simulation of pressure transient behavior for ... · simulation of pressure transient behavior for...

Transp Porous Med (2013) 97:353–372DOI 10.1007/s11242-013-0128-z

Simulation of Pressure Transient Behaviorfor Asymmetrically Finite-Conductivity Fractured Wellsin Coal Reservoirs

Lei Wang · Xiaodong Wang · Junqian Li ·Jiahang Wang

Received: 18 September 2012 / Accepted: 19 January 2013 / Published online: 7 February 2013© Springer Science+Business Media Dordrecht 2013

Abstract Based on Fick’s law in matrix and Darcy flow in cleats and hydraulic fractures, anew semi-analytical model considering the effects of boundary conditions was presented toinvestigate pressure transient behavior for asymmetrically fractured wells in coal reservoirs.The new model is more accurate than previous model proposed by Anbarci and Ertekin, SPEannual technical conference and exhibition, New Orleans, 27–30 Sept 1998 because newmodel is expressed in the form of integral expressions and is validated well through numer-ical simulation. (1) In this paper, the effects of parameters including fracture conductivity,coal reservoir porosity and permeability, fracture asymmetry factor, sorption time constant,fracture half-length, and coalbed methane (CBM) viscosity on bottomhole pressure behaviorwere discussed in detail. (2) Type curves were established to analyze both transient pres-sure behavior and flow characteristics in CBM reservoir. According to the characteristics ofdimensionless pseudo pressure derivative curves, the process of the flow for fractured CBMwells was divided into six sub-stages. (3) This paper showed the comparison of transientsteady state and pseudo steady state models. (4) The effects of parameters including transfercoefficient, wellbore storage coefficient, storage coefficient of cleat, fracture conductivity,fracture asymmetry factor, and rate coefficient on the shape of type curves were also dis-cussed in detail, indicating that it is necessary to keep a bigger fracture conductivity andfracture symmetry for enhancing well production and reducing pressure depletion during thehydraulic fracturing design.

Keywords Semi-analytical model · The effects of boundary conditions ·Pressure transient behavior · Asymmetrically fractured wells · Type curves

L. Wang (B) · X. Wang · J. Li · J. WangSchool of Energy Resources, China University of Geosciences, Beijing100083,People’s Republic of Chinae-mail: [email protected]

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Nomenclature

Dimensionless Variables: Real DomainCfD Dimensionless fracture conductivitytD Dimensionless timeCD Dimensionless wellbore storage coefficientpwD Dimensionless well bottom pressurepD Dimensionless pseudo pressuredpD Dimensionless pseudo pressure derivativepfD Dimensionless pseudo fracture pressureSk Skin factorxD j Midpoint of the j segmentθ Fracture asymmetry factorλ Tranfer coefficient of CBM from matrix to cleatω Storage coefficient of cleat� Rate constant

Dimensionless Variables: Laplace Domain

s Time variable in Laplace domain, dimensionlessp̃D The dimensionless pseudo pressure pD in Laplace domainp̃wD Bottom pressure pwD in Laplace domainp̃fD Dimensionless pseudo fracture pressure pfD in Laplace domainq̃(u) Fracture rate q(x, t) in Laplace domainq̃fD Dimensionless fracture rate qfD in Laplace domain

Field variables

ct Total compressibility, 1/psik Effective permeability, mDp Bottomhole pressure, psipic Initial formation pressure, psiq Rate of per unit fracture length from formation, MMscf/dμ Fluid viscosity, cph Formation thickness, ftφ Porosity, fractionr Reservoir radius, ftre Equivalent drainage radius, ftt Time variable, hh Formation thickness, ftτ Temperature, oRZ Gas compressibility factor, fractionxf Fracture half length, ftw Width of the fracture, ftx ′ Integral variableC Volumetric gas concentration in the micropores, scf/ft3

D Diffusion coefficient, ft2/hT Sorption time constant, hVL Total sorption capacity, scf/ft3

