in the last stage wellZX-12 has a constant pressure boundary implying that thegas production of well ZX-12 is affected by the production ofadjacent well
7 Conclusions
Themathematic model of gas flowing into horizontal wells inCBMreservoirswith complex boundary conditions is derivedbased on the percolation theory The curves of bottom-hole pressure and pressure derivative are obtained by usingboundary element method and Laplace transform The con-clusions are as follows
(1) The fundamental boundary element solutions fortransient pressure response of horizontal wells inCBM reservoirs with complex boundary conditionscould be obtained by using Lord Kelvinrsquos point sourcesolution point source function theory and Poissonrsquossummation formula
(2) Comparison of the typical curves of flow character-istics between horizontal wells in CBM and conven-tional gas reservoirs shows that there is an additionalradial flow which reflects gas flow in fracture
(3) Comparing the typical curves of flow characteris-tics with complex boundary conditions the pressure
and pressure derivative curves with closed boundarywould be upward after pressure transmitting to theboundary The pressure derivative curves with con-stant pressure boundary and mixed boundary wouldbe fallen but the falling range of pressure derivativecurve with mixed boundary is less
Nomenclature
119903119908 Radius m
119896 Permeability mD120601 Porosity fraction119869 Diffusion flux gm2s119904 Laplace transform variableℎ Thickness m119861 Volume factor fraction119902 Influx into the wellbore m3d119871 Half length of horizontal well m120583 Viscosity mPasdots119888 Concentration of coalbed methane kgm3119875119894 Initial pressure MPa
119902 Production of well m3d119881119871 Langmuir volume constant m3ton
119881 Volume of coal matrix m3120595119894 Pseudopressure
120596 Fracture storage ratio fraction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is supported by National Natural Science Foun-dation of China (Grant no 51304032) National MajorScience and Technology Special Project of China (Grantno 2011ZX05037-003) and Research Fund for the Doc-toral Program of Higher Education of China (Grant no20125122110017)
References
[1] J C Cai E Perfect C-L Cheng and X Y Hu ldquoGeneralizedmodeling of spontaneous imbibition based on hagen-poiseuilleflow in tortuous capillaries with variably shaped aperturesrdquoLangmuir vol 30 no 18 pp 5142ndash5151 2014
[2] T Ertekin and W Sung ldquoPressure transient analysis of coalseams in the presence of multi-mechanistic flow and sorptionphenomenardquo in Proceedings of the SPE Gas Technology Sympo-sium pp 469ndash477 June 1989
[3] K Anbarci and T Ertekin ldquoA comprehensive study of pressuretransient analysis with sorption phenomena for single phase gasflow in coal seamsrdquo SPE 20568 1990
[4] C R Clarkson and R M Bustin ldquoEffect of pore structureand gas pressure upon the transport properties of coal alaboratory and modeling study 1 Isotherms and pore volumedistributionsrdquo Fuel vol 78 no 11 pp 1333ndash1344 1999
Journal of Chemistry 11
[5] C R Clarkson and R M Bustin ldquoEffect of pore structure andgas pressure upon the transport properties of coal a laboratoryandmodeling study 2 Adsorption rate modelingrdquo Fuel vol 78no 11 pp 1345ndash1362 1999
[6] S Reeve ldquoAdvanced reservoir modeling in desorption con-trolled reservoirsrdquo in Proceedings of the SPE Rocky MountainPetroleum Technology Conference SPE 71090 May 2001
[7] D K Tong and S Liu ldquoUnsteady percolation flow of coalbedmethane through deformed coal seamrdquo Natural Gas Industryvol 1 no 1 pp 74ndash76 2005
[8] X M Zhang and D K Tong ldquoAnalysis for pressure transient ofcoalbed methane reservoirrdquo Well Testing vol 6 no 17 pp 1ndash42008
[9] X M Zhang and D K Tong ldquoPressure transient analysisfor coal seams with triple-porositydual-permeability modelrdquoChineseQuarterly ofMechanics vol 4 no 29 pp 578ndash582 2008
[10] X H Hu S M Hu D I Zhang et al ldquoApplication of quadraticpressure method in well test interpretation of gas water twophase flow in coal seamrdquo Journal of Oil and Gas Technology vol7 no 33 pp 118ndash122 2011
[11] C R Clarkson C L Jordan D Ilk and T A Blasingame ldquoRate-transient analysis of 2-phase (gas + water) CBM wellsrdquo Journalof Natural Gas Science and Engineering vol 8 pp 106ndash120 2012
[12] K Aminian and S Ameri ldquoPredicting production performanceof CBM reservoirsrdquo Journal of Natural Gas Science and Engi-neering vol 1 no 1-2 pp 25ndash30 2009
[13] J C Cai and BM Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011
[14] W Sung T Ertekin and F C Schwerer ldquoThe developmenttesting and application of a comprehensive coal seam degasi-fication modelrdquo in Proceedings of the SPE Unconventional GasTechnology Symposium SPE 15247 Louisville Ky USA May1986
[15] W Sung and T Ertekin ldquoAn analysis of field developmentstrategies formethane production from coal seamsrdquo in Proceed-ings of the 62nd Annual Technical Conference and ExhibitionConference Paper SPE 16858 Dallas Tex USA 1987
[16] T Ertekin W Sung and F C Schwerer ldquoProduction perfor-mance analysis of horizontal drainage wells for degasificationof coal-seamsrdquo Journal of Petroleum Technology vol 40 no 5pp 625ndash632 1988
[17] T W Engler and J M Rajtar ldquoPressure transient testing ofhorizontal wells in coalbed reservoirsrdquo in Proceedings of theSPE Rocky Mountain Regional Meeting Paper SPE-24374-MSCasper Wyo USA May 1992
[18] P S Sarkar and J M Rajtar ldquoHorizontal well transient pres-sure testing in coalbed reservoirsrdquo in Proceedings of the 3rdLatin AmericaCaribbean Petroleum Engineering ConferenceSPE 26995 Buenos Aires Argentina April 1994
[19] P S Sarkar and J M Rajtar ldquoTransient well testing of coalbedmethane reservoirs with horizontal wellsrdquo in Proceedings of thePermian Basin Oil amp Gas Recovery Conference Paper SPE27681pp 531ndash539 Midland Tex USA March 1994
[20] X H Wang D I Zhang and Y Song ldquoDevelopment mecha-nism of low permeable coalbed methane frommultilateral hor-izontal pinnatewell with non-darcy seepage flow characteristicrdquoActa Geologica Sinica vol 7 no 82 pp 1437ndash1443 2008
[21] R-S Nie Y-F Meng J-C Guo and Y-L Jia ldquoModelingtransient flow behavior of a horizontal well in a coal seamrdquoInternational Journal of Coal Geology vol 92 pp 54ndash68 2012
[22] K L Katsifarakis D K Karpouzos and N TheodossiouldquoCombined use of BEM and genetic algorithms in groundwaterflow and mass transport problemsrdquo Engineering Analysis withBoundary Elements vol 23 no 7 pp 555ndash565 1999
[23] D T Numbere and D Tiab ldquoImproved streamline-generatingtechnique that uses the boundary (integral) element methodrdquoSPE Reservoir Engineering vol 3 no 3 pp 1061ndash1068 1988
[24] J Kikani and R N Horne ldquoApplication of boundary ele-ment method to reservoir engineering problemsrdquo Journal ofPetroleum Science and Engineering vol 3 no 3 pp 229ndash2411989
[25] J Hou YDWang andYMChen ldquoBoundary elementmethodin enhanced oil recoveryrdquo Chinese Journal of Hydrodynamicsvol 3 no 13 pp 23ndash28 2003
[26] K Chaiyo P Rattanadecho and S Chantasiriwan ldquoThemethodof fundamental solutions for solving free boundary saturatedseepage problemrdquo International Communications in Heat andMass Transfer vol 38 no 2 pp 249ndash254 2011
[27] K Rafiezadeh and B Ataie-Ashtiani ldquoSeepage analysis inmulti-domain general anisotropic media by three-dimensionalboundary elementsrdquo Engineering Analysis with Boundary Ele-ments vol 37 no 3 pp 527ndash541 2013
[28] X H Fu Y Qin W H Zhang et al ldquoCoal pore fractal classifi-cation and natural classification research based on the coal-bedgas migrationrdquo Chinese Science Bulletin vol 12 supplement 1pp 51ndash55 2005
[29] J C Cai and S Y Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal ofModern Physics C vol 24 no 8 pp 135ndash156 2013
[30] D M Han The Dynamic Analysis Technology Research onHorizontal Wells of Jingbian Gas Field University of XirsquoanShiyou Xirsquoan China 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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CatalystsJournal of
2 Journal of Chemistry
the fractal theory to study the enhancing recovery mecha-nism in natural gas-saturated porous media by spontaneousimbibition effect
In order to improve the production a large number ofhorizontal wells have been used in the CBM reservoirs Sungand Ertekin [14ndash16] established a two-dimensional two-phasemultiwell gas-water flowmodel and thismodel has theability to simulatemultiple horizontal wells Engler andRajtar[17ndash19] established a mathematical model of single-phasegas flow in the horizontal wells and the analytical solutionsare given for the horizontal well pressure drop and pressurerecovery Wang et al [20] established a mathematical modelin which anisotropy formation heterogeneity permeabilitystress sensitivity and influence of wellbore pressure drop ondirectional pinnate horizontal wells in CBM reservoirs areconsidered Nie et al [21] deduced CBMflow equations basedon Langmuir adsorption inmatrix andDarcy flow in fractureand analyzed the transient transport characteristics of gasfrom CBM reservoirs to horizontal wells
Generally the theories of calculating reservoir and bot-tom-hole pressures which can reflect the gas flow characteris-tics are mostly based on homogeneous reservoirs and regulargeometry such as infinite boundary or circular boundaryOuter boundary conditions of a reservoir have also beensimplified it is generally regarded as simple situations asconstant pressure or closed boundary However influencedby characteristics of geological structures the true reservoirsusually have complex and diversiform boundaries In thiscase the conventional flow theories and solving methodscould do nothing to calculate reservoir or bottom-hole pres-sures with mixed boundary conditions
The boundary element method (BEM) is a numericalcomputational method of solving linear partial differentialequations developed after the better-known finite elementmethod (FEM) and finite difference method (FDM) TheBEM could be able to reduce dimension and save computermemory and running time Couplingwith the BEM [22] bot-tom-hole pressures and complicated flow characteristicsunder the condition of irregularly shaped area with mixedboundary could be calculated and analyzed Numbere andTiab [23] developed a streamline simulationwith the BEM forhomogeneous or partly homogeneous reservoirs with irreg-ular boundary to make the simulation match physical modelbetter Kikani and Horne [24] employed the BEM to analyzethe transient pressure response in reservoirs with arbitraryboundary and developed two formulas namely convolutionformula and Laplace domain space formula to solve tran-sient fluid flow through porous medium in homogeneousreservoirs Hou et al [25] used the BEM to simulate flowlinemap in homogeneous reservoirs with irregular boundaryChaiyo et al [26] used the BEM to solve free boundary sat-urated seepage problem Rafiezadeh and Ataie-Ashtiani [27]studied the flow mechanism in anisotropic media by three-dimensional boundary elements
In this paper a mathematical model is developed todescribe gas flow in horizontal wells in CBM reservoirs basedon the theory of fluid flow through porousThe type curves of
pressure derivative characteristics of gas flow with complexexternal boundary and relevant affecting factors are ana-lyzed
2 Physical Model of Gas Flow
Coal reservoir is a dual porous medium composed of matrixand fracture The matrix is the main reservoir space ofcoalbed methane (Figure 1(a)) and the fracture is the flowchannel of fluid (Figure 1(b)) The average pore size of CBMreservoirs is much smaller than those of conventional reser-voirs Pore size can be divided into three categories [28]macropore (aperture gt 20 nm) mesopore (2 nm lt aperture lt20 nm) andmicropore (aperture lt 2 nm)The small porosityof coal leads to the great specific surface areaHencemethanecould be strongly adsorbed resulting in the fact that thecontent of CBM is far more than its pore volume
According to the reservoir dual pore structure of coal aphysical model is set up for gas flow The migration of gasin coal is shown in Figure 2 In the production of CBM for-mation pressure keeps declining When formation pressuredrops below the critical desorption pressure CBM starts todesorb from the coal matrix surface Meanwhile the originalstate of equilibrium is broken causing the flow of gas in frac-ture This process is very similar to spontaneous imbibitionin fractured porous media [29]
To facilitate the derivation it is assumed that the length ofa horizontal well is119871 the center location of the horizontal wellis (119909119908 119910119908 119911119908) and the thickness of a CBM reservoir is ℎ In
this paper gas flow into horizontal wells with closed bound-ary (Figure 3(a)) constant pressure boundary (Figure 3(b))and mixed boundary (Figure 3(c)) is considered
The fundamental assumptions are as follows
(1) CBMdiffuses directly frommatrix to fracture and theprocess of diffusion is unsteady
(2) Gas flow in fracture is radial laminar flow in agree-ment with Darcyrsquos law
(3) Only single-phase gas flow exists in coal
(4) The effects of gravity and capillary force are negligible
(5) The effects of temperature are negligible
(6) CBM isothermal adsorption process is in line withthe Langmuir isotherm adsorption law and the initialstate conforms to the isothermal adsorption curve
(7) Radius of the gas well is regarded infinitesimal andgas well production is constant
3 Mathematical Model
31 Gas Diffusion Model in Matrix In combination with themass conservation equation with the second Fickrsquos diffusionlaw the change of gas concentration with time is given by
120601
120597 (119888)
120597119905
= nabla (1198631015840
nabla119888) (1)
Journal of Chemistry 3
(a) (b)
Figure 1 Pore characteristics of coal ((a) micropore (b) fracture)
Figure 2 Desorption and diffusion in the CBM reservoirs
L
(a)
L
(b)
L
(c)
Figure 3 Horizontal wells with different boundaries ((a) closed boundary (b) constant pressure boundary (c) mixed boundary)
The planar radial flow equation is as follows
120597119888
120597119905
=
119863
119903119868
119894
120597
120597119903119894
(119903119868
119894
120597119888
120597119903119894
) (2)
The dimensionless equation of gas diffusion in matrix is
1
119903119894119863
2
120597
120597119903119894119863
(119903119894119863
2120597119888119863
120597119903119894119863
) = 1205821
120597119888119863
120597119905119863
(3)
where 119903119894119863
is the dimensionless radius defined by 119903119894119863= 119903119894119877
119888119863is the dimensionless diffusion concentration defined by
119888119863= 119888 minus 119888
119894 119905119863is the dimensionless time defined by 119905
119863=
36119896119905120579119903119908
2 1205821is interporosity flow coefficient defined by
1205821= 36119896119877
2
120579119863119903119908
2 120579 is comprehensive storage coefficientdefined by 120579 = 120601
119891119888119905120583+6119901
119904119888119879119902119863119879119904119888120595119894 119902119863is the dimensionless
production defined by 119902119863= 1842times10
minus3
119902119904119888119901119904119888119879119896ℎ119879
119904119888120595119894 119888119894is
the initial concentration of gas kgm3 119896 is the permeability120583m2 and 120595
119894is the pseudopressure
32 Gas Flow Model in Matrix and Fracture In the processof diffusion of CBM the concentration often changes There-fore unsteady diffusion equation which accords with actualsituation is used
Free gas concentration is as follows
1198881= 1205881120601 =
119872119901120601
119877119879119885
(4)
Concentration of the adsorbed gas is
1198882= 119881119871
119875
119875119871+ 119875
(5)
4 Journal of Chemistry
Gas concentration in coal is
119888 =
119872119901120601
119877119879119885
+ 119881119871
119875
119875119871+ 119875
(6)
Volume of gas desorption from coal is
119902119889= minus
120597119872119889
120597119905
= minus120588119904119888
120597119881119889
120597119905
(7)
where119872119889is the gas mass under standard condition kgm3
119881119889is the volume of gas adsorption in coal under standard
condition m3m3 120588119904119888is the density of gas under standard
condition kgm3Density of gas is defined as
120588119904119888=
119872119901119904119888
119911119877119879119904119888
(8)
Spherical matrix has the following relationship
120597119881119889
120597119905
= minus
119860
119881
119869 = minus
119860
119881
119863
120597119888
120597119903119894
=
3119863
119877
120597119888
120597119903119894
(9)
where119881 is the volume of coalmatrixm3119860 is the surface areaof coal matrix m2 119863 is the mass diffusion coefficient m2s119869 is the diffusion flux gm2s
Combining (7) (8) and (9) the volume of gas desorptionis given by
119902119889=
119872119901119904119888
119877119879119904119888
3119863
119877
120597119888
120597119903119894
(10)
Based on the basic principle of material balance the gov-erning equations of gas flow in fracture system are given by
1
119903
120597
120597119903
(119903120588
119896
120583119892
120597119901
120597119903
) + 119902119889= 120588120601119862
119905
120597119901
120597119905
(11)
Equation (11) is simplified into the following equation
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596
120597120595119863
120597119905119863
(12)
where 120596 is the fracture storage ratio given by 120596 = (120601119888119905120583)120579 120582
is the diffusion coefficient given by 120582 = 36119896120591120579119903119908
2 120595119863is the
dimensionless pseudopressure given by 120595119863= (120595119894minus 120595)120595
119894119902119863
120591 is the adsorption time given by 120591 = 1198772119863
33 Mathematical Model Solution Diffusion and governingequations of gas flow in fracture are combined as follows
1
119903119894119863
2
120597
120597119903119894119863
(119903119894119863
2120597119888119863
120597119903119894119863
) = 120582
120597119888119863
120597119905119863
(diffusion in matrix)
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596
120597120595119863
120597119905119863
(flow in fracture)
(13)
Dimensionless initial and boundary conditions are givenas follows
The initial condition is
120595119863(119903119863 119905119863= 0) = 0 (14)
The boundary condition is
120597120595119863
120597119903119863
(119903119863= 1 119905119863) = minus1 (15)
The infinite outer boundary condition is
120595119863(119903119863997888rarr infin 119905
119863) = 0 (16)
The constant pressure boundary condition is
120595119863(119903119890119863 119905119863) = 0 (17)
The closed boundary condition is
120597120595119863
120597119903119863
(119903119890119863 119905119863) = 0 (18)
When119872 = 119903119894119863119888119863 (13) can be transformed into
1205972
119872
120597119903119894119863
2= 120582
120597119872
120597119905119863
(19)
The Laplace transform of (19) is
1205972
119872
120597119903119894119863
2= 120582119906119872 (20)
The general solution of (20) is
119872 = 119860 sinh (radic120582119906119903119894119863) + 119861 cosh (radic120582119906119903
119894119863) (21)
where sinh(119909) = (119890119909 minus 119890minus119909)2 and cosh(119909) = (119890119909 + 119890minus119909)2The coefficients 119860 and 119861 could be obtained by the initial
and boundary conditions
119861 = 0 119860 =
119872119886
sinh (radic120582119906) (22)
Hence
119872 = 119872119886
sinh (radic120582119906119903119894119863)
sinh (radic120582119906) (23)
Therefore the diffusion equation is transformed into
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= 119888119863(radic120582119906 cothradic120582119906 minus 1) (24)
Combining the dimensionless definition 119888119863and Lang-
muir isothermal adsorption equation 119881 = 119881119871119875119898(119875119871+ 119875119898)
(24) is transformed into
120597119888119863
120597119903119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= 119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] (radic120582119906 cothradic120582119906 minus 1)
(25)
Journal of Chemistry 5
where 119871 is Laplace transform
119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] = 120573119875119863 (26)
So
120597119888119863
120597119903119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= minus120573119875119863(radic120582119906 cothradic120582119906 minus 1) (27)
Laplace transform of (20) is
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596119906120595119863 (28)
The initial conditions are
120595119863
1003816100381610038161003816119906rarrinfin
= 0 (29)
The inner boundary conditions are
119903119863
120597120595119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
= minus
1
119906
(30)
The outer boundary conditions are
120595119863
1003816100381610038161003816119903119863rarrinfin
= 0 (infinite)
120595119863
1003816100381610038161003816119903119863=119903119890119863
= 0 (constant pressure)
120597120595119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=119903119890119863
= 0 (closed)
(31)
The solution of diffusion equation (27) is
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= 119888119863(radic120582119906 cothradic120582119906 minus 1) (32)
Substituting the dimensionless definition 119888119863and Lang-
muir isothermal adsorption equation 119881 = 119881119871119875119898(119875119871+ 119875119898)
in (32) it can be written as follows
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= 119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] (radic120582119906 cothradic120582119906 minus 1)
(33)
Substituting 120595119863= (120595119894minus 120595)120595
119894119902119863in (33) it can be exp-
ressed as follows119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
= minus
120595119871119881119871120595119894119902119863
(120595119871+ 120595) (120595
119871+ 120595119894) (120595 + 120595
119894)
120595119863 (34)
Defining 120573 = 120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894)
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= minus120573120595119863(radic120582119906 cothradic120582119906 minus 1) (35)
Substituting (35) in (28)
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) = 119891 (119906) 120595119863
119891 (119906) = 120596119906 +
1 minus 120596
120582
120573 (radic120582119906 cothradic120582119906 minus 1)
(36)
4 Equation of Boundary Condition
41 Boundary Integral Equation Applying the theory ofboundary element previous equation can be solved from theintegral transformation of the governing equation based onthe expression of the fundamental solutions and differentialequation of fluid flow through porous medium
int
Ω
[120595119863(119875 119906) nabla
2
119866 (119875119876 119906) minus 119866 (119875 119876 119906) nabla2
120595119863(119875 119906)
+ 120575 (119875 119876) 120595119863(119875 119906) minus
1
119906
119873119908
sum
119894=1
119902119863119894120575 (119909119863minus 119909119863119894 119910119863minus 119910119863119894)
sdot 119866 (119875 119876 119906)] 119889Ω = 0
(37)
where 119866(119875119876 119906) is the fundamental solution of horizontalwells in complex boundary reservoirs
According to the properties of 120575 function and the secondorder of Green formula it can be simplified to the boundaryintegral equation
120595119863(119876119896 119906) = int
Γ
[119866 (119875119876119896 119906)
120597120595119863(1198751015840
119906)
120597119899
minus120595119863(119875 119906)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889Γ (1198751015840
)
+
1
119906
119873119908
sum
119894=1
119902119863119894119866 (119875119876
119894 119906)
(38)
The boundary Γ is divided into119873119887cells which are located
at the end point and are taken as the nodes of boundaryelements Cell properties are assumed as linear distributionMeanwhile the boundary sections near nodes are assumed asarcs with nodes as their centers the resulting boundary integ-ral equation is
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
int
Γ119894
[119866 (1198751015840
119876119896 119906)
120597120595119863(1198751015840
119906)
120597119899
minus120595119863(1198751015840
119906)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889Γ119894(1198751015840
)
+
1
119906
119873119908
sum
119894=1
119902119863119894119866 (119875119876
119894 119906)
(39)
6 Journal of Chemistry
where 120579119896represents interior angles between any two adjacent
boundary elements Consider
120579119896=
1 the point in domain 120579119894= 2120587
05 the point at smooth boundary 120579119894= 120587
120579119894
2120587
the point at smoothless boundary
(40)
Using the linear interpolation in boundary element theboundary integral formula is deformed as follows
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
119897119894
2
int
1
minus1
[119866 (1198751015840
119876119896 119906) (120593
1(120585)
120597120595119863119894
120597119899
+ 1205932(120585)
120597120595119863119894+1
120597119899
)
minus (1205931(120585) 120595119863119894+ 1205932(120585) 120595119863119894+1
)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889120585
+
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876119894 119906)
(41)
where 1206011(120585) = (1 minus 120585)2 and 120601
2(120585) = (1 + 120585)2 are linear
interpolation formula 119897119894= radic(119909
119894+1minus 119909119894)2
+ (119910119894+1minus 119910119894)2 minus1 lt
120585 lt 1 and Γ119894is the length of linearity cell
42 Fundamental Solution of Boundary Element IntegralEquation It is crucial to find its fundamental solution whenthe horizontal flow problem in complex reservoirs is resolvedusing the boundary element method According to the prop-erties of the boundary element and the mathematic equationgoverning pressure transmission in porous medium thefundamental solution must satisfy the modified Helmholtzoperator The equation is given by
1
119903
120597
120597119903
(119903
120597119866
120597119903
) minus 119891 (119906)119866 = minus2120587120575 (1198721198631198721015840
119863) (42)
With Lord Kelvin point source solution the fundamentalsolution of (42) can be derived as follows
120574 =
exp (minus120588119863radic119891 (119906))
4120587120588119863
(43)
With mirror image method of fluid mechanics in porousmedium the transient point source fundamental solution ofclosed boundary is
120574 =
1
4120587
+infin
sum
minusinfin
exp (minusradic119891 (119906)radic1198772119863+ (119885119863+ 1198851015840
119863minus 2119899119885
119890119863)2
)
radic1198772
119863+ (119885119863+ 1198851015840
119863minus 2119899119885
119890119863)2
+
exp (minusradic119891 (119906)radic1198772119863+(119885119863minus1198851015840
119863minus2119899119885
119890119863)2
)
radic1198772
119863+(119885119863minus 1198851015840
119863minus2119899119885
119890119863)2
(44)
With Poisson superposition formula (44) can be simpli-fied and the transient point source fundamental solution ofsealed boundary at 119885 = 0 and 119885 = 119885119890 is
120574 =
1
2120587119885119890119863
[1198700(119877119863radic119891 (119906))
+ 2
119899=infin
sum
119899=1
1198700(119877119863radic119891 (119906) +
1198992
1205872
119885119890119863
2)
sdot cos(119899120587 119885119863119885119890119863
) cos(1198991205871198851015840
119863
119885119890119863
)]
(45)
The fundamental boundary element solution of horizon-tal wells in a reservoir with closed top and bottom boundariesis
119866(1198751015840
119876 119906)
=
1
2
int
1
minus1
1198700(119877119863radic119891 (119906)) 119889120572
+
119899=infin
sum
119899=1
cos (119899120587119911119863) cos (119899120587119911
119908119863)
sdot int
1
minus1
1198700(radic(119909
119863minus 120572)2
+ 119910119863
2radic119891 (119906) +
1198992
1205872
119885119890119863
2)119889120572
120597119866 (1198751015840
119876 119906)
120597119899
= minus
1
2
int
1
minus1
radic119891 (119906)1198701((119903119863minus 120572)radic119891 (119906))
120597119903119863
120597119899
119889120572
minus
119899=infin
sum
119899=1
cos (119899120587119911119863)
sdot cos (119899120587119911119908119863) int
1
minus1
radic119891 (119906) +
1198992
1205872
119885119890119863
2
sdot 1198701((119903119863minus 120572)
sdotradic119891 (119906) +
1198992
1205872
119885119890119863
2)
120597119903119863
120597119899
119889120572
(46)
where120597119903119863
120597119899
= plusmn
100381610038161003816100381610038161003816100381610038161003816
