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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International Journal of

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Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Quantum Chemistry

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Organic Chemistry International

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

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Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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Analytical Methods in Chemistry

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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Journal of

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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 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

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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

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Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

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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

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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 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

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Journal of Chemistry 7

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