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Simulation of Pressure Transient Behavior 355

Special functions

K0(x) Modified Bessel function (2nd kind, zero order)K1(x) Modified Bessel function (2nd kind, first order)I0(x) Modified Bessel function (1st kind, zero order)I1(x) Modified Bessel function (1st kind, first order)

Special Subscripts

a Macropore propertyi Micropore propertyf Fracture propertyD Dimensionlessg Gas propertysc Standard conditionic Initial conditionw Wellbore property

1 Introduction

Coal seams are categorized as unconventional gas reservoirs together with tight gas sands,devonian shales, geopressured aquifers and hydrate (Guo et al. 2003). As an alternativeenergy of conventional petroleum-gas resources, coalbed methane (CBM) has been studiedglobally. Particularly in China, a numerous CBM resources of 36.81 × 108m3 within thecoal reservoirs of <2,000 m burial depth has been found which nearly equals to conventionalnatural gas resources on land (Li et al. 2011; Yao et al. 2008, 2009).

Coal seam reservoirs are the dual-porosity media gas reservoirs composing of well-definedmacropore and micropore structures. The macropore system is occupied by the butt andface cleats while the micropore structure is made up of rock matrix. The face cleat in themacropore system is continuous throughout the whole coal seam while the butt cleat inmany cases is discontinuous, ending at an intersection with the face cleat (Ertekin and Sung1989). Generally, the face and butt cleats intersect at right angles (Guo et al. 2003). For thesake of the CBM mainly absorbing on the coal grains surface, the transport mechanism inCBM reservoirs is different from that in conventional gas reservoirs. Two types of modelscan be often used to describe the process of CBM sorption: equilibrium sorption model andnon-equilibrium sorption model. CBM sorption in equilibrium model is assumed to be onlypressure-dependent, while in non-equilibrium model it is assumed to be both pressure- andtime-dependent (Guo et al. 2003; Ertekin and Sung 1989). Langmuir adsorption isotherm isoften used to describe the process of equilibrium sorption, while Fick’s laws of diffusion canbe used to describe the model of non-equilibrium sorption/diffusion. Therefore, based on thesorption phenomenon above, mathematical model of fractured wells in CBM reservoirs willbe more complex than those in conventional reservoirs.

Massive hydraulic fracturing (MHF) in coal seams is an effective technique for produc-tivity enhancement of CBM wells. Therefore, evaluating coal seams properties of CBMfractured wells is an important task. During the past decades, there has been a continuouslyincreasing interest in the determination of conventional formation properties from transientpressure test or flow rate data analysis (Mcguire and Sikora 1960; Prats 1961; Raghavan

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et al. 1972). Gringarten et al. (1975) had made an extraordinary contribution to both thedevelopment of transient pressure analysis and type-curves analysis of fractured wells. Inhis work, three basic solutions were presented: the infinite fracture conductivity solution,the uniform flux solution for vertical fractures, and the uniform flux solution for horizon-tal fractures. Since fracture conductivity could not be ignored, a semi-analytical solutionand an analytical solution for a finite-conductivity vertical fractured well were presented(Cinco-Ley et al. 1978; Cinco-Ley and Fernando Samaniego 1981). These solutions werequite significant to the later analysis of productivity and well test data for fractured wells (Tiab1995; Agarwal et al. 1998; Pratikno et al. 2003; Tiab 2005; Lei et al. 2007; Jacques 2008).However, few papers have been reported about CBM wells. Sung et al. (1986) developeda coal seam degasification model and gave application of this model. Anbarci and Ertekin(1991) presented a simplified analysis technique to determine the desorption characteristicsof coal seams using pressure transient analysis. Anbarci and Ertekin (1990) developed somespecialized solutions for pressure transient analysis with sorption phenomenon for single-phase gas flow in coal seams. Aminian and Ameri (2009) developed a numerical CBM modelto predict production behavior of CBM reservoirs. Clarkson et al. (2009) made productiondata analysis of fractured and horizontal CBM wells. Nie et al. (2012) presented an analyticsolution and modeled transient flow behavior of a horizontal CBM well in a coal seam. Kinget al. (1986) developed a numerical method for a finite or infinite conductivity vertical fracturein coal seams to simulate the transient behavior of coal seal degasification wells. Anbarci andErtekin (1992) established a mathematical model of infinite conductivity vertical fracture inCBM reservoirs and discussed the pressure transient behavior using the model.