((119909120585minus 119909) (119910
119894minus 119910119894+1) minus (119910
120585minus 119910) (119909
119894minus 119909119894+1))
sdot (radic(119909119894minus 119909119894+1)2
+ (119910119894minus 119910119894+1)2
)
minus1100381610038161003816100381610038161003816100381610038161003816
sdot (radic(119909120585minus 119909119894)2
+ (119910120585minus 119910119894)2
)
minus1
(47)
Journal of Chemistry 7
If 119899 and 1198751015840 are in the same direction then 120597119903119863120597119899 is posi-
tive otherwise it is negative
43 Solving of Integral Equation The integral boundary equa-tion (37) can be simplified as
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(48)
where
1198671015840
1=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206011(120585) 119889120585
1198671015840
2=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206012(120585) 119889120585
1198671015840
3=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206011(120585) 119889120585
1198671015840
4=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206012(120585) 119889120585
(49)
The unknown variable in the integral boundary equationis 120597120595119863119894120597119899 or 120595
119863119894 The number of the nodes at the boundary
is 119873119887 so 119873
119887equation with form of (48) can be established
When the boundary properties are known there are just 119873119887
unknown variables So we can solve the set of equationswhose matrix expression is
[[[[[[[
[
11986711
11986712
sdot sdot sdot 1198671119873119887
11986721
11986722
sdot sdot sdot 1198672119873119887
1198671198731198871
1198671198731198872
sdot sdot sdot 119867119873119887119873119887
]]]]]]]
]
[[[[[[
[
1199091
1199092
119909119873119887
]]]]]]
]
=
[[[[[[
[
1198651
1198652
119865119873119887
]]]]]]
]
(50)
where119909119894is 120597120595119863119894120597119899 or120595
119863119894and119865119894is (1119906)sum119873119908
119894=1119902119863119894119866(1198751015840
119876 119906)Once the unknown variables are acquired we can solve
119875119863
of arbitrary point in the research domain using theboundary integral equation (50)
120595119863(119876 119906) =
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894
+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(51)
5 Validation of the Model
In order to validate the gas flowmodel proposed in this paperwe test the adsorption and desorption data of coal Then wecompare the boundary element model with those publishedworks
001 01 1 10 100
1E16
1E17
120595D
120595998400D
TD
120595D
and120595998400 D
Figure 4 Double logarithmic curve of pressure drop of test data
51 Validation of the Gas FlowModel in CBM Reservoirs Theadsorption and desorption data of coal are tested in the labo-ratory the pressure continues to drop as the coal continuesto adsorb methane We deal with the experimental data inlog-log coordinates as shown in Figure 4 It can be seen fromthe diagram that the results of the numerical simulation arein good match with the experimental data and are consistentwith the gas flow law in CBM reservoirs
52 Validation of Gas Flow in a Horizontal Well with ComplexBoundary A pressure buildup test example of a horizontalgas well in tight gas reservoir has been presented byHan [30]the production and pressure data fitting results show that thehorizontal gas well has a closed boundary
Figure 5 is the comparison of pressure curves betweenhorizontal wells in CBM and conventional gas reservoirs andthe values of parameters are fromHanrsquos paper [30] Accordingto the analysis of flow characteristics there is an additionalradial flowwhich reflects gas flow in fracture After this stagethe pressure transmits from fracture to matrix and gas startsto desorbTheflowcharacteristics of this stagewould be influ-enced by desorption velocity and desorption quantity Thelast stage of pressure derivative curves in different reservoirsis in good match which reflects the type of boundary
6 Analysis of Flow Characteristics andField Application
61 Flow Regime Recognition Type curves for horizontalwells in a CBM reservoir with complex boundary calculatedby the BEM in this paper are shown in Figure 6 The figureshows that flow characteristics can be divided into sevenflow periods (1) wellbore storage period influenced by earlywellbore storage effect the curves of pressure and pressurederivative are straight lines with the same slope in this period(2) the first transition flow section which reflects the degreeof pollution near the bottom and is mainly affected by skin
8 Journal of Chemistry
001 0
1 1 10 100
1000
1000
0
1000
00
01
1
10
100
1Eminus4
1Eminus3
120595D of conventional gas reservoir horizontal well120595998400D of conventional gas reservoir horizontal well
120595D of CBM horizontal well120595998400D of CBM horizontal well
TD
120595D
and120595998400 D
Figure 5 Comparison of typical curves between horizontal wells inCBM and conventional gas reservoirs
001 0
1 1 10 100
1000
1000
0
1000
00
001
01
1
10
100
Closed boundaryConstant pressure boundaryMixed boundary
1Eminus6
1Eminus5
1Eminus4
1Eminus3
I II III IV V VI VII
TD
120595D
and120595998400 D
Figure 6 Typical curve of CBM horizontal well with complexboundary conditions (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
factor 119878 (3) the first radial flow section which is perpendic-ular to the horizontal wellbore the pressure derivative curveis shown as a horizontal line which reflects the early radialflow perpendicular to the horizontal wellbore (4) the secondtransition section which is the transition stage of radial flowfrom wellbore to fracture (5) the second radial flow stagewhich reflects the free gas flow in fracture this period isaffected by the flow of free gas and adsorbed gas diffusioncoefficient 120582 and Langmuir adsorption parameter 120573 (6) thethird radial flow section which reflects radial flow of whole
001 1 100 10000 1000000
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
120582 = 1
120582 = 100
120582 = 10000
Figure 7 Influence of diffusion coefficient 120582 on bottom-holepressure with mixed boundary (120596 = 001 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
system after pressure balance and pressure derivative curveappears as a flat behavior (7) boundary response sectionThepressure and pressure derivative curves with closed boundarywould be upward after pressure transmitting to the boundaryThe pressure derivative curve with constant pressure bound-ary would be in decline after pressure transmitting to theboundary Compared with the constant pressure boundarythe falling range of pressure derivative curve with mixedboundary is less
62 Parameter Sensitivity Analysis Figure 7 shows the influ-ence of diffusion coefficient 120582 on the bottom-hole pressurewith mixed boundary There is relevance between diffusioncoefficient 120582 = 120572119896120591120579119871
2 and adsorption time 120591 = 1198772
119863The smaller the 120591 the shorter the time of desorption-diffusionand the shorter the time of pressure balance between fractureand matrix As shown in the diagram diffusion coefficient 120582mainly affects the duration of the second radial flow periodand appearance of third radial flow The greater the diffusioncoefficient 120582 the longer the duration of second radial flowperiod and the later the 119905 appearance of third radial flow andvice versa If 120582 is small enough the second radial flow periodwould be covered
Figure 8 shows the influence of Langmuir parameter 120573on the bottom-hole pressure with mixed boundary 120573 =
120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894) is related to Langmuir
adsorption pressure 119901119871and Langmuir adsorption volume119881
119871
The larger the 120573 the greater the adsorption ability of coalAs shown in the graph 120573 only affects the second radial flowsection The greater the 120573 the smaller the pressure derivativevalues of second radial flow period
Figure 9 shows the influence of fracture storage ratio120596 onthe bottom-hole pressure with mixed boundary Its expres-sion 120596 = 120601
119891119888119905120583(120601119891119888119905120583 + 6119901
119904119888119879119902119863119879119904119888120595119894) indicates that the
smaller the 120601119891119888119905 the smaller the 120596 and the more the radial
Journal of Chemistry 9
001 1 100 10000 1000000
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
0011E8
120573 = 01
120573 = 1
120573 = 10
Figure 8 Influence of 120573 on the bottom-hole pressure with mixedboundary (120596 = 001 120582 = 300 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
1E8
120596 = 04
120596 = 01
120596 = 001
Figure 9 Influence of 120596 on the bottom-hole pressure with mixedboundary (120582 = 300 120573 = 50 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
flow in fracture From Figure 8 almost all of the flow periodswould be affected by 120596 except for wellbore storage periodThe greater the 120596 the smaller the hump value of pressurederivative and the longer the duration of first radial flowperiod the greater the pressure derivative value in secondradial flow period
Figure 10 shows the influence of eccentricity of horizontalwell 119885
119908119889on the bottom-hole pressure with mixed boundary
The streamline shape and pressure distribution of a horizon-tal well would be affected by the size of 119885
119908119889 The smaller
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
Zwd = 01
Zwd = 03
Zwd = 05
Figure 10 Influence of119885119908119889
on the bottom-hole pressure withmixedboundary (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25 119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
LD = 1
LD = 25
LD = 50
Figure 11 Influence of 119871119863on the bottom-hole pressure with mixed
boundary (120596 = 001 120582 = 300 120573 = 50119885119908119889= 05119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
the119885119908119889 the larger the pressure derivative value in first radial
flow periodFigure 11 shows the influence of length of horizontal well
119871119863on the bottom-hole pressure with mixed boundary It
shows large influence of 119871119863on the pressure derivative value
in first radial flow period The smaller the 119871119863 the larger the
pressure derivative value in first radial flow period
63 Field Example To demonstrate the application of themodel proposed in this paper a horizontal well ZX-12 in aCBM reservoir is studied The vertical depth and horizontallength of this well are 689m and 536m respectively withwell radius 119903
119908of 01m coal thickness of 45m and initial
10 Journal of Chemistry
10 100100
1000
1000
TD
120595D
and120595998400 D
120595D of CBM horizontal well fitted curve120595998400D CBM horizontal well fitted curve
120595D of CBM horizontal well product curve120595998400D of CBM horizontal well product data curve
Figure 12 Fitted curves of test data from an actual well
pressure of 55MPa According to test results the Langmuirvolume 119881
119871is 3275m3t and the Langmuir pressure 119875
119871is
249MPa Figure 12 is the fitted curves which were obtainedby calculating the actual production and pressure data Asshown in Figure 12 the field data are in good match with thetheoretical results calculated by the new model in this paper
According to the actual field data some of reservoirparameters are fitted as follows the porosity is 004 and thepermeability is 3mD The stage of desorption and boundaryresponse can be clearly seen in Figure 12 Because of thefalling of the pressure derivative curve in the last stage wellZX-12 has a constant pressure boundary implying that thegas production of well ZX-12 is affected by the production ofadjacent well
7 Conclusions
Themathematic model of gas flowing into horizontal wells inCBMreservoirswith complex boundary conditions is derivedbased on the percolation theory The curves of bottom-hole pressure and pressure derivative are obtained by usingboundary element method and Laplace transform The con-clusions are as follows
(1) The fundamental boundary element solutions fortransient pressure response of horizontal wells inCBM reservoirs with complex boundary conditionscould be obtained by using Lord Kelvinrsquos point sourcesolution point source function theory and Poissonrsquossummation formula
(2) Comparison of the typical curves of flow character-istics between horizontal wells in CBM and conven-tional gas reservoirs shows that there is an additionalradial flow which reflects gas flow in fracture
(3) Comparing the typical curves of flow characteris-tics with complex boundary conditions the pressure
and pressure derivative curves with closed boundarywould be upward after pressure transmitting to theboundary The pressure derivative curves with con-stant pressure boundary and mixed boundary wouldbe fallen but the falling range of pressure derivativecurve with mixed boundary is less
Nomenclature
119903119908 Radius m
119896 Permeability mD120601 Porosity fraction119869 Diffusion flux gm2s119904 Laplace transform variableℎ Thickness m119861 Volume factor fraction119902 Influx into the wellbore m3d119871 Half length of horizontal well m120583 Viscosity mPasdots119888 Concentration of coalbed methane kgm3119875119894 Initial pressure MPa
119902 Production of well m3d119881119871 Langmuir volume constant m3ton
119881 Volume of coal matrix m3120595119894 Pseudopressure
120596 Fracture storage ratio fraction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is supported by National Natural Science Foun-dation of China (Grant no 51304032) National MajorScience and Technology Special Project of China (Grantno 2011ZX05037-003) and Research Fund for the Doc-toral Program of Higher Education of China (Grant no20125122110017)
References
[1] J C Cai E Perfect C-L Cheng and X Y Hu ldquoGeneralizedmodeling of spontaneous imbibition based on hagen-poiseuilleflow in tortuous capillaries with variably shaped aperturesrdquoLangmuir vol 30 no 18 pp 5142ndash5151 2014
[2] T Ertekin and W Sung ldquoPressure transient analysis of coalseams in the presence of multi-mechanistic flow and sorptionphenomenardquo in Proceedings of the SPE Gas Technology Sympo-sium pp 469ndash477 June 1989
[3] K Anbarci and T Ertekin ldquoA comprehensive study of pressuretransient analysis with sorption phenomena for single phase gasflow in coal seamsrdquo SPE 20568 1990
[4] C R Clarkson and R M Bustin ldquoEffect of pore structureand gas pressure upon the transport properties of coal alaboratory and modeling study 1 Isotherms and pore volumedistributionsrdquo Fuel vol 78 no 11 pp 1333ndash1344 1999
Journal of Chemistry 11
[5] C R Clarkson and R M Bustin ldquoEffect of pore structure andgas pressure upon the transport properties of coal a laboratoryandmodeling study 2 Adsorption rate modelingrdquo Fuel vol 78no 11 pp 1345ndash1362 1999
[6] S Reeve ldquoAdvanced reservoir modeling in desorption con-trolled reservoirsrdquo in Proceedings of the SPE Rocky MountainPetroleum Technology Conference SPE 71090 May 2001
[7] D K Tong and S Liu ldquoUnsteady percolation flow of coalbedmethane through deformed coal seamrdquo Natural Gas Industryvol 1 no 1 pp 74ndash76 2005
[8] X M Zhang and D K Tong ldquoAnalysis for pressure transient ofcoalbed methane reservoirrdquo Well Testing vol 6 no 17 pp 1ndash42008
[9] X M Zhang and D K Tong ldquoPressure transient analysisfor coal seams with triple-porositydual-permeability modelrdquoChineseQuarterly ofMechanics vol 4 no 29 pp 578ndash582 2008
[10] X H Hu S M Hu D I Zhang et al ldquoApplication of quadraticpressure method in well test interpretation of gas water twophase flow in coal seamrdquo Journal of Oil and Gas Technology vol7 no 33 pp 118ndash122 2011
[11] C R Clarkson C L Jordan D Ilk and T A Blasingame ldquoRate-transient analysis of 2-phase (gas + water) CBM wellsrdquo Journalof Natural Gas Science and Engineering vol 8 pp 106ndash120 2012
[12] K Aminian and S Ameri ldquoPredicting production performanceof CBM reservoirsrdquo Journal of Natural Gas Science and Engi-neering vol 1 no 1-2 pp 25ndash30 2009
[13] J C Cai and BM Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011
[14] W Sung T Ertekin and F C Schwerer ldquoThe developmenttesting and application of a comprehensive coal seam degasi-fication modelrdquo in Proceedings of the SPE Unconventional GasTechnology Symposium SPE 15247 Louisville Ky USA May1986
[15] W Sung and T Ertekin ldquoAn analysis of field developmentstrategies formethane production from coal seamsrdquo in Proceed-ings of the 62nd Annual Technical Conference and ExhibitionConference Paper SPE 16858 Dallas Tex USA 1987
[16] T Ertekin W Sung and F C Schwerer ldquoProduction perfor-mance analysis of horizontal drainage wells for degasificationof coal-seamsrdquo Journal of Petroleum Technology vol 40 no 5pp 625ndash632 1988
[17] T W Engler and J M Rajtar ldquoPressure transient testing ofhorizontal wells in coalbed reservoirsrdquo in Proceedings of theSPE Rocky Mountain Regional Meeting Paper SPE-24374-MSCasper Wyo USA May 1992
[18] P S Sarkar and J M Rajtar ldquoHorizontal well transient pres-sure testing in coalbed reservoirsrdquo in Proceedings of the 3rdLatin AmericaCaribbean Petroleum Engineering ConferenceSPE 26995 Buenos Aires Argentina April 1994
[19] P S Sarkar and J M Rajtar ldquoTransient well testing of coalbedmethane reservoirs with horizontal wellsrdquo in Proceedings of thePermian Basin Oil amp Gas Recovery Conference Paper SPE27681pp 531ndash539 Midland Tex USA March 1994
[20] X H Wang D I Zhang and Y Song ldquoDevelopment mecha-nism of low permeable coalbed methane frommultilateral hor-izontal pinnatewell with non-darcy seepage flow characteristicrdquoActa Geologica Sinica vol 7 no 82 pp 1437ndash1443 2008
[21] R-S Nie Y-F Meng J-C Guo and Y-L Jia ldquoModelingtransient flow behavior of a horizontal well in a coal seamrdquoInternational Journal of Coal Geology vol 92 pp 54ndash68 2012
[22] K L Katsifarakis D K Karpouzos and N TheodossiouldquoCombined use of BEM and genetic algorithms in groundwaterflow and mass transport problemsrdquo Engineering Analysis withBoundary Elements vol 23 no 7 pp 555ndash565 1999
[23] D T Numbere and D Tiab ldquoImproved streamline-generatingtechnique that uses the boundary (integral) element methodrdquoSPE Reservoir Engineering vol 3 no 3 pp 1061ndash1068 1988
[24] J Kikani and R N Horne ldquoApplication of boundary ele-ment method to reservoir engineering problemsrdquo Journal ofPetroleum Science and Engineering vol 3 no 3 pp 229ndash2411989
[25] J Hou YDWang andYMChen ldquoBoundary elementmethodin enhanced oil recoveryrdquo Chinese Journal of Hydrodynamicsvol 3 no 13 pp 23ndash28 2003
[26] K Chaiyo P Rattanadecho and S Chantasiriwan ldquoThemethodof fundamental solutions for solving free boundary saturatedseepage problemrdquo International Communications in Heat andMass Transfer vol 38 no 2 pp 249ndash254 2011
[27] K Rafiezadeh and B Ataie-Ashtiani ldquoSeepage analysis inmulti-domain general anisotropic media by three-dimensionalboundary elementsrdquo Engineering Analysis with Boundary Ele-ments vol 37 no 3 pp 527ndash541 2013
[28] X H Fu Y Qin W H Zhang et al ldquoCoal pore fractal classifi-cation and natural classification research based on the coal-bedgas migrationrdquo Chinese Science Bulletin vol 12 supplement 1pp 51ndash55 2005
[29] J C Cai and S Y Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal ofModern Physics C vol 24 no 8 pp 135ndash156 2013
[30] D M Han The Dynamic Analysis Technology Research onHorizontal Wells of Jingbian Gas Field University of XirsquoanShiyou Xirsquoan China 2011
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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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
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Journal of
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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 3
(a) (b)
Figure 1 Pore characteristics of coal ((a) micropore (b) fracture)
Figure 2 Desorption and diffusion in the CBM reservoirs
L
(a)
L
(b)
L
(c)
Figure 3 Horizontal wells with different boundaries ((a) closed boundary (b) constant pressure boundary (c) mixed boundary)
The planar radial flow equation is as follows
120597119888
120597119905
=
119863
119903119868
119894
120597
120597119903119894
(119903119868
119894
120597119888
120597119903119894
) (2)
The dimensionless equation of gas diffusion in matrix is
1
119903119894119863
2
120597
120597119903119894119863
(119903119894119863
2120597119888119863
120597119903119894119863
) = 1205821
120597119888119863
120597119905119863
(3)
where 119903119894119863
is the dimensionless radius defined by 119903119894119863= 119903119894119877
119888119863is the dimensionless diffusion concentration defined by
119888119863= 119888 minus 119888
119894 119905119863is the dimensionless time defined by 119905
119863=
36119896119905120579119903119908
2 1205821is interporosity flow coefficient defined by
1205821= 36119896119877
2
120579119863119903119908
2 120579 is comprehensive storage coefficientdefined by 120579 = 120601
119891119888119905120583+6119901
119904119888119879119902119863119879119904119888120595119894 119902119863is the dimensionless
production defined by 119902119863= 1842times10
minus3
119902119904119888119901119904119888119879119896ℎ119879
119904119888120595119894 119888119894is
the initial concentration of gas kgm3 119896 is the permeability120583m2 and 120595
119894is the pseudopressure
32 Gas Flow Model in Matrix and Fracture In the processof diffusion of CBM the concentration often changes There-fore unsteady diffusion equation which accords with actualsituation is used
Free gas concentration is as follows
1198881= 1205881120601 =
119872119901120601
119877119879119885
(4)
Concentration of the adsorbed gas is
1198882= 119881119871
119875
119875119871+ 119875
(5)
4 Journal of Chemistry
Gas concentration in coal is
119888 =
119872119901120601
119877119879119885
+ 119881119871
119875
119875119871+ 119875
(6)
Volume of gas desorption from coal is
119902119889= minus
120597119872119889
120597119905
= minus120588119904119888
120597119881119889
120597119905
(7)
where119872119889is the gas mass under standard condition kgm3
119881119889is the volume of gas adsorption in coal under standard
condition m3m3 120588119904119888is the density of gas under standard
condition kgm3Density of gas is defined as
120588119904119888=
119872119901119904119888
119911119877119879119904119888
(8)
Spherical matrix has the following relationship
120597119881119889
120597119905
= minus
119860
119881
119869 = minus
119860
119881
119863
120597119888
120597119903119894
=
3119863
119877
120597119888
120597119903119894
(9)
where119881 is the volume of coalmatrixm3119860 is the surface areaof coal matrix m2 119863 is the mass diffusion coefficient m2s119869 is the diffusion flux gm2s
Combining (7) (8) and (9) the volume of gas desorptionis given by
119902119889=
119872119901119904119888
119877119879119904119888
3119863
119877
120597119888
120597119903119894
(10)
Based on the basic principle of material balance the gov-erning equations of gas flow in fracture system are given by
1
119903
120597
120597119903
(119903120588
119896
120583119892
120597119901
120597119903
) + 119902119889= 120588120601119862
119905
120597119901
120597119905
(11)
Equation (11) is simplified into the following equation
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596
120597120595119863
120597119905119863
(12)
where 120596 is the fracture storage ratio given by 120596 = (120601119888119905120583)120579 120582
is the diffusion coefficient given by 120582 = 36119896120591120579119903119908
2 120595119863is the
dimensionless pseudopressure given by 120595119863= (120595119894minus 120595)120595
119894119902119863
120591 is the adsorption time given by 120591 = 1198772119863
33 Mathematical Model Solution Diffusion and governingequations of gas flow in fracture are combined as follows
1
119903119894119863
2
120597
120597119903119894119863
(119903119894119863
2120597119888119863
120597119903119894119863
) = 120582
120597119888119863
120597119905119863
(diffusion in matrix)
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596
120597120595119863
120597119905119863
(flow in fracture)
(13)
Dimensionless initial and boundary conditions are givenas follows
The initial condition is
120595119863(119903119863 119905119863= 0) = 0 (14)
The boundary condition is
120597120595119863
120597119903119863
(119903119863= 1 119905119863) = minus1 (15)
The infinite outer boundary condition is
120595119863(119903119863997888rarr infin 119905
119863) = 0 (16)
The constant pressure boundary condition is
120595119863(119903119890119863 119905119863) = 0 (17)
The closed boundary condition is
120597120595119863
120597119903119863
(119903119890119863 119905119863) = 0 (18)
When119872 = 119903119894119863119888119863 (13) can be transformed into
1205972
119872
120597119903119894119863
2= 120582
120597119872
120597119905119863
(19)
The Laplace transform of (19) is
1205972
119872
120597119903119894119863
2= 120582119906119872 (20)
The general solution of (20) is
119872 = 119860 sinh (radic120582119906119903119894119863) + 119861 cosh (radic120582119906119903
119894119863) (21)
where sinh(119909) = (119890119909 minus 119890minus119909)2 and cosh(119909) = (119890119909 + 119890minus119909)2The coefficients 119860 and 119861 could be obtained by the initial
and boundary conditions
119861 = 0 119860 =
119872119886
sinh (radic120582119906) (22)
Hence
119872 = 119872119886
sinh (radic120582119906119903119894119863)
sinh (radic120582119906) (23)
Therefore the diffusion equation is transformed into
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= 119888119863(radic120582119906 cothradic120582119906 minus 1) (24)
Combining the dimensionless definition 119888119863and Lang-
muir isothermal adsorption equation 119881 = 119881119871119875119898(119875119871+ 119875119898)
(24) is transformed into
120597119888119863
120597119903119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= 119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] (radic120582119906 cothradic120582119906 minus 1)
(25)
Journal of Chemistry 5
where 119871 is Laplace transform
119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] = 120573119875119863 (26)
So
120597119888119863
120597119903119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= minus120573119875119863(radic120582119906 cothradic120582119906 minus 1) (27)
Laplace transform of (20) is
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596119906120595119863 (28)
The initial conditions are
120595119863
1003816100381610038161003816119906rarrinfin
= 0 (29)
The inner boundary conditions are
119903119863
120597120595119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
= minus
1
119906
(30)
The outer boundary conditions are
120595119863
1003816100381610038161003816119903119863rarrinfin
= 0 (infinite)
120595119863
1003816100381610038161003816119903119863=119903119890119863
= 0 (constant pressure)
120597120595119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=119903119890119863
= 0 (closed)
(31)
The solution of diffusion equation (27) is
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= 119888119863(radic120582119906 cothradic120582119906 minus 1) (32)
Substituting the dimensionless definition 119888119863and Lang-
muir isothermal adsorption equation 119881 = 119881119871119875119898(119875119871+ 119875119898)
in (32) it can be written as follows
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= 119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] (radic120582119906 cothradic120582119906 minus 1)
(33)
Substituting 120595119863= (120595119894minus 120595)120595
119894119902119863in (33) it can be exp-
ressed as follows119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
= minus
120595119871119881119871120595119894119902119863
(120595119871+ 120595) (120595
119871+ 120595119894) (120595 + 120595
119894)
120595119863 (34)
Defining 120573 = 120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894)
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= minus120573120595119863(radic120582119906 cothradic120582119906 minus 1) (35)
Substituting (35) in (28)
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) = 119891 (119906) 120595119863
119891 (119906) = 120596119906 +
1 minus 120596
120582
120573 (radic120582119906 cothradic120582119906 minus 1)
(36)
4 Equation of Boundary Condition