Most of the above literatures are modeling the CBM transport characteristics of verticalwell and horizontal wells (Sung et al. 1986; Anbarci and Ertekin 1990, 1991, 1992; Aminianand Ameri 2009; Clarkson et al. 2009; Nie et al. 2012), while few reports are related to CBMfractured wells (Anbarci and Ertekin 1992; Clarkson et al. 2009). At present, there is noliterature about complete analytical or semi-analytical model for a finite conductivity CBMfractured well. Objectives of this paper are to establish a complete semi-analytical model fora finite conductivity CBM fractured well and discuss the transient pressure behavior for CBMfractured wells. There are several advantages in this semi-analytical model. First, the newmodel is in Laplace space, so it is unnecessary to scatter time, thereby reducing the amountof computation and improving the computational efficiency. Second, it is convenient to addwellbore storage effects into the constant rate solution. Third, it is generally known that thereports above about MHF technology are all based on the assumption that the well is at thecenter of the fracture. However, the actual fractures obtained from MHF are asymmetric tothe well. Thus, in this new model we consider fracture asymmetry.

2 Physical Model

Physical model is assumed as follows:

(1) An isotropic, horizontal, slap CBM reservoir is bounded by overlying and underlyingimpermeable strata (see Fig. 1).

(2) The CBM reservoir has uniform thickness h, permeability k, and porosity φ, which donot change along with pressure.

(3) The CBM production is assumed to be an isothermal process. Flow in coal matrix obeysFick’s law. Darcy flow is considered in the cleats (butt and face cleats) and hydraulicfracture (see Figs. 2 and 3). We also assume that a linear flow occurs in the fracture andthere is no fluid through two endpoints of the fracture (see Fig. 3).

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Fig. 1 A finite conductivityvertical fracture in a boundedslap reservoir

Fig. 2 Darcy flow in the cleatand pseudo-steady diffusion inmatrix

Fig. 3 Darcy flow in thehydraulic fracture

(4) The CBM reservoir contains a slightly compressible fluid with constant compressibilityc and viscosity μ.

(5) CBM is produced through a vertically fractured well intersected by a fully penetrating,finite-conductivity fracture, which has a constant half length xf , width w, permeabilitykf , and porosity φf .

(6) The CBM well with asymmetric two wings produces in a condition of a constant-volumeproduction (see Fig. 4). The outer boundary of CBM reservoir may be infinite or closed,or constant pressure considered in this paper.

(7) Wellbore storage effect and skin effect are considered in this model.

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Fig. 4 The CBM well withasymmetric two wings producesin a condition of aconstant-volume production

3 Mathematical Model

3.1 CBM Fracture Model

In this section, based on the assumptions above, a two-dimensional fracture flow model isestablished. The fracture is saturated with a single phase gas (CBM). The CBM well is pro-duced under constant laminar flow rate condition. The equation describing two-dimensionalfracture flow model is stated as:

∂(−kf pf

∂pf∂x

)

∂x+

∂(−kf pf

∂pf∂y

)

∂y+ psc Z T μqsc

Tsc Zscwhδ(x − xw) = 0 (1)

where, xw is well position within the fracture.On the surface of the fracture there must be continuity in flux and pressure. Seen from the

reservoir, the fracture can be assumed to have zero thickness. We can, therefore, assume that

pf

(x,

w

2, t

)= pf

(x,−w

2, t

)= p(x, 0, t) (2)

and

kf∂pf

∂y |y= w2

= k∂p

∂y |y=0+= −k

∂p

∂y |y=0−= −kf

∂pf

∂y |y=− w2

(3)

Taking the integral average of pressure across the fracture width (in the y direction), andcombining Eqs. (2) and (3), Eq. (1) could be replaced by

∂(

pf∂pf∂x

)

∂x+ 1

w

2k

kf

(p∂p

∂y

)

|y=0− psc Z T μqsc

kf Tsc Zscwhδ(x − xw) = 0 (4)

where

pf (x, t) = 1

w

w2∫

− w2

pf (x, y, t)dy (5)

pf is the integral average of pressure in the y direction. On the surface of the fracture, it musthave the following flux relationship

qf

2= khTsc Zsc

pscT Zμp∂p

∂y(6)

Outer boundary conditions in the fracture can be given(

∂pf

∂x

)

x=−xf

= 0 (7)

(∂pf

∂x

)

x=xf

= 0 (8)

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Now we introduce dimensionless groups given as

θ = xw

xfqfD = 2xfqf

qscxD = x

xfyD = x

xf

PfD = P2ic − P2

f

P2icqD

qD = βT μZ

khp2ic

qsc CfD = kfw

kxf

Then flow equation in the fracture can be written in dimensionless as follows:

∂2 pfD

∂x2D

+ 2

CfD

∂pD

∂yD |yD=0+ 2π

CfDδ(xD − θ) = 0 (9)

Flux condition is

∂pD

∂yD |yD=0= −π

2qfD (10)

Outer boundary conditions are(

∂pfD

∂xD

)

xD=−1= 0 (11)

(∂pfD

∂xD

)

xD=1= 0 (12)

We give the Eqs. (9)–(12) a Laplace domain solution by means of Green’s functions andLaplace transform methods, which is written as

p̃fD(xD, s) = [ p̃fD(s)]avg + π

CfD

1∫

−1

N (x ′, xD)q̃fD(x ′, s) − 2π

CfDsN (θ, xD) (13)

In the Eq. (13), [ p̃fD(s)]avg is the average pressure for any time in the fracture, x ′ is the integralvariable, and N (x ′, xD) is the second Green’s function, which is a piecewise function and isdefined as

N (x ′, xD) = −1

4

[(x ′ + 1)2 + (xD − 1)2 − 4

3

]− 1 ≤ x ′ < xD (14)

N (x ′, xD) = −1

4

[(x ′ − 1)2 + (xD + 1)2 − 4

3

]xD < x ′ ≤ 1 (15)

3.2 CBM Reservoir Model

Anbarci and Ertekin (1990) had already obtained a series of line source solutions whichdescribed the pressure transient behavior in coal seams for various combinations of outer andinner boundary conditions. However, as for the present solutions of fractured CBM wells,we must change the inner boundary conditions. In this paper, we use Langmuir’s theoryin approximating the equilibrium isotherm to describe the model. We obtained the solutionof both unsteady state and pseudo-steady state sorption/diffusion models. Spherical systemelements can be used in the unsteady state model (see Fig. 5a).

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Fig. 5 Unsteady state and pseudo-steady state sorption/diffusion models: a spherical system elements. b Cubesystem elements

Macropore transport equation in the cleat can be obtained by performing a mass balanceon a radial elemental volume. It is similar to the one proposed by Anbarci and Ertekin (1990)

1

ra

∂

∂ra(ra

∂p2

∂ra) = φμcg

αk

∂p2

∂t+ 2pscT μZ

αkTsc

∂V

∂t(16)

Based on the description above, the unsteady state sorption equation for coal matrix can bewritten as

∂V

∂t= 3D

R

∂C

∂ri |ri=R(17)

Substituting Eq. (17) into Eq. (16), we obtain

1

ra

∂

∂ra(ra

∂p2

∂ra) = φμcg

αk

∂p2

∂t+ 6pscT μZ D

αkTsc R

∂C

∂ri |ri=R(18)