41 Boundary Integral Equation Applying the theory ofboundary element previous equation can be solved from theintegral transformation of the governing equation based onthe expression of the fundamental solutions and differentialequation of fluid flow through porous medium
int
Ω
[120595119863(119875 119906) nabla
2
119866 (119875119876 119906) minus 119866 (119875 119876 119906) nabla2
120595119863(119875 119906)
+ 120575 (119875 119876) 120595119863(119875 119906) minus
1
119906
119873119908
sum
119894=1
119902119863119894120575 (119909119863minus 119909119863119894 119910119863minus 119910119863119894)
sdot 119866 (119875 119876 119906)] 119889Ω = 0
(37)
where 119866(119875119876 119906) is the fundamental solution of horizontalwells in complex boundary reservoirs
According to the properties of 120575 function and the secondorder of Green formula it can be simplified to the boundaryintegral equation
120595119863(119876119896 119906) = int
Γ
[119866 (119875119876119896 119906)
120597120595119863(1198751015840
119906)
120597119899
minus120595119863(119875 119906)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889Γ (1198751015840
)
+
1
119906
119873119908
sum
119894=1
119902119863119894119866 (119875119876
119894 119906)
(38)
The boundary Γ is divided into119873119887cells which are located
at the end point and are taken as the nodes of boundaryelements Cell properties are assumed as linear distributionMeanwhile the boundary sections near nodes are assumed asarcs with nodes as their centers the resulting boundary integ-ral equation is
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
int
Γ119894
[119866 (1198751015840
119876119896 119906)
120597120595119863(1198751015840
119906)
120597119899
minus120595119863(1198751015840
119906)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889Γ119894(1198751015840
)
+
1
119906
119873119908
sum
119894=1
119902119863119894119866 (119875119876
119894 119906)
(39)
6 Journal of Chemistry
where 120579119896represents interior angles between any two adjacent
boundary elements Consider
120579119896=
1 the point in domain 120579119894= 2120587
05 the point at smooth boundary 120579119894= 120587
120579119894
2120587
the point at smoothless boundary
(40)
Using the linear interpolation in boundary element theboundary integral formula is deformed as follows
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
119897119894
2
int
1
minus1
[119866 (1198751015840
119876119896 119906) (120593
1(120585)
120597120595119863119894
120597119899
+ 1205932(120585)
120597120595119863119894+1
120597119899
)
minus (1205931(120585) 120595119863119894+ 1205932(120585) 120595119863119894+1
)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889120585
+
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876119894 119906)
(41)
where 1206011(120585) = (1 minus 120585)2 and 120601
2(120585) = (1 + 120585)2 are linear
interpolation formula 119897119894= radic(119909
119894+1minus 119909119894)2
+ (119910119894+1minus 119910119894)2 minus1 lt
120585 lt 1 and Γ119894is the length of linearity cell
42 Fundamental Solution of Boundary Element IntegralEquation It is crucial to find its fundamental solution whenthe horizontal flow problem in complex reservoirs is resolvedusing the boundary element method According to the prop-erties of the boundary element and the mathematic equationgoverning pressure transmission in porous medium thefundamental solution must satisfy the modified Helmholtzoperator The equation is given by
1
119903
120597
120597119903
(119903
120597119866
120597119903
) minus 119891 (119906)119866 = minus2120587120575 (1198721198631198721015840
119863) (42)
With Lord Kelvin point source solution the fundamentalsolution of (42) can be derived as follows
120574 =
exp (minus120588119863radic119891 (119906))
4120587120588119863
(43)
With mirror image method of fluid mechanics in porousmedium the transient point source fundamental solution ofclosed boundary is
120574 =
1
4120587
+infin
sum
minusinfin
exp (minusradic119891 (119906)radic1198772119863+ (119885119863+ 1198851015840
119863minus 2119899119885
119890119863)2
)
radic1198772
119863+ (119885119863+ 1198851015840
119863minus 2119899119885
119890119863)2
+
exp (minusradic119891 (119906)radic1198772119863+(119885119863minus1198851015840
119863minus2119899119885
119890119863)2
)
radic1198772
119863+(119885119863minus 1198851015840
119863minus2119899119885
119890119863)2
(44)
With Poisson superposition formula (44) can be simpli-fied and the transient point source fundamental solution ofsealed boundary at 119885 = 0 and 119885 = 119885119890 is
120574 =
1
2120587119885119890119863
[1198700(119877119863radic119891 (119906))
+ 2
119899=infin
sum
119899=1
1198700(119877119863radic119891 (119906) +
1198992
1205872
119885119890119863
2)
sdot cos(119899120587 119885119863119885119890119863
) cos(1198991205871198851015840
119863
119885119890119863
)]
(45)
The fundamental boundary element solution of horizon-tal wells in a reservoir with closed top and bottom boundariesis
119866(1198751015840
119876 119906)
=
1
2
int
1
minus1
1198700(119877119863radic119891 (119906)) 119889120572
+
119899=infin
sum
119899=1
cos (119899120587119911119863) cos (119899120587119911
119908119863)
sdot int
1
minus1
1198700(radic(119909
119863minus 120572)2
+ 119910119863
2radic119891 (119906) +
1198992
1205872
119885119890119863
2)119889120572
120597119866 (1198751015840
119876 119906)
120597119899
= minus
1
2
int
1
minus1
radic119891 (119906)1198701((119903119863minus 120572)radic119891 (119906))
120597119903119863
120597119899
119889120572
minus
119899=infin
sum
119899=1
cos (119899120587119911119863)
sdot cos (119899120587119911119908119863) int
1
minus1
radic119891 (119906) +
1198992
1205872
119885119890119863
2
sdot 1198701((119903119863minus 120572)
sdotradic119891 (119906) +
1198992
1205872
119885119890119863
2)
120597119903119863
120597119899
119889120572
(46)
where120597119903119863
120597119899
= plusmn
100381610038161003816100381610038161003816100381610038161003816
((119909120585minus 119909) (119910
119894minus 119910119894+1) minus (119910
120585minus 119910) (119909
119894minus 119909119894+1))
sdot (radic(119909119894minus 119909119894+1)2
+ (119910119894minus 119910119894+1)2
)
minus1100381610038161003816100381610038161003816100381610038161003816
sdot (radic(119909120585minus 119909119894)2
+ (119910120585minus 119910119894)2
)
minus1
(47)
Journal of Chemistry 7
If 119899 and 1198751015840 are in the same direction then 120597119903119863120597119899 is posi-
tive otherwise it is negative
43 Solving of Integral Equation The integral boundary equa-tion (37) can be simplified as
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(48)
where
1198671015840
1=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206011(120585) 119889120585
1198671015840
2=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206012(120585) 119889120585
1198671015840
3=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206011(120585) 119889120585
1198671015840
4=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206012(120585) 119889120585
(49)
The unknown variable in the integral boundary equationis 120597120595119863119894120597119899 or 120595
119863119894 The number of the nodes at the boundary
is 119873119887 so 119873
119887equation with form of (48) can be established
When the boundary properties are known there are just 119873119887
unknown variables So we can solve the set of equationswhose matrix expression is
[[[[[[[
[
11986711
11986712
sdot sdot sdot 1198671119873119887
11986721
11986722
sdot sdot sdot 1198672119873119887
1198671198731198871
1198671198731198872
sdot sdot sdot 119867119873119887119873119887
]]]]]]]
]
[[[[[[
[
1199091
1199092
119909119873119887
]]]]]]
]
=
[[[[[[
[
1198651
1198652
119865119873119887
]]]]]]
]
(50)
where119909119894is 120597120595119863119894120597119899 or120595
119863119894and119865119894is (1119906)sum119873119908
119894=1119902119863119894119866(1198751015840
119876 119906)Once the unknown variables are acquired we can solve
119875119863
of arbitrary point in the research domain using theboundary integral equation (50)
120595119863(119876 119906) =
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894
+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(51)
5 Validation of the Model
In order to validate the gas flowmodel proposed in this paperwe test the adsorption and desorption data of coal Then wecompare the boundary element model with those publishedworks
001 01 1 10 100
1E16
1E17
120595D
120595998400D
TD
120595D
and120595998400 D
Figure 4 Double logarithmic curve of pressure drop of test data
51 Validation of the Gas FlowModel in CBM Reservoirs Theadsorption and desorption data of coal are tested in the labo-ratory the pressure continues to drop as the coal continuesto adsorb methane We deal with the experimental data inlog-log coordinates as shown in Figure 4 It can be seen fromthe diagram that the results of the numerical simulation arein good match with the experimental data and are consistentwith the gas flow law in CBM reservoirs
52 Validation of Gas Flow in a Horizontal Well with ComplexBoundary A pressure buildup test example of a horizontalgas well in tight gas reservoir has been presented byHan [30]the production and pressure data fitting results show that thehorizontal gas well has a closed boundary
Figure 5 is the comparison of pressure curves betweenhorizontal wells in CBM and conventional gas reservoirs andthe values of parameters are fromHanrsquos paper [30] Accordingto the analysis of flow characteristics there is an additionalradial flowwhich reflects gas flow in fracture After this stagethe pressure transmits from fracture to matrix and gas startsto desorbTheflowcharacteristics of this stagewould be influ-enced by desorption velocity and desorption quantity Thelast stage of pressure derivative curves in different reservoirsis in good match which reflects the type of boundary
6 Analysis of Flow Characteristics andField Application
61 Flow Regime Recognition Type curves for horizontalwells in a CBM reservoir with complex boundary calculatedby the BEM in this paper are shown in Figure 6 The figureshows that flow characteristics can be divided into sevenflow periods (1) wellbore storage period influenced by earlywellbore storage effect the curves of pressure and pressurederivative are straight lines with the same slope in this period(2) the first transition flow section which reflects the degreeof pollution near the bottom and is mainly affected by skin
8 Journal of Chemistry
001 0
1 1 10 100
1000
1000
0
1000
00
01
1
10
100
1Eminus4
1Eminus3
120595D of conventional gas reservoir horizontal well120595998400D of conventional gas reservoir horizontal well
120595D of CBM horizontal well120595998400D of CBM horizontal well
TD
120595D
and120595998400 D
Figure 5 Comparison of typical curves between horizontal wells inCBM and conventional gas reservoirs
001 0
1 1 10 100
1000
1000
0
1000
00
001
01
1
10
100
Closed boundaryConstant pressure boundaryMixed boundary
1Eminus6
1Eminus5
1Eminus4
1Eminus3
I II III IV V VI VII
TD
120595D
and120595998400 D
Figure 6 Typical curve of CBM horizontal well with complexboundary conditions (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
factor 119878 (3) the first radial flow section which is perpendic-ular to the horizontal wellbore the pressure derivative curveis shown as a horizontal line which reflects the early radialflow perpendicular to the horizontal wellbore (4) the secondtransition section which is the transition stage of radial flowfrom wellbore to fracture (5) the second radial flow stagewhich reflects the free gas flow in fracture this period isaffected by the flow of free gas and adsorbed gas diffusioncoefficient 120582 and Langmuir adsorption parameter 120573 (6) thethird radial flow section which reflects radial flow of whole
001 1 100 10000 1000000
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
120582 = 1
120582 = 100
120582 = 10000
Figure 7 Influence of diffusion coefficient 120582 on bottom-holepressure with mixed boundary (120596 = 001 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
system after pressure balance and pressure derivative curveappears as a flat behavior (7) boundary response sectionThepressure and pressure derivative curves with closed boundarywould be upward after pressure transmitting to the boundaryThe pressure derivative curve with constant pressure bound-ary would be in decline after pressure transmitting to theboundary Compared with the constant pressure boundarythe falling range of pressure derivative curve with mixedboundary is less
62 Parameter Sensitivity Analysis Figure 7 shows the influ-ence of diffusion coefficient 120582 on the bottom-hole pressurewith mixed boundary There is relevance between diffusioncoefficient 120582 = 120572119896120591120579119871
2 and adsorption time 120591 = 1198772
119863The smaller the 120591 the shorter the time of desorption-diffusionand the shorter the time of pressure balance between fractureand matrix As shown in the diagram diffusion coefficient 120582mainly affects the duration of the second radial flow periodand appearance of third radial flow The greater the diffusioncoefficient 120582 the longer the duration of second radial flowperiod and the later the 119905 appearance of third radial flow andvice versa If 120582 is small enough the second radial flow periodwould be covered
Figure 8 shows the influence of Langmuir parameter 120573on the bottom-hole pressure with mixed boundary 120573 =
120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894) is related to Langmuir
adsorption pressure 119901119871and Langmuir adsorption volume119881
119871
The larger the 120573 the greater the adsorption ability of coalAs shown in the graph 120573 only affects the second radial flowsection The greater the 120573 the smaller the pressure derivativevalues of second radial flow period
Figure 9 shows the influence of fracture storage ratio120596 onthe bottom-hole pressure with mixed boundary Its expres-sion 120596 = 120601
119891119888119905120583(120601119891119888119905120583 + 6119901
119904119888119879119902119863119879119904119888120595119894) indicates that the
smaller the 120601119891119888119905 the smaller the 120596 and the more the radial
Journal of Chemistry 9
001 1 100 10000 1000000
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
0011E8
120573 = 01
120573 = 1
120573 = 10
Figure 8 Influence of 120573 on the bottom-hole pressure with mixedboundary (120596 = 001 120582 = 300 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
1E8
120596 = 04
120596 = 01
120596 = 001
Figure 9 Influence of 120596 on the bottom-hole pressure with mixedboundary (120582 = 300 120573 = 50 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
flow in fracture From Figure 8 almost all of the flow periodswould be affected by 120596 except for wellbore storage periodThe greater the 120596 the smaller the hump value of pressurederivative and the longer the duration of first radial flowperiod the greater the pressure derivative value in secondradial flow period
Figure 10 shows the influence of eccentricity of horizontalwell 119885
119908119889on the bottom-hole pressure with mixed boundary
The streamline shape and pressure distribution of a horizon-tal well would be affected by the size of 119885
119908119889 The smaller
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
Zwd = 01
Zwd = 03
Zwd = 05
Figure 10 Influence of119885119908119889
on the bottom-hole pressure withmixedboundary (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25 119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
LD = 1
LD = 25
LD = 50
Figure 11 Influence of 119871119863on the bottom-hole pressure with mixed
boundary (120596 = 001 120582 = 300 120573 = 50119885119908119889= 05119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
the119885119908119889 the larger the pressure derivative value in first radial
flow periodFigure 11 shows the influence of length of horizontal well
119871119863on the bottom-hole pressure with mixed boundary It
shows large influence of 119871119863on the pressure derivative value
in first radial flow period The smaller the 119871119863 the larger the
pressure derivative value in first radial flow period
63 Field Example To demonstrate the application of themodel proposed in this paper a horizontal well ZX-12 in aCBM reservoir is studied The vertical depth and horizontallength of this well are 689m and 536m respectively withwell radius 119903
119908of 01m coal thickness of 45m and initial
10 Journal of Chemistry
10 100100
1000
1000
TD
120595D
and120595998400 D
120595D of CBM horizontal well fitted curve120595998400D CBM horizontal well fitted curve
120595D of CBM horizontal well product curve120595998400D of CBM horizontal well product data curve
Figure 12 Fitted curves of test data from an actual well
pressure of 55MPa According to test results the Langmuirvolume 119881
119871is 3275m3t and the Langmuir pressure 119875
119871is
249MPa Figure 12 is the fitted curves which were obtainedby calculating the actual production and pressure data Asshown in Figure 12 the field data are in good match with thetheoretical results calculated by the new model in this paper
According to the actual field data some of reservoirparameters are fitted as follows the porosity is 004 and thepermeability is 3mD The stage of desorption and boundaryresponse can be clearly seen in Figure 12 Because of thefalling of the pressure derivative curve in the last stage wellZX-12 has a constant pressure boundary implying that thegas production of well ZX-12 is affected by the production ofadjacent well
7 Conclusions
Themathematic model of gas flowing into horizontal wells inCBMreservoirswith complex boundary conditions is derivedbased on the percolation theory The curves of bottom-hole pressure and pressure derivative are obtained by usingboundary element method and Laplace transform The con-clusions are as follows
(1) The fundamental boundary element solutions fortransient pressure response of horizontal wells inCBM reservoirs with complex boundary conditionscould be obtained by using Lord Kelvinrsquos point sourcesolution point source function theory and Poissonrsquossummation formula
(2) Comparison of the typical curves of flow character-istics between horizontal wells in CBM and conven-tional gas reservoirs shows that there is an additionalradial flow which reflects gas flow in fracture
(3) Comparing the typical curves of flow characteris-tics with complex boundary conditions the pressure
and pressure derivative curves with closed boundarywould be upward after pressure transmitting to theboundary The pressure derivative curves with con-stant pressure boundary and mixed boundary wouldbe fallen but the falling range of pressure derivativecurve with mixed boundary is less
Nomenclature
119903119908 Radius m
119896 Permeability mD120601 Porosity fraction119869 Diffusion flux gm2s119904 Laplace transform variableℎ Thickness m119861 Volume factor fraction119902 Influx into the wellbore m3d119871 Half length of horizontal well m120583 Viscosity mPasdots119888 Concentration of coalbed methane kgm3119875119894 Initial pressure MPa
119902 Production of well m3d119881119871 Langmuir volume constant m3ton
119881 Volume of coal matrix m3120595119894 Pseudopressure
120596 Fracture storage ratio fraction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is supported by National Natural Science Foun-dation of China (Grant no 51304032) National MajorScience and Technology Special Project of China (Grantno 2011ZX05037-003) and Research Fund for the Doc-toral Program of Higher Education of China (Grant no20125122110017)
References
[1] J C Cai E Perfect C-L Cheng and X Y Hu ldquoGeneralizedmodeling of spontaneous imbibition based on hagen-poiseuilleflow in tortuous capillaries with variably shaped aperturesrdquoLangmuir vol 30 no 18 pp 5142ndash5151 2014
[2] T Ertekin and W Sung ldquoPressure transient analysis of coalseams in the presence of multi-mechanistic flow and sorptionphenomenardquo in Proceedings of the SPE Gas Technology Sympo-sium pp 469ndash477 June 1989
[3] K Anbarci and T Ertekin ldquoA comprehensive study of pressuretransient analysis with sorption phenomena for single phase gasflow in coal seamsrdquo SPE 20568 1990
[4] C R Clarkson and R M Bustin ldquoEffect of pore structureand gas pressure upon the transport properties of coal alaboratory and modeling study 1 Isotherms and pore volumedistributionsrdquo Fuel vol 78 no 11 pp 1333ndash1344 1999
Journal of Chemistry 11
[5] C R Clarkson and R M Bustin ldquoEffect of pore structure andgas pressure upon the transport properties of coal a laboratoryandmodeling study 2 Adsorption rate modelingrdquo Fuel vol 78no 11 pp 1345ndash1362 1999
[6] S Reeve ldquoAdvanced reservoir modeling in desorption con-trolled reservoirsrdquo in Proceedings of the SPE Rocky MountainPetroleum Technology Conference SPE 71090 May 2001
[7] D K Tong and S Liu ldquoUnsteady percolation flow of coalbedmethane through deformed coal seamrdquo Natural Gas Industryvol 1 no 1 pp 74ndash76 2005
[8] X M Zhang and D K Tong ldquoAnalysis for pressure transient ofcoalbed methane reservoirrdquo Well Testing vol 6 no 17 pp 1ndash42008
[9] X M Zhang and D K Tong ldquoPressure transient analysisfor coal seams with triple-porositydual-permeability modelrdquoChineseQuarterly ofMechanics vol 4 no 29 pp 578ndash582 2008
[10] X H Hu S M Hu D I Zhang et al ldquoApplication of quadraticpressure method in well test interpretation of gas water twophase flow in coal seamrdquo Journal of Oil and Gas Technology vol7 no 33 pp 118ndash122 2011
[11] C R Clarkson C L Jordan D Ilk and T A Blasingame ldquoRate-transient analysis of 2-phase (gas + water) CBM wellsrdquo Journalof Natural Gas Science and Engineering vol 8 pp 106ndash120 2012
[12] K Aminian and S Ameri ldquoPredicting production performanceof CBM reservoirsrdquo Journal of Natural Gas Science and Engi-neering vol 1 no 1-2 pp 25ndash30 2009
[13] J C Cai and BM Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011
[14] W Sung T Ertekin and F C Schwerer ldquoThe developmenttesting and application of a comprehensive coal seam degasi-fication modelrdquo in Proceedings of the SPE Unconventional GasTechnology Symposium SPE 15247 Louisville Ky USA May1986
[15] W Sung and T Ertekin ldquoAn analysis of field developmentstrategies formethane production from coal seamsrdquo in Proceed-ings of the 62nd Annual Technical Conference and ExhibitionConference Paper SPE 16858 Dallas Tex USA 1987
[16] T Ertekin W Sung and F C Schwerer ldquoProduction perfor-mance analysis of horizontal drainage wells for degasificationof coal-seamsrdquo Journal of Petroleum Technology vol 40 no 5pp 625ndash632 1988
[17] T W Engler and J M Rajtar ldquoPressure transient testing ofhorizontal wells in coalbed reservoirsrdquo in Proceedings of theSPE Rocky Mountain Regional Meeting Paper SPE-24374-MSCasper Wyo USA May 1992
[18] P S Sarkar and J M Rajtar ldquoHorizontal well transient pres-sure testing in coalbed reservoirsrdquo in Proceedings of the 3rdLatin AmericaCaribbean Petroleum Engineering ConferenceSPE 26995 Buenos Aires Argentina April 1994
[19] P S Sarkar and J M Rajtar ldquoTransient well testing of coalbedmethane reservoirs with horizontal wellsrdquo in Proceedings of thePermian Basin Oil amp Gas Recovery Conference Paper SPE27681pp 531ndash539 Midland Tex USA March 1994
[20] X H Wang D I Zhang and Y Song ldquoDevelopment mecha-nism of low permeable coalbed methane frommultilateral hor-izontal pinnatewell with non-darcy seepage flow characteristicrdquoActa Geologica Sinica vol 7 no 82 pp 1437ndash1443 2008
[21] R-S Nie Y-F Meng J-C Guo and Y-L Jia ldquoModelingtransient flow behavior of a horizontal well in a coal seamrdquoInternational Journal of Coal Geology vol 92 pp 54ndash68 2012
[22] K L Katsifarakis D K Karpouzos and N TheodossiouldquoCombined use of BEM and genetic algorithms in groundwaterflow and mass transport problemsrdquo Engineering Analysis withBoundary Elements vol 23 no 7 pp 555ndash565 1999
[23] D T Numbere and D Tiab ldquoImproved streamline-generatingtechnique that uses the boundary (integral) element methodrdquoSPE Reservoir Engineering vol 3 no 3 pp 1061ndash1068 1988
[24] J Kikani and R N Horne ldquoApplication of boundary ele-ment method to reservoir engineering problemsrdquo Journal ofPetroleum Science and Engineering vol 3 no 3 pp 229ndash2411989
[25] J Hou YDWang andYMChen ldquoBoundary elementmethodin enhanced oil recoveryrdquo Chinese Journal of Hydrodynamicsvol 3 no 13 pp 23ndash28 2003
[26] K Chaiyo P Rattanadecho and S Chantasiriwan ldquoThemethodof fundamental solutions for solving free boundary saturatedseepage problemrdquo International Communications in Heat andMass Transfer vol 38 no 2 pp 249ndash254 2011
[27] K Rafiezadeh and B Ataie-Ashtiani ldquoSeepage analysis inmulti-domain general anisotropic media by three-dimensionalboundary elementsrdquo Engineering Analysis with Boundary Ele-ments vol 37 no 3 pp 527ndash541 2013
[28] X H Fu Y Qin W H Zhang et al ldquoCoal pore fractal classifi-cation and natural classification research based on the coal-bedgas migrationrdquo Chinese Science Bulletin vol 12 supplement 1pp 51ndash55 2005
[29] J C Cai and S Y Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal ofModern Physics C vol 24 no 8 pp 135ndash156 2013
[30] D M Han The Dynamic Analysis Technology Research onHorizontal Wells of Jingbian Gas Field University of XirsquoanShiyou Xirsquoan China 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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CatalystsJournal of
4 Journal of Chemistry
Gas concentration in coal is
119888 =
119872119901120601
119877119879119885
+ 119881119871
119875
119875119871+ 119875
(6)
Volume of gas desorption from coal is
119902119889= minus
120597119872119889
120597119905
= minus120588119904119888
120597119881119889
120597119905
(7)
where119872119889is the gas mass under standard condition kgm3
119881119889is the volume of gas adsorption in coal under standard
condition m3m3 120588119904119888is the density of gas under standard
condition kgm3Density of gas is defined as
120588119904119888=
119872119901119904119888
119911119877119879119904119888
(8)
Spherical matrix has the following relationship
120597119881119889
120597119905
= minus
119860
119881
119869 = minus
119860
119881
119863
120597119888
120597119903119894
=
3119863
119877
120597119888
120597119903119894
(9)
where119881 is the volume of coalmatrixm3119860 is the surface areaof coal matrix m2 119863 is the mass diffusion coefficient m2s119869 is the diffusion flux gm2s
Combining (7) (8) and (9) the volume of gas desorptionis given by
119902119889=
119872119901119904119888
119877119879119904119888
3119863
119877
120597119888
120597119903119894
(10)
Based on the basic principle of material balance the gov-erning equations of gas flow in fracture system are given by
1
119903
120597
120597119903
(119903120588
119896
120583119892
120597119901
120597119903
) + 119902119889= 120588120601119862
119905
120597119901
120597119905
(11)
Equation (11) is simplified into the following equation
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596
120597120595119863
120597119905119863
(12)
where 120596 is the fracture storage ratio given by 120596 = (120601119888119905120583)120579 120582
is the diffusion coefficient given by 120582 = 36119896120591120579119903119908
2 120595119863is the
dimensionless pseudopressure given by 120595119863= (120595119894minus 120595)120595
119894119902119863
120591 is the adsorption time given by 120591 = 1198772119863
33 Mathematical Model Solution Diffusion and governingequations of gas flow in fracture are combined as follows
1
119903119894119863
2
120597
120597119903119894119863
(119903119894119863
2120597119888119863
120597119903119894119863
) = 120582
120597119888119863
120597119905119863
(diffusion in matrix)
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596
120597120595119863
120597119905119863
(flow in fracture)
(13)
Dimensionless initial and boundary conditions are givenas follows
The initial condition is
120595119863(119903119863 119905119863= 0) = 0 (14)
The boundary condition is
120597120595119863
120597119903119863
(119903119863= 1 119905119863) = minus1 (15)
The infinite outer boundary condition is
120595119863(119903119863997888rarr infin 119905
119863) = 0 (16)
The constant pressure boundary condition is
120595119863(119903119890119863 119905119863) = 0 (17)
The closed boundary condition is
120597120595119863
120597119903119863
(119903119890119863 119905119863) = 0 (18)
When119872 = 119903119894119863119888119863 (13) can be transformed into
1205972
119872
120597119903119894119863
2= 120582
120597119872
120597119905119863
(19)
The Laplace transform of (19) is
1205972
119872
120597119903119894119863
2= 120582119906119872 (20)
The general solution of (20) is
119872 = 119860 sinh (radic120582119906119903119894119863) + 119861 cosh (radic120582119906119903
119894119863) (21)
where sinh(119909) = (119890119909 minus 119890minus119909)2 and cosh(119909) = (119890119909 + 119890minus119909)2The coefficients 119860 and 119861 could be obtained by the initial
and boundary conditions
119861 = 0 119860 =
119872119886
sinh (radic120582119906) (22)
Hence
119872 = 119872119886
sinh (radic120582119906119903119894119863)
sinh (radic120582119906) (23)
Therefore the diffusion equation is transformed into
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= 119888119863(radic120582119906 cothradic120582119906 minus 1) (24)
Combining the dimensionless definition 119888119863and Lang-
muir isothermal adsorption equation 119881 = 119881119871119875119898(119875119871+ 119875119898)
(24) is transformed into
120597119888119863
120597119903119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= 119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] (radic120582119906 cothradic120582119906 minus 1)
(25)
Journal of Chemistry 5
where 119871 is Laplace transform
119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] = 120573119875119863 (26)
So
120597119888119863
120597119903119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= minus120573119875119863(radic120582119906 cothradic120582119906 minus 1) (27)
Laplace transform of (20) is
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596119906120595119863 (28)
The initial conditions are
120595119863
1003816100381610038161003816119906rarrinfin
= 0 (29)
The inner boundary conditions are
119903119863
120597120595119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
= minus
1
119906
(30)
The outer boundary conditions are
120595119863
1003816100381610038161003816119903119863rarrinfin
= 0 (infinite)
120595119863
1003816100381610038161003816119903119863=119903119890119863
= 0 (constant pressure)
120597120595119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=119903119890119863
= 0 (closed)
(31)
The solution of diffusion equation (27) is
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= 119888119863(radic120582119906 cothradic120582119906 minus 1) (32)
Substituting the dimensionless definition 119888119863and Lang-
muir isothermal adsorption equation 119881 = 119881119871119875119898(119875119871+ 119875119898)
in (32) it can be written as follows
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= 119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] (radic120582119906 cothradic120582119906 minus 1)
(33)
Substituting 120595119863= (120595119894minus 120595)120595
119894119902119863in (33) it can be exp-
ressed as follows119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
= minus
120595119871119881119871120595119894119902119863
(120595119871+ 120595) (120595
119871+ 120595119894) (120595 + 120595
119894)
120595119863 (34)
Defining 120573 = 120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894)
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= minus120573120595119863(radic120582119906 cothradic120582119906 minus 1) (35)
Substituting (35) in (28)
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) = 119891 (119906) 120595119863
119891 (119906) = 120596119906 +
1 minus 120596
120582
120573 (radic120582119906 cothradic120582119906 minus 1)
(36)
4 Equation of Boundary Condition
41 Boundary Integral Equation Applying the theory ofboundary element previous equation can be solved from theintegral transformation of the governing equation based onthe expression of the fundamental solutions and differentialequation of fluid flow through porous medium
int
Ω
[120595119863(119875 119906) nabla
2
119866 (119875119876 119906) minus 119866 (119875 119876 119906) nabla2
120595119863(119875 119906)
+ 120575 (119875 119876) 120595119863(119875 119906) minus
1
119906
119873119908
sum
119894=1
119902119863119894120575 (119909119863minus 119909119863119894 119910119863minus 119910119863119894)
sdot 119866 (119875 119876 119906)] 119889Ω = 0
(37)
where 119866(119875119876 119906) is the fundamental solution of horizontalwells in complex boundary reservoirs
According to the properties of 120575 function and the secondorder of Green formula it can be simplified to the boundaryintegral equation
120595119863(119876119896 119906) = int
Γ
[119866 (119875119876119896 119906)
120597120595119863(1198751015840
119906)
120597119899
minus120595119863(119875 119906)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889Γ (1198751015840
)
+
1
119906
119873119908
sum
119894=1
119902119863119894119866 (119875119876
119894 119906)
(38)
The boundary Γ is divided into119873119887cells which are located
at the end point and are taken as the nodes of boundaryelements Cell properties are assumed as linear distributionMeanwhile the boundary sections near nodes are assumed asarcs with nodes as their centers the resulting boundary integ-ral equation is
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
int
Γ119894
[119866 (1198751015840
119876119896 119906)
120597120595119863(1198751015840
119906)
120597119899
minus120595119863(1198751015840
119906)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889Γ119894(1198751015840
)
+
1
119906
119873119908
sum
119894=1
119902119863119894119866 (119875119876
119894 119906)
(39)
6 Journal of Chemistry
where 120579119896represents interior angles between any two adjacent
boundary elements Consider
120579119896=
1 the point in domain 120579119894= 2120587
05 the point at smooth boundary 120579119894= 120587
120579119894
2120587
the point at smoothless boundary
(40)
Using the linear interpolation in boundary element theboundary integral formula is deformed as follows
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
119897119894
2
int
1
minus1
[119866 (1198751015840
119876119896 119906) (120593
1(120585)
120597120595119863119894
120597119899
+ 1205932(120585)
120597120595119863119894+1
120597119899
)
minus (1205931(120585) 120595119863119894+ 1205932(120585) 120595119863119894+1
)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889120585
+
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876119894 119906)
(41)
where 1206011(120585) = (1 minus 120585)2 and 120601
2(120585) = (1 + 120585)2 are linear
interpolation formula 119897119894= radic(119909
119894+1minus 119909119894)2
+ (119910119894+1minus 119910119894)2 minus1 lt
120585 lt 1 and Γ119894is the length of linearity cell
42 Fundamental Solution of Boundary Element IntegralEquation It is crucial to find its fundamental solution whenthe horizontal flow problem in complex reservoirs is resolvedusing the boundary element method According to the prop-erties of the boundary element and the mathematic equationgoverning pressure transmission in porous medium thefundamental solution must satisfy the modified Helmholtzoperator The equation is given by
1
119903
120597
120597119903
(119903
120597119866
120597119903
) minus 119891 (119906)119866 = minus2120587120575 (1198721198631198721015840
119863) (42)
With Lord Kelvin point source solution the fundamentalsolution of (42) can be derived as follows
120574 =
exp (minus120588119863radic119891 (119906))
4120587120588119863
(43)
With mirror image method of fluid mechanics in porousmedium the transient point source fundamental solution ofclosed boundary is
120574 =
1
4120587
+infin
sum
minusinfin
exp (minusradic119891 (119906)radic1198772119863+ (119885119863+ 1198851015840
119863minus 2119899119885
119890119863)2
)
radic1198772
119863+ (119885119863+ 1198851015840
119863minus 2119899119885
119890119863)2
+
exp (minusradic119891 (119906)radic1198772119863+(119885119863minus1198851015840
119863minus2119899119885
119890119863)2
)
radic1198772
119863+(119885119863minus 1198851015840
119863minus2119899119885
119890119863)2
(44)
With Poisson superposition formula (44) can be simpli-fied and the transient point source fundamental solution ofsealed boundary at 119885 = 0 and 119885 = 119885119890 is
120574 =
1
2120587119885119890119863
[1198700(119877119863radic119891 (119906))
+ 2
119899=infin
sum
119899=1
1198700(119877119863radic119891 (119906) +
1198992
1205872
119885119890119863
2)
sdot cos(119899120587 119885119863119885119890119863
) cos(1198991205871198851015840
119863
119885119890119863
)]
(45)
The fundamental boundary element solution of horizon-tal wells in a reservoir with closed top and bottom boundariesis
119866(1198751015840
119876 119906)
=
1
2
int
1
minus1
1198700(119877119863radic119891 (119906)) 119889120572
+
119899=infin
sum
119899=1
cos (119899120587119911119863) cos (119899120587119911
119908119863)
sdot int
1
minus1
1198700(radic(119909
119863minus 120572)2
+ 119910119863
2radic119891 (119906) +
1198992
1205872
119885119890119863
2)119889120572
120597119866 (1198751015840
119876 119906)
120597119899
= minus
1
2
int
1
minus1
radic119891 (119906)1198701((119903119863minus 120572)radic119891 (119906))
120597119903119863
120597119899
119889120572
minus
119899=infin
sum
119899=1
cos (119899120587119911119863)
sdot cos (119899120587119911119908119863) int
1
minus1
radic119891 (119906) +
1198992
1205872
119885119890119863
2
sdot 1198701((119903119863minus 120572)
sdotradic119891 (119906) +
1198992
1205872
119885119890119863
2)
120597119903119863
120597119899
119889120572
(46)
where120597119903119863
120597119899
= plusmn
100381610038161003816100381610038161003816100381610038161003816
((119909120585minus 119909) (119910
119894minus 119910119894+1) minus (119910
120585minus 119910) (119909
119894minus 119909119894+1))
sdot (radic(119909119894minus 119909119894+1)2
+ (119910119894minus 119910119894+1)2
)
minus1100381610038161003816100381610038161003816100381610038161003816
sdot (radic(119909120585minus 119909119894)2
+ (119910120585minus 119910119894)2
)
minus1
(47)
Journal of Chemistry 7
If 119899 and 1198751015840 are in the same direction then 120597119903119863120597119899 is posi-
tive otherwise it is negative
43 Solving of Integral Equation The integral boundary equa-tion (37) can be simplified as
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(48)
where
1198671015840
1=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206011(120585) 119889120585
1198671015840
2=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206012(120585) 119889120585
1198671015840
3=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206011(120585) 119889120585
1198671015840
4=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206012(120585) 119889120585
(49)
The unknown variable in the integral boundary equationis 120597120595119863119894120597119899 or 120595
119863119894 The number of the nodes at the boundary
is 119873119887 so 119873
119887equation with form of (48) can be established
When the boundary properties are known there are just 119873119887
unknown variables So we can solve the set of equationswhose matrix expression is
[[[[[[[
[
11986711
11986712
sdot sdot sdot 1198671119873119887
11986721
11986722
sdot sdot sdot 1198672119873119887
1198671198731198871
1198671198731198872
sdot sdot sdot 119867119873119887119873119887
]]]]]]]
]
[[[[[[
[
1199091
1199092
119909119873119887
]]]]]]
]
=
[[[[[[
[
1198651
1198652
119865119873119887
]]]]]]
]
(50)
where119909119894is 120597120595119863119894120597119899 or120595
119863119894and119865119894is (1119906)sum119873119908
119894=1119902119863119894119866(1198751015840
119876 119906)Once the unknown variables are acquired we can solve
119875119863
of arbitrary point in the research domain using theboundary integral equation (50)
120595119863(119876 119906) =
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894
+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(51)
5 Validation of the Model
In order to validate the gas flowmodel proposed in this paperwe test the adsorption and desorption data of coal Then wecompare the boundary element model with those publishedworks
001 01 1 10 100
1E16
1E17
120595D
120595998400D
TD
120595D
and120595998400 D
Figure 4 Double logarithmic curve of pressure drop of test data
51 Validation of the Gas FlowModel in CBM Reservoirs Theadsorption and desorption data of coal are tested in the labo-ratory the pressure continues to drop as the coal continuesto adsorb methane We deal with the experimental data inlog-log coordinates as shown in Figure 4 It can be seen fromthe diagram that the results of the numerical simulation arein good match with the experimental data and are consistentwith the gas flow law in CBM reservoirs
52 Validation of Gas Flow in a Horizontal Well with ComplexBoundary A pressure buildup test example of a horizontalgas well in tight gas reservoir has been presented byHan [30]the production and pressure data fitting results show that thehorizontal gas well has a closed boundary
Figure 5 is the comparison of pressure curves betweenhorizontal wells in CBM and conventional gas reservoirs andthe values of parameters are fromHanrsquos paper [30] Accordingto the analysis of flow characteristics there is an additionalradial flowwhich reflects gas flow in fracture After this stagethe pressure transmits from fracture to matrix and gas startsto desorbTheflowcharacteristics of this stagewould be influ-enced by desorption velocity and desorption quantity Thelast stage of pressure derivative curves in different reservoirsis in good match which reflects the type of boundary
6 Analysis of Flow Characteristics andField Application
61 Flow Regime Recognition Type curves for horizontalwells in a CBM reservoir with complex boundary calculatedby the BEM in this paper are shown in Figure 6 The figureshows that flow characteristics can be divided into sevenflow periods (1) wellbore storage period influenced by earlywellbore storage effect the curves of pressure and pressurederivative are straight lines with the same slope in this period(2) the first transition flow section which reflects the degreeof pollution near the bottom and is mainly affected by skin
8 Journal of Chemistry
001 0
1 1 10 100
1000
1000
0
1000
00
01
1
10
100
1Eminus4
1Eminus3
120595D of conventional gas reservoir horizontal well120595998400D of conventional gas reservoir horizontal well
120595D of CBM horizontal well120595998400D of CBM horizontal well
TD
120595D
and120595998400 D
Figure 5 Comparison of typical curves between horizontal wells inCBM and conventional gas reservoirs
001 0
1 1 10 100
1000
1000
0
1000
00
001
01
1
10
100
Closed boundaryConstant pressure boundaryMixed boundary
1Eminus6
1Eminus5
1Eminus4
1Eminus3
I II III IV V VI VII
TD
120595D
and120595998400 D
Figure 6 Typical curve of CBM horizontal well with complexboundary conditions (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
factor 119878 (3) the first radial flow section which is perpendic-ular to the horizontal wellbore the pressure derivative curveis shown as a horizontal line which reflects the early radialflow perpendicular to the horizontal wellbore (4) the secondtransition section which is the transition stage of radial flowfrom wellbore to fracture (5) the second radial flow stagewhich reflects the free gas flow in fracture this period isaffected by the flow of free gas and adsorbed gas diffusioncoefficient 120582 and Langmuir adsorption parameter 120573 (6) thethird radial flow section which reflects radial flow of whole
001 1 100 10000 1000000
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
120582 = 1
120582 = 100
120582 = 10000
Figure 7 Influence of diffusion coefficient 120582 on bottom-holepressure with mixed boundary (120596 = 001 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
system after pressure balance and pressure derivative curveappears as a flat behavior (7) boundary response sectionThepressure and pressure derivative curves with closed boundarywould be upward after pressure transmitting to the boundaryThe pressure derivative curve with constant pressure bound-ary would be in decline after pressure transmitting to theboundary Compared with the constant pressure boundarythe falling range of pressure derivative curve with mixedboundary is less
62 Parameter Sensitivity Analysis Figure 7 shows the influ-ence of diffusion coefficient 120582 on the bottom-hole pressurewith mixed boundary There is relevance between diffusioncoefficient 120582 = 120572119896120591120579119871
2 and adsorption time 120591 = 1198772
119863The smaller the 120591 the shorter the time of desorption-diffusionand the shorter the time of pressure balance between fractureand matrix As shown in the diagram diffusion coefficient 120582mainly affects the duration of the second radial flow periodand appearance of third radial flow The greater the diffusioncoefficient 120582 the longer the duration of second radial flowperiod and the later the 119905 appearance of third radial flow andvice versa If 120582 is small enough the second radial flow periodwould be covered
Figure 8 shows the influence of Langmuir parameter 120573on the bottom-hole pressure with mixed boundary 120573 =
120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894) is related to Langmuir
adsorption pressure 119901119871and Langmuir adsorption volume119881
119871
The larger the 120573 the greater the adsorption ability of coalAs shown in the graph 120573 only affects the second radial flowsection The greater the 120573 the smaller the pressure derivativevalues of second radial flow period
Figure 9 shows the influence of fracture storage ratio120596 onthe bottom-hole pressure with mixed boundary Its expres-sion 120596 = 120601
119891119888119905120583(120601119891119888119905120583 + 6119901
119904119888119879119902119863119879119904119888120595119894) indicates that the
smaller the 120601119891119888119905 the smaller the 120596 and the more the radial
Journal of Chemistry 9
001 1 100 10000 1000000
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
0011E8
120573 = 01
120573 = 1
120573 = 10
Figure 8 Influence of 120573 on the bottom-hole pressure with mixedboundary (120596 = 001 120582 = 300 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
1E8
120596 = 04
120596 = 01
120596 = 001
Figure 9 Influence of 120596 on the bottom-hole pressure with mixedboundary (120582 = 300 120573 = 50 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
flow in fracture From Figure 8 almost all of the flow periodswould be affected by 120596 except for wellbore storage periodThe greater the 120596 the smaller the hump value of pressurederivative and the longer the duration of first radial flowperiod the greater the pressure derivative value in secondradial flow period
Figure 10 shows the influence of eccentricity of horizontalwell 119885
119908119889on the bottom-hole pressure with mixed boundary
The streamline shape and pressure distribution of a horizon-tal well would be affected by the size of 119885
119908119889 The smaller
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
Zwd = 01
Zwd = 03
Zwd = 05
Figure 10 Influence of119885119908119889
on the bottom-hole pressure withmixedboundary (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25 119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
LD = 1
LD = 25
LD = 50
Figure 11 Influence of 119871119863on the bottom-hole pressure with mixed
boundary (120596 = 001 120582 = 300 120573 = 50119885119908119889= 05119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
the119885119908119889 the larger the pressure derivative value in first radial
flow periodFigure 11 shows the influence of length of horizontal well
119871119863on the bottom-hole pressure with mixed boundary It
shows large influence of 119871119863on the pressure derivative value
in first radial flow period The smaller the 119871119863 the larger the
pressure derivative value in first radial flow period
63 Field Example To demonstrate the application of themodel proposed in this paper a horizontal well ZX-12 in aCBM reservoir is studied The vertical depth and horizontallength of this well are 689m and 536m respectively withwell radius 119903
119908of 01m coal thickness of 45m and initial
10 Journal of Chemistry
10 100100
1000
1000
TD
120595D
and120595998400 D
120595D of CBM horizontal well fitted curve120595998400D CBM horizontal well fitted curve
120595D of CBM horizontal well product curve120595998400D of CBM horizontal well product data curve
Figure 12 Fitted curves of test data from an actual well
pressure of 55MPa According to test results the Langmuirvolume 119881
119871is 3275m3t and the Langmuir pressure 119875
119871is
249MPa Figure 12 is the fitted curves which were obtainedby calculating the actual production and pressure data Asshown in Figure 12 the field data are in good match with thetheoretical results calculated by the new model in this paper
According to the actual field data some of reservoirparameters are fitted as follows the porosity is 004 and thepermeability is 3mD The stage of desorption and boundaryresponse can be clearly seen in Figure 12 Because of thefalling of the pressure derivative curve in the last stage wellZX-12 has a constant pressure boundary implying that thegas production of well ZX-12 is affected by the production ofadjacent well
7 Conclusions
Themathematic model of gas flowing into horizontal wells inCBMreservoirswith complex boundary conditions is derivedbased on the percolation theory The curves of bottom-hole pressure and pressure derivative are obtained by usingboundary element method and Laplace transform The con-clusions are as follows
(1) The fundamental boundary element solutions fortransient pressure response of horizontal wells inCBM reservoirs with complex boundary conditionscould be obtained by using Lord Kelvinrsquos point sourcesolution point source function theory and Poissonrsquossummation formula
(2) Comparison of the typical curves of flow character-istics between horizontal wells in CBM and conven-tional gas reservoirs shows that there is an additionalradial flow which reflects gas flow in fracture
(3) Comparing the typical curves of flow characteris-tics with complex boundary conditions the pressure
and pressure derivative curves with closed boundarywould be upward after pressure transmitting to theboundary The pressure derivative curves with con-stant pressure boundary and mixed boundary wouldbe fallen but the falling range of pressure derivativecurve with mixed boundary is less
Nomenclature
119903119908 Radius m
119896 Permeability mD120601 Porosity fraction119869 Diffusion flux gm2s119904 Laplace transform variableℎ Thickness m119861 Volume factor fraction119902 Influx into the wellbore m3d119871 Half length of horizontal well m120583 Viscosity mPasdots119888 Concentration of coalbed methane kgm3119875119894 Initial pressure MPa
119902 Production of well m3d119881119871 Langmuir volume constant m3ton
119881 Volume of coal matrix m3120595119894 Pseudopressure
120596 Fracture storage ratio fraction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is supported by National Natural Science Foun-dation of China (Grant no 51304032) National MajorScience and Technology Special Project of China (Grantno 2011ZX05037-003) and Research Fund for the Doc-toral Program of Higher Education of China (Grant no20125122110017)
References
[1] J C Cai E Perfect C-L Cheng and X Y Hu ldquoGeneralizedmodeling of spontaneous imbibition based on hagen-poiseuilleflow in tortuous capillaries with variably shaped aperturesrdquoLangmuir vol 30 no 18 pp 5142ndash5151 2014
[2] T Ertekin and W Sung ldquoPressure transient analysis of coalseams in the presence of multi-mechanistic flow and sorptionphenomenardquo in Proceedings of the SPE Gas Technology Sympo-sium pp 469ndash477 June 1989
[3] K Anbarci and T Ertekin ldquoA comprehensive study of pressuretransient analysis with sorption phenomena for single phase gasflow in coal seamsrdquo SPE 20568 1990
[4] C R Clarkson and R M Bustin ldquoEffect of pore structureand gas pressure upon the transport properties of coal alaboratory and modeling study 1 Isotherms and pore volumedistributionsrdquo Fuel vol 78 no 11 pp 1333ndash1344 1999
Journal of Chemistry 11
[5] C R Clarkson and R M Bustin ldquoEffect of pore structure andgas pressure upon the transport properties of coal a laboratoryandmodeling study 2 Adsorption rate modelingrdquo Fuel vol 78no 11 pp 1345ndash1362 1999
[6] S Reeve ldquoAdvanced reservoir modeling in desorption con-trolled reservoirsrdquo in Proceedings of the SPE Rocky MountainPetroleum Technology Conference SPE 71090 May 2001
[7] D K Tong and S Liu ldquoUnsteady percolation flow of coalbedmethane through deformed coal seamrdquo Natural Gas Industryvol 1 no 1 pp 74ndash76 2005
[8] X M Zhang and D K Tong ldquoAnalysis for pressure transient ofcoalbed methane reservoirrdquo Well Testing vol 6 no 17 pp 1ndash42008
[9] X M Zhang and D K Tong ldquoPressure transient analysisfor coal seams with triple-porositydual-permeability modelrdquoChineseQuarterly ofMechanics vol 4 no 29 pp 578ndash582 2008
[10] X H Hu S M Hu D I Zhang et al ldquoApplication of quadraticpressure method in well test interpretation of gas water twophase flow in coal seamrdquo Journal of Oil and Gas Technology vol7 no 33 pp 118ndash122 2011
[11] C R Clarkson C L Jordan D Ilk and T A Blasingame ldquoRate-transient analysis of 2-phase (gas + water) CBM wellsrdquo Journalof Natural Gas Science and Engineering vol 8 pp 106ndash120 2012
[12] K Aminian and S Ameri ldquoPredicting production performanceof CBM reservoirsrdquo Journal of Natural Gas Science and Engi-neering vol 1 no 1-2 pp 25ndash30 2009
[13] J C Cai and BM Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011
[14] W Sung T Ertekin and F C Schwerer ldquoThe developmenttesting and application of a comprehensive coal seam degasi-fication modelrdquo in Proceedings of the SPE Unconventional GasTechnology Symposium SPE 15247 Louisville Ky USA May1986
[15] W Sung and T Ertekin ldquoAn analysis of field developmentstrategies formethane production from coal seamsrdquo in Proceed-ings of the 62nd Annual Technical Conference and ExhibitionConference Paper SPE 16858 Dallas Tex USA 1987
[16] T Ertekin W Sung and F C Schwerer ldquoProduction perfor-mance analysis of horizontal drainage wells for degasificationof coal-seamsrdquo Journal of Petroleum Technology vol 40 no 5pp 625ndash632 1988
[17] T W Engler and J M Rajtar ldquoPressure transient testing ofhorizontal wells in coalbed reservoirsrdquo in Proceedings of theSPE Rocky Mountain Regional Meeting Paper SPE-24374-MSCasper Wyo USA May 1992
[18] P S Sarkar and J M Rajtar ldquoHorizontal well transient pres-sure testing in coalbed reservoirsrdquo in Proceedings of the 3rdLatin AmericaCaribbean Petroleum Engineering ConferenceSPE 26995 Buenos Aires Argentina April 1994
[19] P S Sarkar and J M Rajtar ldquoTransient well testing of coalbedmethane reservoirs with horizontal wellsrdquo in Proceedings of thePermian Basin Oil amp Gas Recovery Conference Paper SPE27681pp 531ndash539 Midland Tex USA March 1994
[20] X H Wang D I Zhang and Y Song ldquoDevelopment mecha-nism of low permeable coalbed methane frommultilateral hor-izontal pinnatewell with non-darcy seepage flow characteristicrdquoActa Geologica Sinica vol 7 no 82 pp 1437ndash1443 2008
[21] R-S Nie Y-F Meng J-C Guo and Y-L Jia ldquoModelingtransient flow behavior of a horizontal well in a coal seamrdquoInternational Journal of Coal Geology vol 92 pp 54ndash68 2012
[22] K L Katsifarakis D K Karpouzos and N TheodossiouldquoCombined use of BEM and genetic algorithms in groundwaterflow and mass transport problemsrdquo Engineering Analysis withBoundary Elements vol 23 no 7 pp 555ndash565 1999
[23] D T Numbere and D Tiab ldquoImproved streamline-generatingtechnique that uses the boundary (integral) element methodrdquoSPE Reservoir Engineering vol 3 no 3 pp 1061ndash1068 1988
[24] J Kikani and R N Horne ldquoApplication of boundary ele-ment method to reservoir engineering problemsrdquo Journal ofPetroleum Science and Engineering vol 3 no 3 pp 229ndash2411989