Initial condition is given as

p(ra, 0) = pic (19)

Inner boundary condition is given as

ra∂p

∂ra(rw, t)|rw→0 = βT μZ

2khqsc (20)

Outer boundary conditions for infinite, constant and closed can be given separately as

p(ra → ∞, t) = pic (21)

p(re, t) = pic (22)∂p(re, t)

∂ra= 0 (23)

We introduce the dimensionless groups, which are defined as follows

ω = φμcg

rDa = ra

xfrDi = ri

RtD = αkt

x2f

pfD = p2ic − p2

f

p2icqD

τ = R2

DqD = βT μZ

khp2ic

qsc λ = αkτ

x2f

CD = C − Cic = φμcg + 6pscT μZ

TscqD p2ic

pDa = p2ic − p2

p2icqD

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So, the macropore equations can be given in the dimensionless form as follows

1

rDa

∂

∂rDa(rDa

∂pDa

∂rDa) = ω

∂pDa

∂tD− (1 − ω)

λ

∂CD

∂rDi |rDi=1(24)

Initial condition is

pDa(rDa, 0) = 0 (25)

Inner boundary condition is

rDa∂pDa

∂rDa |rDa→0= −1 (26)

Outer boundary conditions are given separately as follows

pDa(rDa → ∞, t) = 0 (27)

pDa(rDa = reD, t) = 0 (28)∂pDa(reD, t)

∂rDa= 0 (29)

Using Fick’s law of diffusion, the micropore diffusivity equation describing the transport ofCBM in the coal matrix can be written as

1

r2i

∂

∂ri

(r2

i D∂C

∂ri

)= ∂C

∂t(30)

Initial condition in the coal matrix elements is given as:

C(ri, t = 0) = Cic (31)

Using the existing symmetry condition, center of the element could be treated as a no flowboundary:

∂C

∂ri(ri = 0, t) = 0 (32)

Concentration of the CBM on the external surface of the matrix elements is estimated at theCBM pressure in the cleats:

C(ri = R, t) = C|p(ra,t) (33)

Letting

CD = C − Cic rDi = riR = φμcg = 6pscT μZ

TscqD p2ic

tD = αkt x2

f

τ = R2

D rDa = raxf

λ = αkτ

x2f

Then the micropore equation is written as

1

rDi

∂

∂rDi

(rDi

∂CD

∂rDi

)= λ

∂CD

∂tD(34)

Initial condition is

CD(rDi, tD) = 0 (35)

Inner condition becomes

∂CD(rDi = 0, tD)

∂rDi= 0 (36)

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362 L. Wang et al.

Table 1 Coefficients fordifferent outer boundaryconditions

Outer boundary Coefficient A Coefficient B

Infinite 1 0

Constant pressure 1 −K0(reDs1/2)/I0(reDs1/2)

Closed 1 K1(reDs1/2)/I1(reDs1/2)

Outer condition becomes

CD(rDi = 1, tD) = CD|pDa(rDa,tD) (37)

By employing the Lapalace transform, the solution of Eqs. (34)–(37) can be obtained

∂C̃D

∂rDi |rDi=1= −�̃pDa(

√λs coth

√λs − 1) (38)

Now substituting the Eq. (38) into Lapalace transform equation of Eq. (24) and combiningEqs. (25)–(29), we can get the point source solution

p̃Da = AK0{rDa�} + B I0{rDa�} (39)

where

� =√

ws + (1 − w)

λ�(

√λs coth

√λs − 1) (40)

If we would like to gain PSS model (see Fig. 5b), Eq. (28) would be only replaced by

� =√

ws + (1 − w)

λ�

s

s + 1/λ(41)

Through integral of Eq.(39), uniform solution of the fracture can be obtained

p̃fD(xD, s) = 1

2

1∫

−1

q̃fD(x ′, s)[AK0{[(xD − x ′)2]1/2�} + B I0{[(xD − x ′)2]1/2�}]dx ′ (42)