[25] J Hou YDWang andYMChen ldquoBoundary elementmethodin enhanced oil recoveryrdquo Chinese Journal of Hydrodynamicsvol 3 no 13 pp 23ndash28 2003
[26] K Chaiyo P Rattanadecho and S Chantasiriwan ldquoThemethodof fundamental solutions for solving free boundary saturatedseepage problemrdquo International Communications in Heat andMass Transfer vol 38 no 2 pp 249ndash254 2011
[27] K Rafiezadeh and B Ataie-Ashtiani ldquoSeepage analysis inmulti-domain general anisotropic media by three-dimensionalboundary elementsrdquo Engineering Analysis with Boundary Ele-ments vol 37 no 3 pp 527ndash541 2013
[28] X H Fu Y Qin W H Zhang et al ldquoCoal pore fractal classifi-cation and natural classification research based on the coal-bedgas migrationrdquo Chinese Science Bulletin vol 12 supplement 1pp 51ndash55 2005
[29] J C Cai and S Y Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal ofModern Physics C vol 24 no 8 pp 135ndash156 2013
[30] D M Han The Dynamic Analysis Technology Research onHorizontal Wells of Jingbian Gas Field University of XirsquoanShiyou Xirsquoan China 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
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Carbohydrate Chemistry
International Journal of
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Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
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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 5
where 119871 is Laplace transform
119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] = 120573119875119863 (26)
So
120597119888119863
120597119903119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= minus120573119875119863(radic120582119906 cothradic120582119906 minus 1) (27)
Laplace transform of (20) is
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) +(1 minus 120596)
120582
120597119888119863
120597119903119894119863
= 120596119906120595119863 (28)
The initial conditions are
120595119863
1003816100381610038161003816119906rarrinfin
= 0 (29)
The inner boundary conditions are
119903119863
120597120595119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
= minus
1
119906
(30)
The outer boundary conditions are
120595119863
1003816100381610038161003816119903119863rarrinfin
= 0 (infinite)
120595119863
1003816100381610038161003816119903119863=119903119890119863
= 0 (constant pressure)
120597120595119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=119903119890119863
= 0 (closed)
(31)
The solution of diffusion equation (27) is
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= 119888119863(radic120582119906 cothradic120582119906 minus 1) (32)
Substituting the dimensionless definition 119888119863and Lang-
muir isothermal adsorption equation 119881 = 119881119871119875119898(119875119871+ 119875119898)
in (32) it can be written as follows
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119863=1
= 119871 [
119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
] (radic120582119906 cothradic120582119906 minus 1)
(33)
Substituting 120595119863= (120595119894minus 120595)120595
119894119902119863in (33) it can be exp-
ressed as follows119881119871119875
119875119871+ 119875
minus
119881119871119875119894
119875119871+ 119875119894
= minus
120595119871119881119871120595119894119902119863
(120595119871+ 120595) (120595
119871+ 120595119894) (120595 + 120595
119894)
120595119863 (34)
Defining 120573 = 120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894)
120597119888119863
120597119903119894119863
10038161003816100381610038161003816100381610038161003816119903119894119863=1
= minus120573120595119863(radic120582119906 cothradic120582119906 minus 1) (35)
Substituting (35) in (28)
1
119903119863
120597
120597119903119863
(119903119863
120597120595119863
120597119903119863
) = 119891 (119906) 120595119863
119891 (119906) = 120596119906 +
1 minus 120596
120582
120573 (radic120582119906 cothradic120582119906 minus 1)
(36)
4 Equation of Boundary Condition
41 Boundary Integral Equation Applying the theory ofboundary element previous equation can be solved from theintegral transformation of the governing equation based onthe expression of the fundamental solutions and differentialequation of fluid flow through porous medium
int
Ω
[120595119863(119875 119906) nabla
2
119866 (119875119876 119906) minus 119866 (119875 119876 119906) nabla2
120595119863(119875 119906)
+ 120575 (119875 119876) 120595119863(119875 119906) minus
1
119906
119873119908
sum
119894=1
119902119863119894120575 (119909119863minus 119909119863119894 119910119863minus 119910119863119894)
sdot 119866 (119875 119876 119906)] 119889Ω = 0
(37)
where 119866(119875119876 119906) is the fundamental solution of horizontalwells in complex boundary reservoirs
According to the properties of 120575 function and the secondorder of Green formula it can be simplified to the boundaryintegral equation
120595119863(119876119896 119906) = int
Γ
[119866 (119875119876119896 119906)
120597120595119863(1198751015840
119906)
120597119899
minus120595119863(119875 119906)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889Γ (1198751015840
)
+
1
119906
119873119908
sum
119894=1
119902119863119894119866 (119875119876
119894 119906)
(38)
The boundary Γ is divided into119873119887cells which are located
at the end point and are taken as the nodes of boundaryelements Cell properties are assumed as linear distributionMeanwhile the boundary sections near nodes are assumed asarcs with nodes as their centers the resulting boundary integ-ral equation is
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
int
Γ119894
[119866 (1198751015840
119876119896 119906)
120597120595119863(1198751015840
119906)
120597119899
minus120595119863(1198751015840
119906)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889Γ119894(1198751015840
)
+
1
119906
119873119908
sum
119894=1
119902119863119894119866 (119875119876
119894 119906)
(39)
6 Journal of Chemistry
where 120579119896represents interior angles between any two adjacent
boundary elements Consider
120579119896=
1 the point in domain 120579119894= 2120587
05 the point at smooth boundary 120579119894= 120587
120579119894
2120587
the point at smoothless boundary
(40)
Using the linear interpolation in boundary element theboundary integral formula is deformed as follows
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
119897119894
2
int
1
minus1
[119866 (1198751015840
119876119896 119906) (120593
1(120585)
120597120595119863119894
120597119899
+ 1205932(120585)
120597120595119863119894+1
120597119899
)
minus (1205931(120585) 120595119863119894+ 1205932(120585) 120595119863119894+1
)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889120585
+
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876119894 119906)
(41)
where 1206011(120585) = (1 minus 120585)2 and 120601
2(120585) = (1 + 120585)2 are linear
interpolation formula 119897119894= radic(119909
119894+1minus 119909119894)2
+ (119910119894+1minus 119910119894)2 minus1 lt
120585 lt 1 and Γ119894is the length of linearity cell
42 Fundamental Solution of Boundary Element IntegralEquation It is crucial to find its fundamental solution whenthe horizontal flow problem in complex reservoirs is resolvedusing the boundary element method According to the prop-erties of the boundary element and the mathematic equationgoverning pressure transmission in porous medium thefundamental solution must satisfy the modified Helmholtzoperator The equation is given by
1
119903
120597
120597119903
(119903
120597119866
120597119903
) minus 119891 (119906)119866 = minus2120587120575 (1198721198631198721015840
119863) (42)
With Lord Kelvin point source solution the fundamentalsolution of (42) can be derived as follows
120574 =
exp (minus120588119863radic119891 (119906))
4120587120588119863
(43)
With mirror image method of fluid mechanics in porousmedium the transient point source fundamental solution ofclosed boundary is
120574 =
1
4120587
+infin
sum
minusinfin
exp (minusradic119891 (119906)radic1198772119863+ (119885119863+ 1198851015840
119863minus 2119899119885
119890119863)2
)
radic1198772
119863+ (119885119863+ 1198851015840
119863minus 2119899119885
119890119863)2
+
exp (minusradic119891 (119906)radic1198772119863+(119885119863minus1198851015840
119863minus2119899119885
119890119863)2
)
radic1198772
119863+(119885119863minus 1198851015840
119863minus2119899119885
119890119863)2
(44)
With Poisson superposition formula (44) can be simpli-fied and the transient point source fundamental solution ofsealed boundary at 119885 = 0 and 119885 = 119885119890 is
120574 =
1
2120587119885119890119863
[1198700(119877119863radic119891 (119906))
+ 2
119899=infin
sum
119899=1
1198700(119877119863radic119891 (119906) +
1198992
1205872
119885119890119863
2)
sdot cos(119899120587 119885119863119885119890119863
) cos(1198991205871198851015840
119863
119885119890119863
)]
(45)
The fundamental boundary element solution of horizon-tal wells in a reservoir with closed top and bottom boundariesis
119866(1198751015840
119876 119906)
=
1
2
int
1
minus1
1198700(119877119863radic119891 (119906)) 119889120572
+
119899=infin
sum
119899=1
cos (119899120587119911119863) cos (119899120587119911
119908119863)
sdot int
1
minus1
1198700(radic(119909
119863minus 120572)2
+ 119910119863
2radic119891 (119906) +
1198992
1205872
119885119890119863
2)119889120572
120597119866 (1198751015840
119876 119906)
120597119899
= minus
1
2
int
1
minus1
radic119891 (119906)1198701((119903119863minus 120572)radic119891 (119906))
120597119903119863
120597119899
119889120572
minus
119899=infin
sum
119899=1
cos (119899120587119911119863)
sdot cos (119899120587119911119908119863) int
1
minus1
radic119891 (119906) +
1198992
1205872
119885119890119863
2
sdot 1198701((119903119863minus 120572)
sdotradic119891 (119906) +
1198992
1205872
119885119890119863
2)
120597119903119863
120597119899
119889120572
(46)
where120597119903119863
120597119899
= plusmn
100381610038161003816100381610038161003816100381610038161003816
((119909120585minus 119909) (119910
119894minus 119910119894+1) minus (119910
120585minus 119910) (119909
119894minus 119909119894+1))
sdot (radic(119909119894minus 119909119894+1)2
+ (119910119894minus 119910119894+1)2
)
minus1100381610038161003816100381610038161003816100381610038161003816
sdot (radic(119909120585minus 119909119894)2
+ (119910120585minus 119910119894)2
)
minus1
(47)
Journal of Chemistry 7
If 119899 and 1198751015840 are in the same direction then 120597119903119863120597119899 is posi-
tive otherwise it is negative
43 Solving of Integral Equation The integral boundary equa-tion (37) can be simplified as
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(48)
where
1198671015840
1=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206011(120585) 119889120585
1198671015840
2=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206012(120585) 119889120585
1198671015840
3=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206011(120585) 119889120585
1198671015840
4=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206012(120585) 119889120585
(49)
The unknown variable in the integral boundary equationis 120597120595119863119894120597119899 or 120595
119863119894 The number of the nodes at the boundary
is 119873119887 so 119873
119887equation with form of (48) can be established
When the boundary properties are known there are just 119873119887
unknown variables So we can solve the set of equationswhose matrix expression is
[[[[[[[
[
11986711
11986712
sdot sdot sdot 1198671119873119887
11986721
11986722
sdot sdot sdot 1198672119873119887
1198671198731198871
1198671198731198872
sdot sdot sdot 119867119873119887119873119887
]]]]]]]
]
[[[[[[
[
1199091
1199092
119909119873119887
]]]]]]
]
=
[[[[[[
[
1198651
1198652
119865119873119887
]]]]]]
]
(50)
where119909119894is 120597120595119863119894120597119899 or120595
119863119894and119865119894is (1119906)sum119873119908
119894=1119902119863119894119866(1198751015840
119876 119906)Once the unknown variables are acquired we can solve
119875119863
of arbitrary point in the research domain using theboundary integral equation (50)
120595119863(119876 119906) =
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894
+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(51)
5 Validation of the Model
In order to validate the gas flowmodel proposed in this paperwe test the adsorption and desorption data of coal Then wecompare the boundary element model with those publishedworks
001 01 1 10 100
1E16
1E17
120595D
120595998400D
TD
120595D
and120595998400 D
Figure 4 Double logarithmic curve of pressure drop of test data
51 Validation of the Gas FlowModel in CBM Reservoirs Theadsorption and desorption data of coal are tested in the labo-ratory the pressure continues to drop as the coal continuesto adsorb methane We deal with the experimental data inlog-log coordinates as shown in Figure 4 It can be seen fromthe diagram that the results of the numerical simulation arein good match with the experimental data and are consistentwith the gas flow law in CBM reservoirs
52 Validation of Gas Flow in a Horizontal Well with ComplexBoundary A pressure buildup test example of a horizontalgas well in tight gas reservoir has been presented byHan [30]the production and pressure data fitting results show that thehorizontal gas well has a closed boundary
Figure 5 is the comparison of pressure curves betweenhorizontal wells in CBM and conventional gas reservoirs andthe values of parameters are fromHanrsquos paper [30] Accordingto the analysis of flow characteristics there is an additionalradial flowwhich reflects gas flow in fracture After this stagethe pressure transmits from fracture to matrix and gas startsto desorbTheflowcharacteristics of this stagewould be influ-enced by desorption velocity and desorption quantity Thelast stage of pressure derivative curves in different reservoirsis in good match which reflects the type of boundary
6 Analysis of Flow Characteristics andField Application
61 Flow Regime Recognition Type curves for horizontalwells in a CBM reservoir with complex boundary calculatedby the BEM in this paper are shown in Figure 6 The figureshows that flow characteristics can be divided into sevenflow periods (1) wellbore storage period influenced by earlywellbore storage effect the curves of pressure and pressurederivative are straight lines with the same slope in this period(2) the first transition flow section which reflects the degreeof pollution near the bottom and is mainly affected by skin
8 Journal of Chemistry
001 0
1 1 10 100
1000
1000
0
1000
00
01
1
10
100
1Eminus4
1Eminus3
120595D of conventional gas reservoir horizontal well120595998400D of conventional gas reservoir horizontal well
120595D of CBM horizontal well120595998400D of CBM horizontal well
TD
120595D
and120595998400 D
Figure 5 Comparison of typical curves between horizontal wells inCBM and conventional gas reservoirs
001 0
1 1 10 100
1000
1000
0
1000
00
001
01
1
10
100
Closed boundaryConstant pressure boundaryMixed boundary
1Eminus6
1Eminus5
1Eminus4
1Eminus3
I II III IV V VI VII
TD
120595D
and120595998400 D
Figure 6 Typical curve of CBM horizontal well with complexboundary conditions (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
factor 119878 (3) the first radial flow section which is perpendic-ular to the horizontal wellbore the pressure derivative curveis shown as a horizontal line which reflects the early radialflow perpendicular to the horizontal wellbore (4) the secondtransition section which is the transition stage of radial flowfrom wellbore to fracture (5) the second radial flow stagewhich reflects the free gas flow in fracture this period isaffected by the flow of free gas and adsorbed gas diffusioncoefficient 120582 and Langmuir adsorption parameter 120573 (6) thethird radial flow section which reflects radial flow of whole
001 1 100 10000 1000000
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
120582 = 1
120582 = 100
120582 = 10000
Figure 7 Influence of diffusion coefficient 120582 on bottom-holepressure with mixed boundary (120596 = 001 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
system after pressure balance and pressure derivative curveappears as a flat behavior (7) boundary response sectionThepressure and pressure derivative curves with closed boundarywould be upward after pressure transmitting to the boundaryThe pressure derivative curve with constant pressure bound-ary would be in decline after pressure transmitting to theboundary Compared with the constant pressure boundarythe falling range of pressure derivative curve with mixedboundary is less
62 Parameter Sensitivity Analysis Figure 7 shows the influ-ence of diffusion coefficient 120582 on the bottom-hole pressurewith mixed boundary There is relevance between diffusioncoefficient 120582 = 120572119896120591120579119871
2 and adsorption time 120591 = 1198772
119863The smaller the 120591 the shorter the time of desorption-diffusionand the shorter the time of pressure balance between fractureand matrix As shown in the diagram diffusion coefficient 120582mainly affects the duration of the second radial flow periodand appearance of third radial flow The greater the diffusioncoefficient 120582 the longer the duration of second radial flowperiod and the later the 119905 appearance of third radial flow andvice versa If 120582 is small enough the second radial flow periodwould be covered
Figure 8 shows the influence of Langmuir parameter 120573on the bottom-hole pressure with mixed boundary 120573 =
120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894) is related to Langmuir
adsorption pressure 119901119871and Langmuir adsorption volume119881
119871
The larger the 120573 the greater the adsorption ability of coalAs shown in the graph 120573 only affects the second radial flowsection The greater the 120573 the smaller the pressure derivativevalues of second radial flow period
Figure 9 shows the influence of fracture storage ratio120596 onthe bottom-hole pressure with mixed boundary Its expres-sion 120596 = 120601
119891119888119905120583(120601119891119888119905120583 + 6119901
119904119888119879119902119863119879119904119888120595119894) indicates that the
smaller the 120601119891119888119905 the smaller the 120596 and the more the radial
Journal of Chemistry 9
001 1 100 10000 1000000
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
0011E8
120573 = 01
120573 = 1
120573 = 10
Figure 8 Influence of 120573 on the bottom-hole pressure with mixedboundary (120596 = 001 120582 = 300 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
1E8
120596 = 04
120596 = 01
120596 = 001
Figure 9 Influence of 120596 on the bottom-hole pressure with mixedboundary (120582 = 300 120573 = 50 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
flow in fracture From Figure 8 almost all of the flow periodswould be affected by 120596 except for wellbore storage periodThe greater the 120596 the smaller the hump value of pressurederivative and the longer the duration of first radial flowperiod the greater the pressure derivative value in secondradial flow period
Figure 10 shows the influence of eccentricity of horizontalwell 119885
119908119889on the bottom-hole pressure with mixed boundary
The streamline shape and pressure distribution of a horizon-tal well would be affected by the size of 119885
119908119889 The smaller
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
Zwd = 01
Zwd = 03
Zwd = 05
Figure 10 Influence of119885119908119889
on the bottom-hole pressure withmixedboundary (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25 119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
LD = 1
LD = 25
LD = 50
Figure 11 Influence of 119871119863on the bottom-hole pressure with mixed
boundary (120596 = 001 120582 = 300 120573 = 50119885119908119889= 05119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
the119885119908119889 the larger the pressure derivative value in first radial
flow periodFigure 11 shows the influence of length of horizontal well
119871119863on the bottom-hole pressure with mixed boundary It
shows large influence of 119871119863on the pressure derivative value
in first radial flow period The smaller the 119871119863 the larger the
pressure derivative value in first radial flow period
63 Field Example To demonstrate the application of themodel proposed in this paper a horizontal well ZX-12 in aCBM reservoir is studied The vertical depth and horizontallength of this well are 689m and 536m respectively withwell radius 119903
119908of 01m coal thickness of 45m and initial
10 Journal of Chemistry
10 100100
1000
1000
TD
120595D
and120595998400 D
120595D of CBM horizontal well fitted curve120595998400D CBM horizontal well fitted curve
120595D of CBM horizontal well product curve120595998400D of CBM horizontal well product data curve
Figure 12 Fitted curves of test data from an actual well
pressure of 55MPa According to test results the Langmuirvolume 119881
119871is 3275m3t and the Langmuir pressure 119875
119871is
249MPa Figure 12 is the fitted curves which were obtainedby calculating the actual production and pressure data Asshown in Figure 12 the field data are in good match with thetheoretical results calculated by the new model in this paper
According to the actual field data some of reservoirparameters are fitted as follows the porosity is 004 and thepermeability is 3mD The stage of desorption and boundaryresponse can be clearly seen in Figure 12 Because of thefalling of the pressure derivative curve in the last stage wellZX-12 has a constant pressure boundary implying that thegas production of well ZX-12 is affected by the production ofadjacent well
7 Conclusions
Themathematic model of gas flowing into horizontal wells inCBMreservoirswith complex boundary conditions is derivedbased on the percolation theory The curves of bottom-hole pressure and pressure derivative are obtained by usingboundary element method and Laplace transform The con-clusions are as follows
(1) The fundamental boundary element solutions fortransient pressure response of horizontal wells inCBM reservoirs with complex boundary conditionscould be obtained by using Lord Kelvinrsquos point sourcesolution point source function theory and Poissonrsquossummation formula
(2) Comparison of the typical curves of flow character-istics between horizontal wells in CBM and conven-tional gas reservoirs shows that there is an additionalradial flow which reflects gas flow in fracture
(3) Comparing the typical curves of flow characteris-tics with complex boundary conditions the pressure
and pressure derivative curves with closed boundarywould be upward after pressure transmitting to theboundary The pressure derivative curves with con-stant pressure boundary and mixed boundary wouldbe fallen but the falling range of pressure derivativecurve with mixed boundary is less
Nomenclature
119903119908 Radius m
119896 Permeability mD120601 Porosity fraction119869 Diffusion flux gm2s119904 Laplace transform variableℎ Thickness m119861 Volume factor fraction119902 Influx into the wellbore m3d119871 Half length of horizontal well m120583 Viscosity mPasdots119888 Concentration of coalbed methane kgm3119875119894 Initial pressure MPa
119902 Production of well m3d119881119871 Langmuir volume constant m3ton
119881 Volume of coal matrix m3120595119894 Pseudopressure
120596 Fracture storage ratio fraction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is supported by National Natural Science Foun-dation of China (Grant no 51304032) National MajorScience and Technology Special Project of China (Grantno 2011ZX05037-003) and Research Fund for the Doc-toral Program of Higher Education of China (Grant no20125122110017)
References
[1] J C Cai E Perfect C-L Cheng and X Y Hu ldquoGeneralizedmodeling of spontaneous imbibition based on hagen-poiseuilleflow in tortuous capillaries with variably shaped aperturesrdquoLangmuir vol 30 no 18 pp 5142ndash5151 2014
[2] T Ertekin and W Sung ldquoPressure transient analysis of coalseams in the presence of multi-mechanistic flow and sorptionphenomenardquo in Proceedings of the SPE Gas Technology Sympo-sium pp 469ndash477 June 1989
[3] K Anbarci and T Ertekin ldquoA comprehensive study of pressuretransient analysis with sorption phenomena for single phase gasflow in coal seamsrdquo SPE 20568 1990
[4] C R Clarkson and R M Bustin ldquoEffect of pore structureand gas pressure upon the transport properties of coal alaboratory and modeling study 1 Isotherms and pore volumedistributionsrdquo Fuel vol 78 no 11 pp 1333ndash1344 1999
Journal of Chemistry 11
[5] C R Clarkson and R M Bustin ldquoEffect of pore structure andgas pressure upon the transport properties of coal a laboratoryandmodeling study 2 Adsorption rate modelingrdquo Fuel vol 78no 11 pp 1345ndash1362 1999
[6] S Reeve ldquoAdvanced reservoir modeling in desorption con-trolled reservoirsrdquo in Proceedings of the SPE Rocky MountainPetroleum Technology Conference SPE 71090 May 2001
[7] D K Tong and S Liu ldquoUnsteady percolation flow of coalbedmethane through deformed coal seamrdquo Natural Gas Industryvol 1 no 1 pp 74ndash76 2005
[8] X M Zhang and D K Tong ldquoAnalysis for pressure transient ofcoalbed methane reservoirrdquo Well Testing vol 6 no 17 pp 1ndash42008
[9] X M Zhang and D K Tong ldquoPressure transient analysisfor coal seams with triple-porositydual-permeability modelrdquoChineseQuarterly ofMechanics vol 4 no 29 pp 578ndash582 2008
[10] X H Hu S M Hu D I Zhang et al ldquoApplication of quadraticpressure method in well test interpretation of gas water twophase flow in coal seamrdquo Journal of Oil and Gas Technology vol7 no 33 pp 118ndash122 2011
[11] C R Clarkson C L Jordan D Ilk and T A Blasingame ldquoRate-transient analysis of 2-phase (gas + water) CBM wellsrdquo Journalof Natural Gas Science and Engineering vol 8 pp 106ndash120 2012
[12] K Aminian and S Ameri ldquoPredicting production performanceof CBM reservoirsrdquo Journal of Natural Gas Science and Engi-neering vol 1 no 1-2 pp 25ndash30 2009
[13] J C Cai and BM Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011
[14] W Sung T Ertekin and F C Schwerer ldquoThe developmenttesting and application of a comprehensive coal seam degasi-fication modelrdquo in Proceedings of the SPE Unconventional GasTechnology Symposium SPE 15247 Louisville Ky USA May1986
[15] W Sung and T Ertekin ldquoAn analysis of field developmentstrategies formethane production from coal seamsrdquo in Proceed-ings of the 62nd Annual Technical Conference and ExhibitionConference Paper SPE 16858 Dallas Tex USA 1987
[16] T Ertekin W Sung and F C Schwerer ldquoProduction perfor-mance analysis of horizontal drainage wells for degasificationof coal-seamsrdquo Journal of Petroleum Technology vol 40 no 5pp 625ndash632 1988
[17] T W Engler and J M Rajtar ldquoPressure transient testing ofhorizontal wells in coalbed reservoirsrdquo in Proceedings of theSPE Rocky Mountain Regional Meeting Paper SPE-24374-MSCasper Wyo USA May 1992
[18] P S Sarkar and J M Rajtar ldquoHorizontal well transient pres-sure testing in coalbed reservoirsrdquo in Proceedings of the 3rdLatin AmericaCaribbean Petroleum Engineering ConferenceSPE 26995 Buenos Aires Argentina April 1994
[19] P S Sarkar and J M Rajtar ldquoTransient well testing of coalbedmethane reservoirs with horizontal wellsrdquo in Proceedings of thePermian Basin Oil amp Gas Recovery Conference Paper SPE27681pp 531ndash539 Midland Tex USA March 1994
[20] X H Wang D I Zhang and Y Song ldquoDevelopment mecha-nism of low permeable coalbed methane frommultilateral hor-izontal pinnatewell with non-darcy seepage flow characteristicrdquoActa Geologica Sinica vol 7 no 82 pp 1437ndash1443 2008
[21] R-S Nie Y-F Meng J-C Guo and Y-L Jia ldquoModelingtransient flow behavior of a horizontal well in a coal seamrdquoInternational Journal of Coal Geology vol 92 pp 54ndash68 2012
[22] K L Katsifarakis D K Karpouzos and N TheodossiouldquoCombined use of BEM and genetic algorithms in groundwaterflow and mass transport problemsrdquo Engineering Analysis withBoundary Elements vol 23 no 7 pp 555ndash565 1999
[23] D T Numbere and D Tiab ldquoImproved streamline-generatingtechnique that uses the boundary (integral) element methodrdquoSPE Reservoir Engineering vol 3 no 3 pp 1061ndash1068 1988
[24] J Kikani and R N Horne ldquoApplication of boundary ele-ment method to reservoir engineering problemsrdquo Journal ofPetroleum Science and Engineering vol 3 no 3 pp 229ndash2411989
[25] J Hou YDWang andYMChen ldquoBoundary elementmethodin enhanced oil recoveryrdquo Chinese Journal of Hydrodynamicsvol 3 no 13 pp 23ndash28 2003
[26] K Chaiyo P Rattanadecho and S Chantasiriwan ldquoThemethodof fundamental solutions for solving free boundary saturatedseepage problemrdquo International Communications in Heat andMass Transfer vol 38 no 2 pp 249ndash254 2011
[27] K Rafiezadeh and B Ataie-Ashtiani ldquoSeepage analysis inmulti-domain general anisotropic media by three-dimensionalboundary elementsrdquo Engineering Analysis with Boundary Ele-ments vol 37 no 3 pp 527ndash541 2013
[28] X H Fu Y Qin W H Zhang et al ldquoCoal pore fractal classifi-cation and natural classification research based on the coal-bedgas migrationrdquo Chinese Science Bulletin vol 12 supplement 1pp 51ndash55 2005
[29] J C Cai and S Y Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal ofModern Physics C vol 24 no 8 pp 135ndash156 2013
[30] D M Han The Dynamic Analysis Technology Research onHorizontal Wells of Jingbian Gas Field University of XirsquoanShiyou Xirsquoan China 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
where 120579119896represents interior angles between any two adjacent
boundary elements Consider
120579119896=
1 the point in domain 120579119894= 2120587
05 the point at smooth boundary 120579119894= 120587
120579119894
2120587
the point at smoothless boundary
(40)
Using the linear interpolation in boundary element theboundary integral formula is deformed as follows
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
119897119894
2
int
1
minus1
[119866 (1198751015840
119876119896 119906) (120593
1(120585)
120597120595119863119894
120597119899
+ 1205932(120585)
120597120595119863119894+1
120597119899
)
minus (1205931(120585) 120595119863119894+ 1205932(120585) 120595119863119894+1
)
120597119866 (1198751015840
119876119896 119906)
120597119899
] 119889120585
+
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876119894 119906)
(41)
where 1206011(120585) = (1 minus 120585)2 and 120601
2(120585) = (1 + 120585)2 are linear
interpolation formula 119897119894= radic(119909
119894+1minus 119909119894)2
+ (119910119894+1minus 119910119894)2 minus1 lt
120585 lt 1 and Γ119894is the length of linearity cell
42 Fundamental Solution of Boundary Element IntegralEquation It is crucial to find its fundamental solution whenthe horizontal flow problem in complex reservoirs is resolvedusing the boundary element method According to the prop-erties of the boundary element and the mathematic equationgoverning pressure transmission in porous medium thefundamental solution must satisfy the modified Helmholtzoperator The equation is given by
1
119903
120597
120597119903
(119903
120597119866
120597119903
) minus 119891 (119906)119866 = minus2120587120575 (1198721198631198721015840
119863) (42)
With Lord Kelvin point source solution the fundamentalsolution of (42) can be derived as follows
120574 =
exp (minus120588119863radic119891 (119906))
4120587120588119863
(43)
With mirror image method of fluid mechanics in porousmedium the transient point source fundamental solution ofclosed boundary is
120574 =
1
4120587
+infin
sum
minusinfin
exp (minusradic119891 (119906)radic1198772119863+ (119885119863+ 1198851015840
119863minus 2119899119885
119890119863)2
)
radic1198772
119863+ (119885119863+ 1198851015840
119863minus 2119899119885
119890119863)2
+
exp (minusradic119891 (119906)radic1198772119863+(119885119863minus1198851015840
119863minus2119899119885
119890119863)2
)
radic1198772
119863+(119885119863minus 1198851015840
119863minus2119899119885
119890119863)2
(44)
With Poisson superposition formula (44) can be simpli-fied and the transient point source fundamental solution ofsealed boundary at 119885 = 0 and 119885 = 119885119890 is
120574 =
1
2120587119885119890119863
[1198700(119877119863radic119891 (119906))
+ 2
119899=infin
sum
119899=1
1198700(119877119863radic119891 (119906) +
1198992
1205872
119885119890119863
2)
sdot cos(119899120587 119885119863119885119890119863
) cos(1198991205871198851015840
119863
119885119890119863
)]
(45)
The fundamental boundary element solution of horizon-tal wells in a reservoir with closed top and bottom boundariesis
119866(1198751015840
119876 119906)
=
1
2
int
1
minus1
1198700(119877119863radic119891 (119906)) 119889120572
+
119899=infin
sum
119899=1
cos (119899120587119911119863) cos (119899120587119911
119908119863)
sdot int
1
minus1
1198700(radic(119909
119863minus 120572)2
+ 119910119863
2radic119891 (119906) +
1198992
1205872
119885119890119863
2)119889120572
120597119866 (1198751015840
119876 119906)
120597119899
= minus
1
2
int
1
minus1
radic119891 (119906)1198701((119903119863minus 120572)radic119891 (119906))
120597119903119863
120597119899
119889120572
minus
119899=infin
sum
119899=1
cos (119899120587119911119863)
sdot cos (119899120587119911119908119863) int
1
minus1
radic119891 (119906) +
1198992
1205872
119885119890119863
2
sdot 1198701((119903119863minus 120572)
sdotradic119891 (119906) +
1198992
1205872
119885119890119863
2)
120597119903119863
120597119899
119889120572
(46)
where120597119903119863
120597119899
= plusmn
100381610038161003816100381610038161003816100381610038161003816
((119909120585minus 119909) (119910
119894minus 119910119894+1) minus (119910
120585minus 119910) (119909
119894minus 119909119894+1))
sdot (radic(119909119894minus 119909119894+1)2
+ (119910119894minus 119910119894+1)2
)
minus1100381610038161003816100381610038161003816100381610038161003816
sdot (radic(119909120585minus 119909119894)2
+ (119910120585minus 119910119894)2
)
minus1
(47)
Journal of Chemistry 7
If 119899 and 1198751015840 are in the same direction then 120597119903119863120597119899 is posi-
tive otherwise it is negative
43 Solving of Integral Equation The integral boundary equa-tion (37) can be simplified as
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(48)
where
1198671015840
1=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206011(120585) 119889120585
1198671015840
2=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206012(120585) 119889120585
1198671015840
3=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206011(120585) 119889120585
1198671015840
4=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206012(120585) 119889120585
(49)
The unknown variable in the integral boundary equationis 120597120595119863119894120597119899 or 120595
119863119894 The number of the nodes at the boundary
is 119873119887 so 119873
119887equation with form of (48) can be established
When the boundary properties are known there are just 119873119887
unknown variables So we can solve the set of equationswhose matrix expression is
[[[[[[[
[
11986711
11986712
sdot sdot sdot 1198671119873119887
11986721
11986722
sdot sdot sdot 1198672119873119887
1198671198731198871
1198671198731198872
sdot sdot sdot 119867119873119887119873119887
]]]]]]]
]
[[[[[[
[
1199091
1199092
119909119873119887
]]]]]]
]
=
[[[[[[
[
1198651
1198652
119865119873119887
]]]]]]
]
(50)
where119909119894is 120597120595119863119894120597119899 or120595
119863119894and119865119894is (1119906)sum119873119908
119894=1119902119863119894119866(1198751015840
119876 119906)Once the unknown variables are acquired we can solve
119875119863
of arbitrary point in the research domain using theboundary integral equation (50)
120595119863(119876 119906) =
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894
+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(51)
5 Validation of the Model
In order to validate the gas flowmodel proposed in this paperwe test the adsorption and desorption data of coal Then wecompare the boundary element model with those publishedworks
001 01 1 10 100
1E16
1E17
120595D
120595998400D
TD
120595D
and120595998400 D
Figure 4 Double logarithmic curve of pressure drop of test data
51 Validation of the Gas FlowModel in CBM Reservoirs Theadsorption and desorption data of coal are tested in the labo-ratory the pressure continues to drop as the coal continuesto adsorb methane We deal with the experimental data inlog-log coordinates as shown in Figure 4 It can be seen fromthe diagram that the results of the numerical simulation arein good match with the experimental data and are consistentwith the gas flow law in CBM reservoirs
52 Validation of Gas Flow in a Horizontal Well with ComplexBoundary A pressure buildup test example of a horizontalgas well in tight gas reservoir has been presented byHan [30]the production and pressure data fitting results show that thehorizontal gas well has a closed boundary
Figure 5 is the comparison of pressure curves betweenhorizontal wells in CBM and conventional gas reservoirs andthe values of parameters are fromHanrsquos paper [30] Accordingto the analysis of flow characteristics there is an additionalradial flowwhich reflects gas flow in fracture After this stagethe pressure transmits from fracture to matrix and gas startsto desorbTheflowcharacteristics of this stagewould be influ-enced by desorption velocity and desorption quantity Thelast stage of pressure derivative curves in different reservoirsis in good match which reflects the type of boundary
6 Analysis of Flow Characteristics andField Application
61 Flow Regime Recognition Type curves for horizontalwells in a CBM reservoir with complex boundary calculatedby the BEM in this paper are shown in Figure 6 The figureshows that flow characteristics can be divided into sevenflow periods (1) wellbore storage period influenced by earlywellbore storage effect the curves of pressure and pressurederivative are straight lines with the same slope in this period(2) the first transition flow section which reflects the degreeof pollution near the bottom and is mainly affected by skin
8 Journal of Chemistry
001 0
1 1 10 100
1000
1000
0
1000
00
01
1
10
100
1Eminus4
1Eminus3
120595D of conventional gas reservoir horizontal well120595998400D of conventional gas reservoir horizontal well
120595D of CBM horizontal well120595998400D of CBM horizontal well
TD
120595D
and120595998400 D
Figure 5 Comparison of typical curves between horizontal wells inCBM and conventional gas reservoirs
001 0
1 1 10 100
1000
1000
0
1000
00
001
01
1
10
100
Closed boundaryConstant pressure boundaryMixed boundary
1Eminus6
1Eminus5
1Eminus4
1Eminus3
I II III IV V VI VII
TD
120595D
and120595998400 D
Figure 6 Typical curve of CBM horizontal well with complexboundary conditions (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
factor 119878 (3) the first radial flow section which is perpendic-ular to the horizontal wellbore the pressure derivative curveis shown as a horizontal line which reflects the early radialflow perpendicular to the horizontal wellbore (4) the secondtransition section which is the transition stage of radial flowfrom wellbore to fracture (5) the second radial flow stagewhich reflects the free gas flow in fracture this period isaffected by the flow of free gas and adsorbed gas diffusioncoefficient 120582 and Langmuir adsorption parameter 120573 (6) thethird radial flow section which reflects radial flow of whole
001 1 100 10000 1000000
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
120582 = 1
120582 = 100
120582 = 10000
Figure 7 Influence of diffusion coefficient 120582 on bottom-holepressure with mixed boundary (120596 = 001 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
system after pressure balance and pressure derivative curveappears as a flat behavior (7) boundary response sectionThepressure and pressure derivative curves with closed boundarywould be upward after pressure transmitting to the boundaryThe pressure derivative curve with constant pressure bound-ary would be in decline after pressure transmitting to theboundary Compared with the constant pressure boundarythe falling range of pressure derivative curve with mixedboundary is less
62 Parameter Sensitivity Analysis Figure 7 shows the influ-ence of diffusion coefficient 120582 on the bottom-hole pressurewith mixed boundary There is relevance between diffusioncoefficient 120582 = 120572119896120591120579119871
2 and adsorption time 120591 = 1198772
119863The smaller the 120591 the shorter the time of desorption-diffusionand the shorter the time of pressure balance between fractureand matrix As shown in the diagram diffusion coefficient 120582mainly affects the duration of the second radial flow periodand appearance of third radial flow The greater the diffusioncoefficient 120582 the longer the duration of second radial flowperiod and the later the 119905 appearance of third radial flow andvice versa If 120582 is small enough the second radial flow periodwould be covered
Figure 8 shows the influence of Langmuir parameter 120573on the bottom-hole pressure with mixed boundary 120573 =
120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894) is related to Langmuir
adsorption pressure 119901119871and Langmuir adsorption volume119881
119871
The larger the 120573 the greater the adsorption ability of coalAs shown in the graph 120573 only affects the second radial flowsection The greater the 120573 the smaller the pressure derivativevalues of second radial flow period
Figure 9 shows the influence of fracture storage ratio120596 onthe bottom-hole pressure with mixed boundary Its expres-sion 120596 = 120601
119891119888119905120583(120601119891119888119905120583 + 6119901
119904119888119879119902119863119879119904119888120595119894) indicates that the
smaller the 120601119891119888119905 the smaller the 120596 and the more the radial
Journal of Chemistry 9
001 1 100 10000 1000000
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
0011E8
120573 = 01
120573 = 1
120573 = 10
Figure 8 Influence of 120573 on the bottom-hole pressure with mixedboundary (120596 = 001 120582 = 300 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
1E8
120596 = 04
120596 = 01
120596 = 001
Figure 9 Influence of 120596 on the bottom-hole pressure with mixedboundary (120582 = 300 120573 = 50 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
flow in fracture From Figure 8 almost all of the flow periodswould be affected by 120596 except for wellbore storage periodThe greater the 120596 the smaller the hump value of pressurederivative and the longer the duration of first radial flowperiod the greater the pressure derivative value in secondradial flow period
Figure 10 shows the influence of eccentricity of horizontalwell 119885
119908119889on the bottom-hole pressure with mixed boundary
The streamline shape and pressure distribution of a horizon-tal well would be affected by the size of 119885
119908119889 The smaller
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
Zwd = 01
Zwd = 03
Zwd = 05
Figure 10 Influence of119885119908119889
on the bottom-hole pressure withmixedboundary (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25 119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
LD = 1
LD = 25
LD = 50
Figure 11 Influence of 119871119863on the bottom-hole pressure with mixed
boundary (120596 = 001 120582 = 300 120573 = 50119885119908119889= 05119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
the119885119908119889 the larger the pressure derivative value in first radial
flow periodFigure 11 shows the influence of length of horizontal well
119871119863on the bottom-hole pressure with mixed boundary It
shows large influence of 119871119863on the pressure derivative value
in first radial flow period The smaller the 119871119863 the larger the
pressure derivative value in first radial flow period
63 Field Example To demonstrate the application of themodel proposed in this paper a horizontal well ZX-12 in aCBM reservoir is studied The vertical depth and horizontallength of this well are 689m and 536m respectively withwell radius 119903
119908of 01m coal thickness of 45m and initial
10 Journal of Chemistry
10 100100
1000
1000
TD
120595D
and120595998400 D
120595D of CBM horizontal well fitted curve120595998400D CBM horizontal well fitted curve
120595D of CBM horizontal well product curve120595998400D of CBM horizontal well product data curve
Figure 12 Fitted curves of test data from an actual well
pressure of 55MPa According to test results the Langmuirvolume 119881
119871is 3275m3t and the Langmuir pressure 119875
119871is
249MPa Figure 12 is the fitted curves which were obtainedby calculating the actual production and pressure data Asshown in Figure 12 the field data are in good match with thetheoretical results calculated by the new model in this paper
According to the actual field data some of reservoirparameters are fitted as follows the porosity is 004 and thepermeability is 3mD The stage of desorption and boundaryresponse can be clearly seen in Figure 12 Because of thefalling of the pressure derivative curve in the last stage wellZX-12 has a constant pressure boundary implying that thegas production of well ZX-12 is affected by the production ofadjacent well
7 Conclusions
Themathematic model of gas flowing into horizontal wells inCBMreservoirswith complex boundary conditions is derivedbased on the percolation theory The curves of bottom-hole pressure and pressure derivative are obtained by usingboundary element method and Laplace transform The con-clusions are as follows
(1) The fundamental boundary element solutions fortransient pressure response of horizontal wells inCBM reservoirs with complex boundary conditionscould be obtained by using Lord Kelvinrsquos point sourcesolution point source function theory and Poissonrsquossummation formula
(2) Comparison of the typical curves of flow character-istics between horizontal wells in CBM and conven-tional gas reservoirs shows that there is an additionalradial flow which reflects gas flow in fracture
(3) Comparing the typical curves of flow characteris-tics with complex boundary conditions the pressure
and pressure derivative curves with closed boundarywould be upward after pressure transmitting to theboundary The pressure derivative curves with con-stant pressure boundary and mixed boundary wouldbe fallen but the falling range of pressure derivativecurve with mixed boundary is less
Nomenclature
119903119908 Radius m
119896 Permeability mD120601 Porosity fraction119869 Diffusion flux gm2s119904 Laplace transform variableℎ Thickness m119861 Volume factor fraction119902 Influx into the wellbore m3d119871 Half length of horizontal well m120583 Viscosity mPasdots119888 Concentration of coalbed methane kgm3119875119894 Initial pressure MPa
119902 Production of well m3d119881119871 Langmuir volume constant m3ton
119881 Volume of coal matrix m3120595119894 Pseudopressure
120596 Fracture storage ratio fraction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is supported by National Natural Science Foun-dation of China (Grant no 51304032) National MajorScience and Technology Special Project of China (Grantno 2011ZX05037-003) and Research Fund for the Doc-toral Program of Higher Education of China (Grant no20125122110017)
References
[1] J C Cai E Perfect C-L Cheng and X Y Hu ldquoGeneralizedmodeling of spontaneous imbibition based on hagen-poiseuilleflow in tortuous capillaries with variably shaped aperturesrdquoLangmuir vol 30 no 18 pp 5142ndash5151 2014
[2] T Ertekin and W Sung ldquoPressure transient analysis of coalseams in the presence of multi-mechanistic flow and sorptionphenomenardquo in Proceedings of the SPE Gas Technology Sympo-sium pp 469ndash477 June 1989
[3] K Anbarci and T Ertekin ldquoA comprehensive study of pressuretransient analysis with sorption phenomena for single phase gasflow in coal seamsrdquo SPE 20568 1990
[4] C R Clarkson and R M Bustin ldquoEffect of pore structureand gas pressure upon the transport properties of coal alaboratory and modeling study 1 Isotherms and pore volumedistributionsrdquo Fuel vol 78 no 11 pp 1333ndash1344 1999
Journal of Chemistry 11
[5] C R Clarkson and R M Bustin ldquoEffect of pore structure andgas pressure upon the transport properties of coal a laboratoryandmodeling study 2 Adsorption rate modelingrdquo Fuel vol 78no 11 pp 1345ndash1362 1999
[6] S Reeve ldquoAdvanced reservoir modeling in desorption con-trolled reservoirsrdquo in Proceedings of the SPE Rocky MountainPetroleum Technology Conference SPE 71090 May 2001
[7] D K Tong and S Liu ldquoUnsteady percolation flow of coalbedmethane through deformed coal seamrdquo Natural Gas Industryvol 1 no 1 pp 74ndash76 2005
[8] X M Zhang and D K Tong ldquoAnalysis for pressure transient ofcoalbed methane reservoirrdquo Well Testing vol 6 no 17 pp 1ndash42008
[9] X M Zhang and D K Tong ldquoPressure transient analysisfor coal seams with triple-porositydual-permeability modelrdquoChineseQuarterly ofMechanics vol 4 no 29 pp 578ndash582 2008
[10] X H Hu S M Hu D I Zhang et al ldquoApplication of quadraticpressure method in well test interpretation of gas water twophase flow in coal seamrdquo Journal of Oil and Gas Technology vol7 no 33 pp 118ndash122 2011
[11] C R Clarkson C L Jordan D Ilk and T A Blasingame ldquoRate-transient analysis of 2-phase (gas + water) CBM wellsrdquo Journalof Natural Gas Science and Engineering vol 8 pp 106ndash120 2012
[12] K Aminian and S Ameri ldquoPredicting production performanceof CBM reservoirsrdquo Journal of Natural Gas Science and Engi-neering vol 1 no 1-2 pp 25ndash30 2009
[13] J C Cai and BM Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011
[14] W Sung T Ertekin and F C Schwerer ldquoThe developmenttesting and application of a comprehensive coal seam degasi-fication modelrdquo in Proceedings of the SPE Unconventional GasTechnology Symposium SPE 15247 Louisville Ky USA May1986
[15] W Sung and T Ertekin ldquoAn analysis of field developmentstrategies formethane production from coal seamsrdquo in Proceed-ings of the 62nd Annual Technical Conference and ExhibitionConference Paper SPE 16858 Dallas Tex USA 1987
[16] T Ertekin W Sung and F C Schwerer ldquoProduction perfor-mance analysis of horizontal drainage wells for degasificationof coal-seamsrdquo Journal of Petroleum Technology vol 40 no 5pp 625ndash632 1988
[17] T W Engler and J M Rajtar ldquoPressure transient testing ofhorizontal wells in coalbed reservoirsrdquo in Proceedings of theSPE Rocky Mountain Regional Meeting Paper SPE-24374-MSCasper Wyo USA May 1992
[18] P S Sarkar and J M Rajtar ldquoHorizontal well transient pres-sure testing in coalbed reservoirsrdquo in Proceedings of the 3rdLatin AmericaCaribbean Petroleum Engineering ConferenceSPE 26995 Buenos Aires Argentina April 1994
[19] P S Sarkar and J M Rajtar ldquoTransient well testing of coalbedmethane reservoirs with horizontal wellsrdquo in Proceedings of thePermian Basin Oil amp Gas Recovery Conference Paper SPE27681pp 531ndash539 Midland Tex USA March 1994
[20] X H Wang D I Zhang and Y Song ldquoDevelopment mecha-nism of low permeable coalbed methane frommultilateral hor-izontal pinnatewell with non-darcy seepage flow characteristicrdquoActa Geologica Sinica vol 7 no 82 pp 1437ndash1443 2008
[21] R-S Nie Y-F Meng J-C Guo and Y-L Jia ldquoModelingtransient flow behavior of a horizontal well in a coal seamrdquoInternational Journal of Coal Geology vol 92 pp 54ndash68 2012
[22] K L Katsifarakis D K Karpouzos and N TheodossiouldquoCombined use of BEM and genetic algorithms in groundwaterflow and mass transport problemsrdquo Engineering Analysis withBoundary Elements vol 23 no 7 pp 555ndash565 1999
[23] D T Numbere and D Tiab ldquoImproved streamline-generatingtechnique that uses the boundary (integral) element methodrdquoSPE Reservoir Engineering vol 3 no 3 pp 1061ndash1068 1988
[24] J Kikani and R N Horne ldquoApplication of boundary ele-ment method to reservoir engineering problemsrdquo Journal ofPetroleum Science and Engineering vol 3 no 3 pp 229ndash2411989
[25] J Hou YDWang andYMChen ldquoBoundary elementmethodin enhanced oil recoveryrdquo Chinese Journal of Hydrodynamicsvol 3 no 13 pp 23ndash28 2003
[26] K Chaiyo P Rattanadecho and S Chantasiriwan ldquoThemethodof fundamental solutions for solving free boundary saturatedseepage problemrdquo International Communications in Heat andMass Transfer vol 38 no 2 pp 249ndash254 2011
[27] K Rafiezadeh and B Ataie-Ashtiani ldquoSeepage analysis inmulti-domain general anisotropic media by three-dimensionalboundary elementsrdquo Engineering Analysis with Boundary Ele-ments vol 37 no 3 pp 527ndash541 2013
[28] X H Fu Y Qin W H Zhang et al ldquoCoal pore fractal classifi-cation and natural classification research based on the coal-bedgas migrationrdquo Chinese Science Bulletin vol 12 supplement 1pp 51ndash55 2005
[29] J C Cai and S Y Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal ofModern Physics C vol 24 no 8 pp 135ndash156 2013
[30] D M Han The Dynamic Analysis Technology Research onHorizontal Wells of Jingbian Gas Field University of XirsquoanShiyou Xirsquoan China 2011
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
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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
If 119899 and 1198751015840 are in the same direction then 120597119903119863120597119899 is posi-
tive otherwise it is negative
43 Solving of Integral Equation The integral boundary equa-tion (37) can be simplified as
120579119896120595119863(119876119896 119906)
=
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(48)
where
1198671015840
1=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206011(120585) 119889120585
1198671015840
2=
119897119894
2
int
1
minus1
119866(1198751015840
119876119896 119906) 1206012(120585) 119889120585
1198671015840
3=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206011(120585) 119889120585
1198671015840
4=
119897119894
2
int
1
minus1
minus
120597119866 (1198751015840
119876119896 119906)
120597119899
1206012(120585) 119889120585
(49)
The unknown variable in the integral boundary equationis 120597120595119863119894120597119899 or 120595
119863119894 The number of the nodes at the boundary
is 119873119887 so 119873
119887equation with form of (48) can be established
When the boundary properties are known there are just 119873119887
unknown variables So we can solve the set of equationswhose matrix expression is
[[[[[[[
[
11986711
11986712
sdot sdot sdot 1198671119873119887
11986721
11986722
sdot sdot sdot 1198672119873119887
1198671198731198871
1198671198731198872
sdot sdot sdot 119867119873119887119873119887
]]]]]]]
]
[[[[[[
[
1199091
1199092
119909119873119887
]]]]]]
]
=
[[[[[[
[
1198651
1198652
119865119873119887
]]]]]]
]
(50)
where119909119894is 120597120595119863119894120597119899 or120595
119863119894and119865119894is (1119906)sum119873119908
119894=1119902119863119894119866(1198751015840
119876 119906)Once the unknown variables are acquired we can solve
119875119863
of arbitrary point in the research domain using theboundary integral equation (50)
120595119863(119876 119906) =
119873119887
sum
119894=1
(1198671015840
1198961
120597120595119863119894
120597119899
+ 1198671015840
1198962
120597120595119863119894+1
120597119899
+ 1198671015840
1198963120595119863119894
+ 1198671015840
1198964120595119863119894+1
) +
1
119906
119873119908
sum
119894=1
119902119863119894119866(1198751015840
119876 119906)
(51)
5 Validation of the Model
In order to validate the gas flowmodel proposed in this paperwe test the adsorption and desorption data of coal Then wecompare the boundary element model with those publishedworks
001 01 1 10 100
1E16
1E17
120595D
120595998400D
TD
120595D
and120595998400 D
Figure 4 Double logarithmic curve of pressure drop of test data
51 Validation of the Gas FlowModel in CBM Reservoirs Theadsorption and desorption data of coal are tested in the labo-ratory the pressure continues to drop as the coal continuesto adsorb methane We deal with the experimental data inlog-log coordinates as shown in Figure 4 It can be seen fromthe diagram that the results of the numerical simulation arein good match with the experimental data and are consistentwith the gas flow law in CBM reservoirs
52 Validation of Gas Flow in a Horizontal Well with ComplexBoundary A pressure buildup test example of a horizontalgas well in tight gas reservoir has been presented byHan [30]the production and pressure data fitting results show that thehorizontal gas well has a closed boundary
Figure 5 is the comparison of pressure curves betweenhorizontal wells in CBM and conventional gas reservoirs andthe values of parameters are fromHanrsquos paper [30] Accordingto the analysis of flow characteristics there is an additionalradial flowwhich reflects gas flow in fracture After this stagethe pressure transmits from fracture to matrix and gas startsto desorbTheflowcharacteristics of this stagewould be influ-enced by desorption velocity and desorption quantity Thelast stage of pressure derivative curves in different reservoirsis in good match which reflects the type of boundary
6 Analysis of Flow Characteristics andField Application
61 Flow Regime Recognition Type curves for horizontalwells in a CBM reservoir with complex boundary calculatedby the BEM in this paper are shown in Figure 6 The figureshows that flow characteristics can be divided into sevenflow periods (1) wellbore storage period influenced by earlywellbore storage effect the curves of pressure and pressurederivative are straight lines with the same slope in this period(2) the first transition flow section which reflects the degreeof pollution near the bottom and is mainly affected by skin
8 Journal of Chemistry
001 0
1 1 10 100
1000
1000
0
1000
00
01
1
10
100
1Eminus4
1Eminus3
120595D of conventional gas reservoir horizontal well120595998400D of conventional gas reservoir horizontal well
120595D of CBM horizontal well120595998400D of CBM horizontal well
TD
120595D
and120595998400 D
Figure 5 Comparison of typical curves between horizontal wells inCBM and conventional gas reservoirs
001 0
1 1 10 100
1000
1000
0
1000
00
001
01
1
10
100
Closed boundaryConstant pressure boundaryMixed boundary
1Eminus6
1Eminus5
1Eminus4
1Eminus3
I II III IV V VI VII
TD
120595D
and120595998400 D
Figure 6 Typical curve of CBM horizontal well with complexboundary conditions (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
factor 119878 (3) the first radial flow section which is perpendic-ular to the horizontal wellbore the pressure derivative curveis shown as a horizontal line which reflects the early radialflow perpendicular to the horizontal wellbore (4) the secondtransition section which is the transition stage of radial flowfrom wellbore to fracture (5) the second radial flow stagewhich reflects the free gas flow in fracture this period isaffected by the flow of free gas and adsorbed gas diffusioncoefficient 120582 and Langmuir adsorption parameter 120573 (6) thethird radial flow section which reflects radial flow of whole
001 1 100 10000 1000000
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
120582 = 1
120582 = 100
120582 = 10000
Figure 7 Influence of diffusion coefficient 120582 on bottom-holepressure with mixed boundary (120596 = 001 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
system after pressure balance and pressure derivative curveappears as a flat behavior (7) boundary response sectionThepressure and pressure derivative curves with closed boundarywould be upward after pressure transmitting to the boundaryThe pressure derivative curve with constant pressure bound-ary would be in decline after pressure transmitting to theboundary Compared with the constant pressure boundarythe falling range of pressure derivative curve with mixedboundary is less
62 Parameter Sensitivity Analysis Figure 7 shows the influ-ence of diffusion coefficient 120582 on the bottom-hole pressurewith mixed boundary There is relevance between diffusioncoefficient 120582 = 120572119896120591120579119871
2 and adsorption time 120591 = 1198772
119863The smaller the 120591 the shorter the time of desorption-diffusionand the shorter the time of pressure balance between fractureand matrix As shown in the diagram diffusion coefficient 120582mainly affects the duration of the second radial flow periodand appearance of third radial flow The greater the diffusioncoefficient 120582 the longer the duration of second radial flowperiod and the later the 119905 appearance of third radial flow andvice versa If 120582 is small enough the second radial flow periodwould be covered
Figure 8 shows the influence of Langmuir parameter 120573on the bottom-hole pressure with mixed boundary 120573 =
120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894) is related to Langmuir
adsorption pressure 119901119871and Langmuir adsorption volume119881
119871
The larger the 120573 the greater the adsorption ability of coalAs shown in the graph 120573 only affects the second radial flowsection The greater the 120573 the smaller the pressure derivativevalues of second radial flow period
Figure 9 shows the influence of fracture storage ratio120596 onthe bottom-hole pressure with mixed boundary Its expres-sion 120596 = 120601
119891119888119905120583(120601119891119888119905120583 + 6119901
119904119888119879119902119863119879119904119888120595119894) indicates that the
smaller the 120601119891119888119905 the smaller the 120596 and the more the radial
Journal of Chemistry 9
001 1 100 10000 1000000
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
0011E8
120573 = 01
120573 = 1
120573 = 10
Figure 8 Influence of 120573 on the bottom-hole pressure with mixedboundary (120596 = 001 120582 = 300 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
1E8
120596 = 04
120596 = 01
120596 = 001
Figure 9 Influence of 120596 on the bottom-hole pressure with mixedboundary (120582 = 300 120573 = 50 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
flow in fracture From Figure 8 almost all of the flow periodswould be affected by 120596 except for wellbore storage periodThe greater the 120596 the smaller the hump value of pressurederivative and the longer the duration of first radial flowperiod the greater the pressure derivative value in secondradial flow period
Figure 10 shows the influence of eccentricity of horizontalwell 119885
119908119889on the bottom-hole pressure with mixed boundary
The streamline shape and pressure distribution of a horizon-tal well would be affected by the size of 119885
119908119889 The smaller
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
Zwd = 01
Zwd = 03
Zwd = 05
Figure 10 Influence of119885119908119889
on the bottom-hole pressure withmixedboundary (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25 119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
LD = 1
LD = 25
LD = 50
Figure 11 Influence of 119871119863on the bottom-hole pressure with mixed
boundary (120596 = 001 120582 = 300 120573 = 50119885119908119889= 05119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
the119885119908119889 the larger the pressure derivative value in first radial
flow periodFigure 11 shows the influence of length of horizontal well
119871119863on the bottom-hole pressure with mixed boundary It
shows large influence of 119871119863on the pressure derivative value
in first radial flow period The smaller the 119871119863 the larger the
pressure derivative value in first radial flow period
63 Field Example To demonstrate the application of themodel proposed in this paper a horizontal well ZX-12 in aCBM reservoir is studied The vertical depth and horizontallength of this well are 689m and 536m respectively withwell radius 119903
119908of 01m coal thickness of 45m and initial
10 Journal of Chemistry
10 100100
1000
1000
TD
120595D
and120595998400 D
120595D of CBM horizontal well fitted curve120595998400D CBM horizontal well fitted curve
120595D of CBM horizontal well product curve120595998400D of CBM horizontal well product data curve
Figure 12 Fitted curves of test data from an actual well
pressure of 55MPa According to test results the Langmuirvolume 119881
119871is 3275m3t and the Langmuir pressure 119875
119871is
249MPa Figure 12 is the fitted curves which were obtainedby calculating the actual production and pressure data Asshown in Figure 12 the field data are in good match with thetheoretical results calculated by the new model in this paper
According to the actual field data some of reservoirparameters are fitted as follows the porosity is 004 and thepermeability is 3mD The stage of desorption and boundaryresponse can be clearly seen in Figure 12 Because of thefalling of the pressure derivative curve in the last stage wellZX-12 has a constant pressure boundary implying that thegas production of well ZX-12 is affected by the production ofadjacent well
7 Conclusions
Themathematic model of gas flowing into horizontal wells inCBMreservoirswith complex boundary conditions is derivedbased on the percolation theory The curves of bottom-hole pressure and pressure derivative are obtained by usingboundary element method and Laplace transform The con-clusions are as follows
(1) The fundamental boundary element solutions fortransient pressure response of horizontal wells inCBM reservoirs with complex boundary conditionscould be obtained by using Lord Kelvinrsquos point sourcesolution point source function theory and Poissonrsquossummation formula
(2) Comparison of the typical curves of flow character-istics between horizontal wells in CBM and conven-tional gas reservoirs shows that there is an additionalradial flow which reflects gas flow in fracture
(3) Comparing the typical curves of flow characteris-tics with complex boundary conditions the pressure
and pressure derivative curves with closed boundarywould be upward after pressure transmitting to theboundary The pressure derivative curves with con-stant pressure boundary and mixed boundary wouldbe fallen but the falling range of pressure derivativecurve with mixed boundary is less
Nomenclature
119903119908 Radius m
119896 Permeability mD120601 Porosity fraction119869 Diffusion flux gm2s119904 Laplace transform variableℎ Thickness m119861 Volume factor fraction119902 Influx into the wellbore m3d119871 Half length of horizontal well m120583 Viscosity mPasdots119888 Concentration of coalbed methane kgm3119875119894 Initial pressure MPa
119902 Production of well m3d119881119871 Langmuir volume constant m3ton
119881 Volume of coal matrix m3120595119894 Pseudopressure
120596 Fracture storage ratio fraction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is supported by National Natural Science Foun-dation of China (Grant no 51304032) National MajorScience and Technology Special Project of China (Grantno 2011ZX05037-003) and Research Fund for the Doc-toral Program of Higher Education of China (Grant no20125122110017)
References
[1] J C Cai E Perfect C-L Cheng and X Y Hu ldquoGeneralizedmodeling of spontaneous imbibition based on hagen-poiseuilleflow in tortuous capillaries with variably shaped aperturesrdquoLangmuir vol 30 no 18 pp 5142ndash5151 2014
[2] T Ertekin and W Sung ldquoPressure transient analysis of coalseams in the presence of multi-mechanistic flow and sorptionphenomenardquo in Proceedings of the SPE Gas Technology Sympo-sium pp 469ndash477 June 1989
[3] K Anbarci and T Ertekin ldquoA comprehensive study of pressuretransient analysis with sorption phenomena for single phase gasflow in coal seamsrdquo SPE 20568 1990
[4] C R Clarkson and R M Bustin ldquoEffect of pore structureand gas pressure upon the transport properties of coal alaboratory and modeling study 1 Isotherms and pore volumedistributionsrdquo Fuel vol 78 no 11 pp 1333ndash1344 1999
Journal of Chemistry 11
[5] C R Clarkson and R M Bustin ldquoEffect of pore structure andgas pressure upon the transport properties of coal a laboratoryandmodeling study 2 Adsorption rate modelingrdquo Fuel vol 78no 11 pp 1345ndash1362 1999
[6] S Reeve ldquoAdvanced reservoir modeling in desorption con-trolled reservoirsrdquo in Proceedings of the SPE Rocky MountainPetroleum Technology Conference SPE 71090 May 2001
[7] D K Tong and S Liu ldquoUnsteady percolation flow of coalbedmethane through deformed coal seamrdquo Natural Gas Industryvol 1 no 1 pp 74ndash76 2005
[8] X M Zhang and D K Tong ldquoAnalysis for pressure transient ofcoalbed methane reservoirrdquo Well Testing vol 6 no 17 pp 1ndash42008
[9] X M Zhang and D K Tong ldquoPressure transient analysisfor coal seams with triple-porositydual-permeability modelrdquoChineseQuarterly ofMechanics vol 4 no 29 pp 578ndash582 2008
[10] X H Hu S M Hu D I Zhang et al ldquoApplication of quadraticpressure method in well test interpretation of gas water twophase flow in coal seamrdquo Journal of Oil and Gas Technology vol7 no 33 pp 118ndash122 2011
[11] C R Clarkson C L Jordan D Ilk and T A Blasingame ldquoRate-transient analysis of 2-phase (gas + water) CBM wellsrdquo Journalof Natural Gas Science and Engineering vol 8 pp 106ndash120 2012
[12] K Aminian and S Ameri ldquoPredicting production performanceof CBM reservoirsrdquo Journal of Natural Gas Science and Engi-neering vol 1 no 1-2 pp 25ndash30 2009
[13] J C Cai and BM Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011
[14] W Sung T Ertekin and F C Schwerer ldquoThe developmenttesting and application of a comprehensive coal seam degasi-fication modelrdquo in Proceedings of the SPE Unconventional GasTechnology Symposium SPE 15247 Louisville Ky USA May1986
[15] W Sung and T Ertekin ldquoAn analysis of field developmentstrategies formethane production from coal seamsrdquo in Proceed-ings of the 62nd Annual Technical Conference and ExhibitionConference Paper SPE 16858 Dallas Tex USA 1987
[16] T Ertekin W Sung and F C Schwerer ldquoProduction perfor-mance analysis of horizontal drainage wells for degasificationof coal-seamsrdquo Journal of Petroleum Technology vol 40 no 5pp 625ndash632 1988
[17] T W Engler and J M Rajtar ldquoPressure transient testing ofhorizontal wells in coalbed reservoirsrdquo in Proceedings of theSPE Rocky Mountain Regional Meeting Paper SPE-24374-MSCasper Wyo USA May 1992
[18] P S Sarkar and J M Rajtar ldquoHorizontal well transient pres-sure testing in coalbed reservoirsrdquo in Proceedings of the 3rdLatin AmericaCaribbean Petroleum Engineering ConferenceSPE 26995 Buenos Aires Argentina April 1994
[19] P S Sarkar and J M Rajtar ldquoTransient well testing of coalbedmethane reservoirs with horizontal wellsrdquo in Proceedings of thePermian Basin Oil amp Gas Recovery Conference Paper SPE27681pp 531ndash539 Midland Tex USA March 1994
[20] X H Wang D I Zhang and Y Song ldquoDevelopment mecha-nism of low permeable coalbed methane frommultilateral hor-izontal pinnatewell with non-darcy seepage flow characteristicrdquoActa Geologica Sinica vol 7 no 82 pp 1437ndash1443 2008
[21] R-S Nie Y-F Meng J-C Guo and Y-L Jia ldquoModelingtransient flow behavior of a horizontal well in a coal seamrdquoInternational Journal of Coal Geology vol 92 pp 54ndash68 2012
[22] K L Katsifarakis D K Karpouzos and N TheodossiouldquoCombined use of BEM and genetic algorithms in groundwaterflow and mass transport problemsrdquo Engineering Analysis withBoundary Elements vol 23 no 7 pp 555ndash565 1999
[23] D T Numbere and D Tiab ldquoImproved streamline-generatingtechnique that uses the boundary (integral) element methodrdquoSPE Reservoir Engineering vol 3 no 3 pp 1061ndash1068 1988
[24] J Kikani and R N Horne ldquoApplication of boundary ele-ment method to reservoir engineering problemsrdquo Journal ofPetroleum Science and Engineering vol 3 no 3 pp 229ndash2411989
[25] J Hou YDWang andYMChen ldquoBoundary elementmethodin enhanced oil recoveryrdquo Chinese Journal of Hydrodynamicsvol 3 no 13 pp 23ndash28 2003
[26] K Chaiyo P Rattanadecho and S Chantasiriwan ldquoThemethodof fundamental solutions for solving free boundary saturatedseepage problemrdquo International Communications in Heat andMass Transfer vol 38 no 2 pp 249ndash254 2011
[27] K Rafiezadeh and B Ataie-Ashtiani ldquoSeepage analysis inmulti-domain general anisotropic media by three-dimensionalboundary elementsrdquo Engineering Analysis with Boundary Ele-ments vol 37 no 3 pp 527ndash541 2013
[28] X H Fu Y Qin W H Zhang et al ldquoCoal pore fractal classifi-cation and natural classification research based on the coal-bedgas migrationrdquo Chinese Science Bulletin vol 12 supplement 1pp 51ndash55 2005
[29] J C Cai and S Y Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal ofModern Physics C vol 24 no 8 pp 135ndash156 2013
[30] D M Han The Dynamic Analysis Technology Research onHorizontal Wells of Jingbian Gas Field University of XirsquoanShiyou Xirsquoan China 2011
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
001 0
1 1 10 100
1000
1000
0
1000
00
01
1
10
100
1Eminus4
1Eminus3
120595D of conventional gas reservoir horizontal well120595998400D of conventional gas reservoir horizontal well
120595D of CBM horizontal well120595998400D of CBM horizontal well
TD
120595D
and120595998400 D
Figure 5 Comparison of typical curves between horizontal wells inCBM and conventional gas reservoirs
001 0
1 1 10 100
1000
1000
0
1000
00
001
01
1
10
100
Closed boundaryConstant pressure boundaryMixed boundary
1Eminus6
1Eminus5
1Eminus4
1Eminus3
I II III IV V VI VII
TD
120595D
and120595998400 D
Figure 6 Typical curve of CBM horizontal well with complexboundary conditions (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
factor 119878 (3) the first radial flow section which is perpendic-ular to the horizontal wellbore the pressure derivative curveis shown as a horizontal line which reflects the early radialflow perpendicular to the horizontal wellbore (4) the secondtransition section which is the transition stage of radial flowfrom wellbore to fracture (5) the second radial flow stagewhich reflects the free gas flow in fracture this period isaffected by the flow of free gas and adsorbed gas diffusioncoefficient 120582 and Langmuir adsorption parameter 120573 (6) thethird radial flow section which reflects radial flow of whole
001 1 100 10000 1000000
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
120582 = 1
120582 = 100
120582 = 10000
Figure 7 Influence of diffusion coefficient 120582 on bottom-holepressure with mixed boundary (120596 = 001 120573 = 50 119871
119863= 25
119885119908119889= 05 119862
119889= 00001 119904 = 1 and 119903
119890119863= 50)
system after pressure balance and pressure derivative curveappears as a flat behavior (7) boundary response sectionThepressure and pressure derivative curves with closed boundarywould be upward after pressure transmitting to the boundaryThe pressure derivative curve with constant pressure bound-ary would be in decline after pressure transmitting to theboundary Compared with the constant pressure boundarythe falling range of pressure derivative curve with mixedboundary is less
62 Parameter Sensitivity Analysis Figure 7 shows the influ-ence of diffusion coefficient 120582 on the bottom-hole pressurewith mixed boundary There is relevance between diffusioncoefficient 120582 = 120572119896120591120579119871
2 and adsorption time 120591 = 1198772
119863The smaller the 120591 the shorter the time of desorption-diffusionand the shorter the time of pressure balance between fractureand matrix As shown in the diagram diffusion coefficient 120582mainly affects the duration of the second radial flow periodand appearance of third radial flow The greater the diffusioncoefficient 120582 the longer the duration of second radial flowperiod and the later the 119905 appearance of third radial flow andvice versa If 120582 is small enough the second radial flow periodwould be covered
Figure 8 shows the influence of Langmuir parameter 120573on the bottom-hole pressure with mixed boundary 120573 =
120595119871119881119871120595119894119902119863(120595119871+ 120595)(120595
119871+ 120595119894)(120595 + 120595
119894) is related to Langmuir
adsorption pressure 119901119871and Langmuir adsorption volume119881
119871
The larger the 120573 the greater the adsorption ability of coalAs shown in the graph 120573 only affects the second radial flowsection The greater the 120573 the smaller the pressure derivativevalues of second radial flow period
Figure 9 shows the influence of fracture storage ratio120596 onthe bottom-hole pressure with mixed boundary Its expres-sion 120596 = 120601
119891119888119905120583(120601119891119888119905120583 + 6119901
119904119888119879119902119863119879119904119888120595119894) indicates that the
smaller the 120601119891119888119905 the smaller the 120596 and the more the radial
Journal of Chemistry 9
001 1 100 10000 1000000
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
0011E8
120573 = 01
120573 = 1
120573 = 10
Figure 8 Influence of 120573 on the bottom-hole pressure with mixedboundary (120596 = 001 120582 = 300 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
1E8
120596 = 04
120596 = 01
120596 = 001
Figure 9 Influence of 120596 on the bottom-hole pressure with mixedboundary (120582 = 300 120573 = 50 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
flow in fracture From Figure 8 almost all of the flow periodswould be affected by 120596 except for wellbore storage periodThe greater the 120596 the smaller the hump value of pressurederivative and the longer the duration of first radial flowperiod the greater the pressure derivative value in secondradial flow period
Figure 10 shows the influence of eccentricity of horizontalwell 119885
119908119889on the bottom-hole pressure with mixed boundary
The streamline shape and pressure distribution of a horizon-tal well would be affected by the size of 119885
119908119889 The smaller
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
Zwd = 01
Zwd = 03
Zwd = 05
Figure 10 Influence of119885119908119889
on the bottom-hole pressure withmixedboundary (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25 119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
LD = 1
LD = 25
LD = 50
Figure 11 Influence of 119871119863on the bottom-hole pressure with mixed
boundary (120596 = 001 120582 = 300 120573 = 50119885119908119889= 05119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
the119885119908119889 the larger the pressure derivative value in first radial
flow periodFigure 11 shows the influence of length of horizontal well
119871119863on the bottom-hole pressure with mixed boundary It
shows large influence of 119871119863on the pressure derivative value
in first radial flow period The smaller the 119871119863 the larger the
pressure derivative value in first radial flow period
63 Field Example To demonstrate the application of themodel proposed in this paper a horizontal well ZX-12 in aCBM reservoir is studied The vertical depth and horizontallength of this well are 689m and 536m respectively withwell radius 119903
119908of 01m coal thickness of 45m and initial
10 Journal of Chemistry
10 100100
1000
1000
TD
120595D
and120595998400 D
120595D of CBM horizontal well fitted curve120595998400D CBM horizontal well fitted curve
120595D of CBM horizontal well product curve120595998400D of CBM horizontal well product data curve
Figure 12 Fitted curves of test data from an actual well
pressure of 55MPa According to test results the Langmuirvolume 119881
119871is 3275m3t and the Langmuir pressure 119875
119871is
249MPa Figure 12 is the fitted curves which were obtainedby calculating the actual production and pressure data Asshown in Figure 12 the field data are in good match with thetheoretical results calculated by the new model in this paper
According to the actual field data some of reservoirparameters are fitted as follows the porosity is 004 and thepermeability is 3mD The stage of desorption and boundaryresponse can be clearly seen in Figure 12 Because of thefalling of the pressure derivative curve in the last stage wellZX-12 has a constant pressure boundary implying that thegas production of well ZX-12 is affected by the production ofadjacent well
7 Conclusions
Themathematic model of gas flowing into horizontal wells inCBMreservoirswith complex boundary conditions is derivedbased on the percolation theory The curves of bottom-hole pressure and pressure derivative are obtained by usingboundary element method and Laplace transform The con-clusions are as follows
(1) The fundamental boundary element solutions fortransient pressure response of horizontal wells inCBM reservoirs with complex boundary conditionscould be obtained by using Lord Kelvinrsquos point sourcesolution point source function theory and Poissonrsquossummation formula
(2) Comparison of the typical curves of flow character-istics between horizontal wells in CBM and conven-tional gas reservoirs shows that there is an additionalradial flow which reflects gas flow in fracture
(3) Comparing the typical curves of flow characteris-tics with complex boundary conditions the pressure
and pressure derivative curves with closed boundarywould be upward after pressure transmitting to theboundary The pressure derivative curves with con-stant pressure boundary and mixed boundary wouldbe fallen but the falling range of pressure derivativecurve with mixed boundary is less
Nomenclature
119903119908 Radius m
119896 Permeability mD120601 Porosity fraction119869 Diffusion flux gm2s119904 Laplace transform variableℎ Thickness m119861 Volume factor fraction119902 Influx into the wellbore m3d119871 Half length of horizontal well m120583 Viscosity mPasdots119888 Concentration of coalbed methane kgm3119875119894 Initial pressure MPa
119902 Production of well m3d119881119871 Langmuir volume constant m3ton
119881 Volume of coal matrix m3120595119894 Pseudopressure
120596 Fracture storage ratio fraction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is supported by National Natural Science Foun-dation of China (Grant no 51304032) National MajorScience and Technology Special Project of China (Grantno 2011ZX05037-003) and Research Fund for the Doc-toral Program of Higher Education of China (Grant no20125122110017)
References
[1] J C Cai E Perfect C-L Cheng and X Y Hu ldquoGeneralizedmodeling of spontaneous imbibition based on hagen-poiseuilleflow in tortuous capillaries with variably shaped aperturesrdquoLangmuir vol 30 no 18 pp 5142ndash5151 2014
[2] T Ertekin and W Sung ldquoPressure transient analysis of coalseams in the presence of multi-mechanistic flow and sorptionphenomenardquo in Proceedings of the SPE Gas Technology Sympo-sium pp 469ndash477 June 1989
[3] K Anbarci and T Ertekin ldquoA comprehensive study of pressuretransient analysis with sorption phenomena for single phase gasflow in coal seamsrdquo SPE 20568 1990
[4] C R Clarkson and R M Bustin ldquoEffect of pore structureand gas pressure upon the transport properties of coal alaboratory and modeling study 1 Isotherms and pore volumedistributionsrdquo Fuel vol 78 no 11 pp 1333ndash1344 1999
Journal of Chemistry 11
[5] C R Clarkson and R M Bustin ldquoEffect of pore structure andgas pressure upon the transport properties of coal a laboratoryandmodeling study 2 Adsorption rate modelingrdquo Fuel vol 78no 11 pp 1345ndash1362 1999
[6] S Reeve ldquoAdvanced reservoir modeling in desorption con-trolled reservoirsrdquo in Proceedings of the SPE Rocky MountainPetroleum Technology Conference SPE 71090 May 2001
[7] D K Tong and S Liu ldquoUnsteady percolation flow of coalbedmethane through deformed coal seamrdquo Natural Gas Industryvol 1 no 1 pp 74ndash76 2005
[8] X M Zhang and D K Tong ldquoAnalysis for pressure transient ofcoalbed methane reservoirrdquo Well Testing vol 6 no 17 pp 1ndash42008
[9] X M Zhang and D K Tong ldquoPressure transient analysisfor coal seams with triple-porositydual-permeability modelrdquoChineseQuarterly ofMechanics vol 4 no 29 pp 578ndash582 2008
[10] X H Hu S M Hu D I Zhang et al ldquoApplication of quadraticpressure method in well test interpretation of gas water twophase flow in coal seamrdquo Journal of Oil and Gas Technology vol7 no 33 pp 118ndash122 2011
[11] C R Clarkson C L Jordan D Ilk and T A Blasingame ldquoRate-transient analysis of 2-phase (gas + water) CBM wellsrdquo Journalof Natural Gas Science and Engineering vol 8 pp 106ndash120 2012
[12] K Aminian and S Ameri ldquoPredicting production performanceof CBM reservoirsrdquo Journal of Natural Gas Science and Engi-neering vol 1 no 1-2 pp 25ndash30 2009
[13] J C Cai and BM Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011
[14] W Sung T Ertekin and F C Schwerer ldquoThe developmenttesting and application of a comprehensive coal seam degasi-fication modelrdquo in Proceedings of the SPE Unconventional GasTechnology Symposium SPE 15247 Louisville Ky USA May1986
[15] W Sung and T Ertekin ldquoAn analysis of field developmentstrategies formethane production from coal seamsrdquo in Proceed-ings of the 62nd Annual Technical Conference and ExhibitionConference Paper SPE 16858 Dallas Tex USA 1987
[16] T Ertekin W Sung and F C Schwerer ldquoProduction perfor-mance analysis of horizontal drainage wells for degasificationof coal-seamsrdquo Journal of Petroleum Technology vol 40 no 5pp 625ndash632 1988
[17] T W Engler and J M Rajtar ldquoPressure transient testing ofhorizontal wells in coalbed reservoirsrdquo in Proceedings of theSPE Rocky Mountain Regional Meeting Paper SPE-24374-MSCasper Wyo USA May 1992
[18] P S Sarkar and J M Rajtar ldquoHorizontal well transient pres-sure testing in coalbed reservoirsrdquo in Proceedings of the 3rdLatin AmericaCaribbean Petroleum Engineering ConferenceSPE 26995 Buenos Aires Argentina April 1994
[19] P S Sarkar and J M Rajtar ldquoTransient well testing of coalbedmethane reservoirs with horizontal wellsrdquo in Proceedings of thePermian Basin Oil amp Gas Recovery Conference Paper SPE27681pp 531ndash539 Midland Tex USA March 1994
[20] X H Wang D I Zhang and Y Song ldquoDevelopment mecha-nism of low permeable coalbed methane frommultilateral hor-izontal pinnatewell with non-darcy seepage flow characteristicrdquoActa Geologica Sinica vol 7 no 82 pp 1437ndash1443 2008
[21] R-S Nie Y-F Meng J-C Guo and Y-L Jia ldquoModelingtransient flow behavior of a horizontal well in a coal seamrdquoInternational Journal of Coal Geology vol 92 pp 54ndash68 2012
[22] K L Katsifarakis D K Karpouzos and N TheodossiouldquoCombined use of BEM and genetic algorithms in groundwaterflow and mass transport problemsrdquo Engineering Analysis withBoundary Elements vol 23 no 7 pp 555ndash565 1999
[23] D T Numbere and D Tiab ldquoImproved streamline-generatingtechnique that uses the boundary (integral) element methodrdquoSPE Reservoir Engineering vol 3 no 3 pp 1061ndash1068 1988
[24] J Kikani and R N Horne ldquoApplication of boundary ele-ment method to reservoir engineering problemsrdquo Journal ofPetroleum Science and Engineering vol 3 no 3 pp 229ndash2411989
[25] J Hou YDWang andYMChen ldquoBoundary elementmethodin enhanced oil recoveryrdquo Chinese Journal of Hydrodynamicsvol 3 no 13 pp 23ndash28 2003
[26] K Chaiyo P Rattanadecho and S Chantasiriwan ldquoThemethodof fundamental solutions for solving free boundary saturatedseepage problemrdquo International Communications in Heat andMass Transfer vol 38 no 2 pp 249ndash254 2011
[27] K Rafiezadeh and B Ataie-Ashtiani ldquoSeepage analysis inmulti-domain general anisotropic media by three-dimensionalboundary elementsrdquo Engineering Analysis with Boundary Ele-ments vol 37 no 3 pp 527ndash541 2013
[28] X H Fu Y Qin W H Zhang et al ldquoCoal pore fractal classifi-cation and natural classification research based on the coal-bedgas migrationrdquo Chinese Science Bulletin vol 12 supplement 1pp 51ndash55 2005
[29] J C Cai and S Y Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal ofModern Physics C vol 24 no 8 pp 135ndash156 2013
[30] D M Han The Dynamic Analysis Technology Research onHorizontal Wells of Jingbian Gas Field University of XirsquoanShiyou Xirsquoan China 2011
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
001 1 100 10000 1000000
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
0011E8
120573 = 01
120573 = 1
120573 = 10
Figure 8 Influence of 120573 on the bottom-hole pressure with mixedboundary (120596 = 001 120582 = 300 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
100
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
1E8
120596 = 04
120596 = 01
120596 = 001
Figure 9 Influence of 120596 on the bottom-hole pressure with mixedboundary (120582 = 300 120573 = 50 119871
119863= 25 119885
119908119889= 05 119862
119889= 00001
119904 = 1 and 119903119890119863= 50)
flow in fracture From Figure 8 almost all of the flow periodswould be affected by 120596 except for wellbore storage periodThe greater the 120596 the smaller the hump value of pressurederivative and the longer the duration of first radial flowperiod the greater the pressure derivative value in secondradial flow period
Figure 10 shows the influence of eccentricity of horizontalwell 119885
119908119889on the bottom-hole pressure with mixed boundary
The streamline shape and pressure distribution of a horizon-tal well would be affected by the size of 119885
119908119889 The smaller
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
Zwd = 01
Zwd = 03
Zwd = 05
Figure 10 Influence of119885119908119889
on the bottom-hole pressure withmixedboundary (120596 = 001 120582 = 300 120573 = 50 119871
119863= 25 119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
001 1 100 10000 1000000001
01
1
10
1E minus 6 1E minus 4
TD
120595D
and120595998400 D
LD = 1
LD = 25
LD = 50
Figure 11 Influence of 119871119863on the bottom-hole pressure with mixed
boundary (120596 = 001 120582 = 300 120573 = 50119885119908119889= 05119862
119889= 00001 119904 = 1
and 119903119890119863= 50)
the119885119908119889 the larger the pressure derivative value in first radial
flow periodFigure 11 shows the influence of length of horizontal well
119871119863on the bottom-hole pressure with mixed boundary It
shows large influence of 119871119863on the pressure derivative value
in first radial flow period The smaller the 119871119863 the larger the
pressure derivative value in first radial flow period
63 Field Example To demonstrate the application of themodel proposed in this paper a horizontal well ZX-12 in aCBM reservoir is studied The vertical depth and horizontallength of this well are 689m and 536m respectively withwell radius 119903
119908of 01m coal thickness of 45m and initial
10 Journal of Chemistry
10 100100
1000
1000
TD
120595D
and120595998400 D
120595D of CBM horizontal well fitted curve120595998400D CBM horizontal well fitted curve
120595D of CBM horizontal well product curve120595998400D of CBM horizontal well product data curve
Figure 12 Fitted curves of test data from an actual well
pressure of 55MPa According to test results the Langmuirvolume 119881
119871is 3275m3t and the Langmuir pressure 119875
119871is
249MPa Figure 12 is the fitted curves which were obtainedby calculating the actual production and pressure data Asshown in Figure 12 the field data are in good match with thetheoretical results calculated by the new model in this paper
According to the actual field data some of reservoirparameters are fitted as follows the porosity is 004 and thepermeability is 3mD The stage of desorption and boundaryresponse can be clearly seen in Figure 12 Because of thefalling of the pressure derivative curve in the last stage wellZX-12 has a constant pressure boundary implying that thegas production of well ZX-12 is affected by the production ofadjacent well
7 Conclusions
Themathematic model of gas flowing into horizontal wells inCBMreservoirswith complex boundary conditions is derivedbased on the percolation theory The curves of bottom-hole pressure and pressure derivative are obtained by usingboundary element method and Laplace transform The con-clusions are as follows
(1) The fundamental boundary element solutions fortransient pressure response of horizontal wells inCBM reservoirs with complex boundary conditionscould be obtained by using Lord Kelvinrsquos point sourcesolution point source function theory and Poissonrsquossummation formula
(2) Comparison of the typical curves of flow character-istics between horizontal wells in CBM and conven-tional gas reservoirs shows that there is an additionalradial flow which reflects gas flow in fracture
(3) Comparing the typical curves of flow characteris-tics with complex boundary conditions the pressure
and pressure derivative curves with closed boundarywould be upward after pressure transmitting to theboundary The pressure derivative curves with con-stant pressure boundary and mixed boundary wouldbe fallen but the falling range of pressure derivativecurve with mixed boundary is less
Nomenclature
119903119908 Radius m
119896 Permeability mD120601 Porosity fraction119869 Diffusion flux gm2s119904 Laplace transform variableℎ Thickness m119861 Volume factor fraction119902 Influx into the wellbore m3d119871 Half length of horizontal well m120583 Viscosity mPasdots119888 Concentration of coalbed methane kgm3119875119894 Initial pressure MPa
119902 Production of well m3d119881119871 Langmuir volume constant m3ton
119881 Volume of coal matrix m3120595119894 Pseudopressure
120596 Fracture storage ratio fraction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is supported by National Natural Science Foun-dation of China (Grant no 51304032) National MajorScience and Technology Special Project of China (Grantno 2011ZX05037-003) and Research Fund for the Doc-toral Program of Higher Education of China (Grant no20125122110017)
References
[1] J C Cai E Perfect C-L Cheng and X Y Hu ldquoGeneralizedmodeling of spontaneous imbibition based on hagen-poiseuilleflow in tortuous capillaries with variably shaped aperturesrdquoLangmuir vol 30 no 18 pp 5142ndash5151 2014
[2] T Ertekin and W Sung ldquoPressure transient analysis of coalseams in the presence of multi-mechanistic flow and sorptionphenomenardquo in Proceedings of the SPE Gas Technology Sympo-sium pp 469ndash477 June 1989
[3] K Anbarci and T Ertekin ldquoA comprehensive study of pressuretransient analysis with sorption phenomena for single phase gasflow in coal seamsrdquo SPE 20568 1990
[4] C R Clarkson and R M Bustin ldquoEffect of pore structureand gas pressure upon the transport properties of coal alaboratory and modeling study 1 Isotherms and pore volumedistributionsrdquo Fuel vol 78 no 11 pp 1333ndash1344 1999
Journal of Chemistry 11
[5] C R Clarkson and R M Bustin ldquoEffect of pore structure andgas pressure upon the transport properties of coal a laboratoryandmodeling study 2 Adsorption rate modelingrdquo Fuel vol 78no 11 pp 1345ndash1362 1999
[6] S Reeve ldquoAdvanced reservoir modeling in desorption con-trolled reservoirsrdquo in Proceedings of the SPE Rocky MountainPetroleum Technology Conference SPE 71090 May 2001
[7] D K Tong and S Liu ldquoUnsteady percolation flow of coalbedmethane through deformed coal seamrdquo Natural Gas Industryvol 1 no 1 pp 74ndash76 2005
[8] X M Zhang and D K Tong ldquoAnalysis for pressure transient ofcoalbed methane reservoirrdquo Well Testing vol 6 no 17 pp 1ndash42008
[9] X M Zhang and D K Tong ldquoPressure transient analysisfor coal seams with triple-porositydual-permeability modelrdquoChineseQuarterly ofMechanics vol 4 no 29 pp 578ndash582 2008
[10] X H Hu S M Hu D I Zhang et al ldquoApplication of quadraticpressure method in well test interpretation of gas water twophase flow in coal seamrdquo Journal of Oil and Gas Technology vol7 no 33 pp 118ndash122 2011
[11] C R Clarkson C L Jordan D Ilk and T A Blasingame ldquoRate-transient analysis of 2-phase (gas + water) CBM wellsrdquo Journalof Natural Gas Science and Engineering vol 8 pp 106ndash120 2012
[12] K Aminian and S Ameri ldquoPredicting production performanceof CBM reservoirsrdquo Journal of Natural Gas Science and Engi-neering vol 1 no 1-2 pp 25ndash30 2009
[13] J C Cai and BM Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011
[14] W Sung T Ertekin and F C Schwerer ldquoThe developmenttesting and application of a comprehensive coal seam degasi-fication modelrdquo in Proceedings of the SPE Unconventional GasTechnology Symposium SPE 15247 Louisville Ky USA May1986
[15] W Sung and T Ertekin ldquoAn analysis of field developmentstrategies formethane production from coal seamsrdquo in Proceed-ings of the 62nd Annual Technical Conference and ExhibitionConference Paper SPE 16858 Dallas Tex USA 1987
[16] T Ertekin W Sung and F C Schwerer ldquoProduction perfor-mance analysis of horizontal drainage wells for degasificationof coal-seamsrdquo Journal of Petroleum Technology vol 40 no 5pp 625ndash632 1988
[17] T W Engler and J M Rajtar ldquoPressure transient testing ofhorizontal wells in coalbed reservoirsrdquo in Proceedings of theSPE Rocky Mountain Regional Meeting Paper SPE-24374-MSCasper Wyo USA May 1992
[18] P S Sarkar and J M Rajtar ldquoHorizontal well transient pres-sure testing in coalbed reservoirsrdquo in Proceedings of the 3rdLatin AmericaCaribbean Petroleum Engineering ConferenceSPE 26995 Buenos Aires Argentina April 1994
[19] P S Sarkar and J M Rajtar ldquoTransient well testing of coalbedmethane reservoirs with horizontal wellsrdquo in Proceedings of thePermian Basin Oil amp Gas Recovery Conference Paper SPE27681pp 531ndash539 Midland Tex USA March 1994
[20] X H Wang D I Zhang and Y Song ldquoDevelopment mecha-nism of low permeable coalbed methane frommultilateral hor-izontal pinnatewell with non-darcy seepage flow characteristicrdquoActa Geologica Sinica vol 7 no 82 pp 1437ndash1443 2008
[21] R-S Nie Y-F Meng J-C Guo and Y-L Jia ldquoModelingtransient flow behavior of a horizontal well in a coal seamrdquoInternational Journal of Coal Geology vol 92 pp 54ndash68 2012
[22] K L Katsifarakis D K Karpouzos and N TheodossiouldquoCombined use of BEM and genetic algorithms in groundwaterflow and mass transport problemsrdquo Engineering Analysis withBoundary Elements vol 23 no 7 pp 555ndash565 1999
[23] D T Numbere and D Tiab ldquoImproved streamline-generatingtechnique that uses the boundary (integral) element methodrdquoSPE Reservoir Engineering vol 3 no 3 pp 1061ndash1068 1988
[24] J Kikani and R N Horne ldquoApplication of boundary ele-ment method to reservoir engineering problemsrdquo Journal ofPetroleum Science and Engineering vol 3 no 3 pp 229ndash2411989
[25] J Hou YDWang andYMChen ldquoBoundary elementmethodin enhanced oil recoveryrdquo Chinese Journal of Hydrodynamicsvol 3 no 13 pp 23ndash28 2003
[26] K Chaiyo P Rattanadecho and S Chantasiriwan ldquoThemethodof fundamental solutions for solving free boundary saturatedseepage problemrdquo International Communications in Heat andMass Transfer vol 38 no 2 pp 249ndash254 2011
[27] K Rafiezadeh and B Ataie-Ashtiani ldquoSeepage analysis inmulti-domain general anisotropic media by three-dimensionalboundary elementsrdquo Engineering Analysis with Boundary Ele-ments vol 37 no 3 pp 527ndash541 2013
[28] X H Fu Y Qin W H Zhang et al ldquoCoal pore fractal classifi-cation and natural classification research based on the coal-bedgas migrationrdquo Chinese Science Bulletin vol 12 supplement 1pp 51ndash55 2005
[29] J C Cai and S Y Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal ofModern Physics C vol 24 no 8 pp 135ndash156 2013
[30] D M Han The Dynamic Analysis Technology Research onHorizontal Wells of Jingbian Gas Field University of XirsquoanShiyou Xirsquoan China 2011
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
10 100100
1000
1000
TD
120595D
and120595998400 D
120595D of CBM horizontal well fitted curve120595998400D CBM horizontal well fitted curve
120595D of CBM horizontal well product curve120595998400D of CBM horizontal well product data curve
Figure 12 Fitted curves of test data from an actual well
pressure of 55MPa According to test results the Langmuirvolume 119881
119871is 3275m3t and the Langmuir pressure 119875
119871is
249MPa Figure 12 is the fitted curves which were obtainedby calculating the actual production and pressure data Asshown in Figure 12 the field data are in good match with thetheoretical results calculated by the new model in this paper
According to the actual field data some of reservoirparameters are fitted as follows the porosity is 004 and thepermeability is 3mD The stage of desorption and boundaryresponse can be clearly seen in Figure 12 Because of thefalling of the pressure derivative curve in the last stage wellZX-12 has a constant pressure boundary implying that thegas production of well ZX-12 is affected by the production ofadjacent well
7 Conclusions
Themathematic model of gas flowing into horizontal wells inCBMreservoirswith complex boundary conditions is derivedbased on the percolation theory The curves of bottom-hole pressure and pressure derivative are obtained by usingboundary element method and Laplace transform The con-clusions are as follows
(1) The fundamental boundary element solutions fortransient pressure response of horizontal wells inCBM reservoirs with complex boundary conditionscould be obtained by using Lord Kelvinrsquos point sourcesolution point source function theory and Poissonrsquossummation formula
(2) Comparison of the typical curves of flow character-istics between horizontal wells in CBM and conven-tional gas reservoirs shows that there is an additionalradial flow which reflects gas flow in fracture
(3) Comparing the typical curves of flow characteris-tics with complex boundary conditions the pressure
and pressure derivative curves with closed boundarywould be upward after pressure transmitting to theboundary The pressure derivative curves with con-stant pressure boundary and mixed boundary wouldbe fallen but the falling range of pressure derivativecurve with mixed boundary is less
Nomenclature
119903119908 Radius m
119896 Permeability mD120601 Porosity fraction119869 Diffusion flux gm2s119904 Laplace transform variableℎ Thickness m119861 Volume factor fraction119902 Influx into the wellbore m3d119871 Half length of horizontal well m120583 Viscosity mPasdots119888 Concentration of coalbed methane kgm3119875119894 Initial pressure MPa
119902 Production of well m3d119881119871 Langmuir volume constant m3ton
119881 Volume of coal matrix m3120595119894 Pseudopressure
120596 Fracture storage ratio fraction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This study is supported by National Natural Science Foun-dation of China (Grant no 51304032) National MajorScience and Technology Special Project of China (Grantno 2011ZX05037-003) and Research Fund for the Doc-toral Program of Higher Education of China (Grant no20125122110017)
References
[1] J C Cai E Perfect C-L Cheng and X Y Hu ldquoGeneralizedmodeling of spontaneous imbibition based on hagen-poiseuilleflow in tortuous capillaries with variably shaped aperturesrdquoLangmuir vol 30 no 18 pp 5142ndash5151 2014
[2] T Ertekin and W Sung ldquoPressure transient analysis of coalseams in the presence of multi-mechanistic flow and sorptionphenomenardquo in Proceedings of the SPE Gas Technology Sympo-sium pp 469ndash477 June 1989
[3] K Anbarci and T Ertekin ldquoA comprehensive study of pressuretransient analysis with sorption phenomena for single phase gasflow in coal seamsrdquo SPE 20568 1990
[4] C R Clarkson and R M Bustin ldquoEffect of pore structureand gas pressure upon the transport properties of coal alaboratory and modeling study 1 Isotherms and pore volumedistributionsrdquo Fuel vol 78 no 11 pp 1333ndash1344 1999
Journal of Chemistry 11
[5] C R Clarkson and R M Bustin ldquoEffect of pore structure andgas pressure upon the transport properties of coal a laboratoryandmodeling study 2 Adsorption rate modelingrdquo Fuel vol 78no 11 pp 1345ndash1362 1999
[6] S Reeve ldquoAdvanced reservoir modeling in desorption con-trolled reservoirsrdquo in Proceedings of the SPE Rocky MountainPetroleum Technology Conference SPE 71090 May 2001
[7] D K Tong and S Liu ldquoUnsteady percolation flow of coalbedmethane through deformed coal seamrdquo Natural Gas Industryvol 1 no 1 pp 74ndash76 2005
[8] X M Zhang and D K Tong ldquoAnalysis for pressure transient ofcoalbed methane reservoirrdquo Well Testing vol 6 no 17 pp 1ndash42008
[9] X M Zhang and D K Tong ldquoPressure transient analysisfor coal seams with triple-porositydual-permeability modelrdquoChineseQuarterly ofMechanics vol 4 no 29 pp 578ndash582 2008
[10] X H Hu S M Hu D I Zhang et al ldquoApplication of quadraticpressure method in well test interpretation of gas water twophase flow in coal seamrdquo Journal of Oil and Gas Technology vol7 no 33 pp 118ndash122 2011
[11] C R Clarkson C L Jordan D Ilk and T A Blasingame ldquoRate-transient analysis of 2-phase (gas + water) CBM wellsrdquo Journalof Natural Gas Science and Engineering vol 8 pp 106ndash120 2012
[12] K Aminian and S Ameri ldquoPredicting production performanceof CBM reservoirsrdquo Journal of Natural Gas Science and Engi-neering vol 1 no 1-2 pp 25ndash30 2009
[13] J C Cai and BM Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011
[14] W Sung T Ertekin and F C Schwerer ldquoThe developmenttesting and application of a comprehensive coal seam degasi-fication modelrdquo in Proceedings of the SPE Unconventional GasTechnology Symposium SPE 15247 Louisville Ky USA May1986
[15] W Sung and T Ertekin ldquoAn analysis of field developmentstrategies formethane production from coal seamsrdquo in Proceed-ings of the 62nd Annual Technical Conference and ExhibitionConference Paper SPE 16858 Dallas Tex USA 1987
[16] T Ertekin W Sung and F C Schwerer ldquoProduction perfor-mance analysis of horizontal drainage wells for degasificationof coal-seamsrdquo Journal of Petroleum Technology vol 40 no 5pp 625ndash632 1988
[17] T W Engler and J M Rajtar ldquoPressure transient testing ofhorizontal wells in coalbed reservoirsrdquo in Proceedings of theSPE Rocky Mountain Regional Meeting Paper SPE-24374-MSCasper Wyo USA May 1992
[18] P S Sarkar and J M Rajtar ldquoHorizontal well transient pres-sure testing in coalbed reservoirsrdquo in Proceedings of the 3rdLatin AmericaCaribbean Petroleum Engineering ConferenceSPE 26995 Buenos Aires Argentina April 1994
[19] P S Sarkar and J M Rajtar ldquoTransient well testing of coalbedmethane reservoirs with horizontal wellsrdquo in Proceedings of thePermian Basin Oil amp Gas Recovery Conference Paper SPE27681pp 531ndash539 Midland Tex USA March 1994
[20] X H Wang D I Zhang and Y Song ldquoDevelopment mecha-nism of low permeable coalbed methane frommultilateral hor-izontal pinnatewell with non-darcy seepage flow characteristicrdquoActa Geologica Sinica vol 7 no 82 pp 1437ndash1443 2008
[21] R-S Nie Y-F Meng J-C Guo and Y-L Jia ldquoModelingtransient flow behavior of a horizontal well in a coal seamrdquoInternational Journal of Coal Geology vol 92 pp 54ndash68 2012
[22] K L Katsifarakis D K Karpouzos and N TheodossiouldquoCombined use of BEM and genetic algorithms in groundwaterflow and mass transport problemsrdquo Engineering Analysis withBoundary Elements vol 23 no 7 pp 555ndash565 1999
[23] D T Numbere and D Tiab ldquoImproved streamline-generatingtechnique that uses the boundary (integral) element methodrdquoSPE Reservoir Engineering vol 3 no 3 pp 1061ndash1068 1988
[24] J Kikani and R N Horne ldquoApplication of boundary ele-ment method to reservoir engineering problemsrdquo Journal ofPetroleum Science and Engineering vol 3 no 3 pp 229ndash2411989
[25] J Hou YDWang andYMChen ldquoBoundary elementmethodin enhanced oil recoveryrdquo Chinese Journal of Hydrodynamicsvol 3 no 13 pp 23ndash28 2003
[26] K Chaiyo P Rattanadecho and S Chantasiriwan ldquoThemethodof fundamental solutions for solving free boundary saturatedseepage problemrdquo International Communications in Heat andMass Transfer vol 38 no 2 pp 249ndash254 2011
[27] K Rafiezadeh and B Ataie-Ashtiani ldquoSeepage analysis inmulti-domain general anisotropic media by three-dimensionalboundary elementsrdquo Engineering Analysis with Boundary Ele-ments vol 37 no 3 pp 527ndash541 2013
[28] X H Fu Y Qin W H Zhang et al ldquoCoal pore fractal classifi-cation and natural classification research based on the coal-bedgas migrationrdquo Chinese Science Bulletin vol 12 supplement 1pp 51ndash55 2005
[29] J C Cai and S Y Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal ofModern Physics C vol 24 no 8 pp 135ndash156 2013
[30] D M Han The Dynamic Analysis Technology Research onHorizontal Wells of Jingbian Gas Field University of XirsquoanShiyou Xirsquoan China 2011
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 11
[5] C R Clarkson and R M Bustin ldquoEffect of pore structure andgas pressure upon the transport properties of coal a laboratoryandmodeling study 2 Adsorption rate modelingrdquo Fuel vol 78no 11 pp 1345ndash1362 1999
[6] S Reeve ldquoAdvanced reservoir modeling in desorption con-trolled reservoirsrdquo in Proceedings of the SPE Rocky MountainPetroleum Technology Conference SPE 71090 May 2001
[7] D K Tong and S Liu ldquoUnsteady percolation flow of coalbedmethane through deformed coal seamrdquo Natural Gas Industryvol 1 no 1 pp 74ndash76 2005
[8] X M Zhang and D K Tong ldquoAnalysis for pressure transient ofcoalbed methane reservoirrdquo Well Testing vol 6 no 17 pp 1ndash42008
[9] X M Zhang and D K Tong ldquoPressure transient analysisfor coal seams with triple-porositydual-permeability modelrdquoChineseQuarterly ofMechanics vol 4 no 29 pp 578ndash582 2008
[10] X H Hu S M Hu D I Zhang et al ldquoApplication of quadraticpressure method in well test interpretation of gas water twophase flow in coal seamrdquo Journal of Oil and Gas Technology vol7 no 33 pp 118ndash122 2011
[11] C R Clarkson C L Jordan D Ilk and T A Blasingame ldquoRate-transient analysis of 2-phase (gas + water) CBM wellsrdquo Journalof Natural Gas Science and Engineering vol 8 pp 106ndash120 2012
[12] K Aminian and S Ameri ldquoPredicting production performanceof CBM reservoirsrdquo Journal of Natural Gas Science and Engi-neering vol 1 no 1-2 pp 25ndash30 2009
[13] J C Cai and BM Yu ldquoA discussion of the effect of tortuosity onthe capillary imbibition in porous mediardquo Transport in PorousMedia vol 89 no 2 pp 251ndash263 2011
[14] W Sung T Ertekin and F C Schwerer ldquoThe developmenttesting and application of a comprehensive coal seam degasi-fication modelrdquo in Proceedings of the SPE Unconventional GasTechnology Symposium SPE 15247 Louisville Ky USA May1986
[15] W Sung and T Ertekin ldquoAn analysis of field developmentstrategies formethane production from coal seamsrdquo in Proceed-ings of the 62nd Annual Technical Conference and ExhibitionConference Paper SPE 16858 Dallas Tex USA 1987
[16] T Ertekin W Sung and F C Schwerer ldquoProduction perfor-mance analysis of horizontal drainage wells for degasificationof coal-seamsrdquo Journal of Petroleum Technology vol 40 no 5pp 625ndash632 1988
[17] T W Engler and J M Rajtar ldquoPressure transient testing ofhorizontal wells in coalbed reservoirsrdquo in Proceedings of theSPE Rocky Mountain Regional Meeting Paper SPE-24374-MSCasper Wyo USA May 1992
[18] P S Sarkar and J M Rajtar ldquoHorizontal well transient pres-sure testing in coalbed reservoirsrdquo in Proceedings of the 3rdLatin AmericaCaribbean Petroleum Engineering ConferenceSPE 26995 Buenos Aires Argentina April 1994
[19] P S Sarkar and J M Rajtar ldquoTransient well testing of coalbedmethane reservoirs with horizontal wellsrdquo in Proceedings of thePermian Basin Oil amp Gas Recovery Conference Paper SPE27681pp 531ndash539 Midland Tex USA March 1994
[20] X H Wang D I Zhang and Y Song ldquoDevelopment mecha-nism of low permeable coalbed methane frommultilateral hor-izontal pinnatewell with non-darcy seepage flow characteristicrdquoActa Geologica Sinica vol 7 no 82 pp 1437ndash1443 2008
[21] R-S Nie Y-F Meng J-C Guo and Y-L Jia ldquoModelingtransient flow behavior of a horizontal well in a coal seamrdquoInternational Journal of Coal Geology vol 92 pp 54ndash68 2012
[22] K L Katsifarakis D K Karpouzos and N TheodossiouldquoCombined use of BEM and genetic algorithms in groundwaterflow and mass transport problemsrdquo Engineering Analysis withBoundary Elements vol 23 no 7 pp 555ndash565 1999
[23] D T Numbere and D Tiab ldquoImproved streamline-generatingtechnique that uses the boundary (integral) element methodrdquoSPE Reservoir Engineering vol 3 no 3 pp 1061ndash1068 1988
[24] J Kikani and R N Horne ldquoApplication of boundary ele-ment method to reservoir engineering problemsrdquo Journal ofPetroleum Science and Engineering vol 3 no 3 pp 229ndash2411989
[25] J Hou YDWang andYMChen ldquoBoundary elementmethodin enhanced oil recoveryrdquo Chinese Journal of Hydrodynamicsvol 3 no 13 pp 23ndash28 2003
[26] K Chaiyo P Rattanadecho and S Chantasiriwan ldquoThemethodof fundamental solutions for solving free boundary saturatedseepage problemrdquo International Communications in Heat andMass Transfer vol 38 no 2 pp 249ndash254 2011
[27] K Rafiezadeh and B Ataie-Ashtiani ldquoSeepage analysis inmulti-domain general anisotropic media by three-dimensionalboundary elementsrdquo Engineering Analysis with Boundary Ele-ments vol 37 no 3 pp 527ndash541 2013
[28] X H Fu Y Qin W H Zhang et al ldquoCoal pore fractal classifi-cation and natural classification research based on the coal-bedgas migrationrdquo Chinese Science Bulletin vol 12 supplement 1pp 51ndash55 2005
[29] J C Cai and S Y Sun ldquoFractal analysis of fracture increasingspontaneous imbibition in porous media with gas-saturatedrdquoInternational Journal ofModern Physics C vol 24 no 8 pp 135ndash156 2013
[30] D M Han The Dynamic Analysis Technology Research onHorizontal Wells of Jingbian Gas Field University of XirsquoanShiyou Xirsquoan China 2011
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