In Eqs. (39) and (42), the coefficients A and B are listed in Table 1 for different boundarycombinations. Substituting the Eqs. (42) into Eq. (13) will yield

1

2

1∫

−1

q̃fD(x ′, s)[AK0{[(xD − x ′)2]1/2�} + B I0{[(xD − x ′)2]1/2�}]dx ′ = [ p̃fD(s)]avg

+ π

CfD

1∫

−1

N (x ′, xD)q̃fD(x ′, s) − 2π

CfDsN (θ, xD) (43)

Eq. (43) is a new semi-analytical model for an asymmetric fracture in CBM reservoirs, andwe need to discrete the Eq. (43) to get its solution.

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3.3 Constant Bottom-Hole Rate Solution

Assuming the fracture can be divided into 2N segments, integral of the left side in the Eq.(43) would have the following transformation,

1

2

1∫

−1

q̃fD(x ′, s)[AK0{[(xD − x ′)2]1/2�} + B I0{[(xD − x ′)2]1/2�}]dx ′

= 1

2A

2N∑i=N+1

q̃fDi

xDi+1∫

xDi

K0[∣∣xD j − x ′∣∣ �]dx ′ + 1

2A

1∑i=N

q̃fDi

xDi+1∫

xDi

K0[∣∣xD j + x ′∣∣ �]dx ′

+1

2B

2N∑i=N+1

q̃fDi

xDi+1∫

xDi

I0[∣∣xD j − x ′∣∣�]dx ′ + 1

2B

×1∑

i=N

q̃fDi

x Di+1∫

xDi

I0[∣∣xD j + x ′∣∣�]dx ′ (44)

Integral of the right side in the Eq. (43) would be transformed as,

π

CfD

1∫

−1

N (x ′, xD j )q̃fD(x ′, s)dx ′ = π

CfD

2N∑i=1

q̃fDi

xDi+1∫

xDi

N (x ′, xD j )dx ′ (45)

where xD j is the midpoint of the j segment. With the descriptions above about steady flow,we can know

1

2�x

2N∑i=1

q̃fDi(s) = 1

s(46)

The unknowns q̃fDi(s) and [ p̃fD(s)]avg can be obtained from Eqs. (44)–(46). Then take q̃fDi(s)and [ p̃fD(s)]avg back into Eq. (42) and make xD = θ to get the bottom-hole pressure solutionin Laplace domains. Then, by Stehfest numerical algorithm, p̃fD(θ, s) · qfD(tD) and pfD(tD)

can be figured out for any given tD. To obtain the solution including wellbore storage andskin effect, we need the relationship given as

p̃wD = 1

s2CD + s/[s p̃fD(θ, s) + Sk] (47)

If substituting the solved p̃fD(θ, s) into Eq. (47), we can obtain the solution including wellborestorage and skin effect.

4 Results and Discussion

4.1 Accuracy of the Solution in this Paper

To validate our results, the data reported in the literature (Anbarci and Ertekin 1992) wereselected, as listed in Table 2. In our model, CfD = 1, 000 and θ = 0 were proposed con-sidering the neglect of fracture conductivity and fracture asymmetry in Anbarci’s model.

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364 L. Wang et al.

Table 2 Basic data used in thispaper τ = 3.2899 × 105 h xf = 100 ft φ = 0.01

μ = 0.01082 cp ct = 2.234 × 10−3psi−1 K = 26 md

T = 530 ◦R Z = 0.9404 h = 6 ft

qsc = 0.2 MMscf/d VL = 18.632 scf/ft3 pL = 167.58 psi

pic = 447.7 psi

Fig. 6 Comparison of our resultsagainst the results in the literature

As shown in Fig. 6, there is a better agreement between the solution obtained in this workand the results of numerical simulation. Anbarci’s results are relatively lower than that fromnumerical simulation. It is also verified that Green’s Function method which is used to solvethe fracture model in this paper is more accurate than other methods in the literatures.

4.2 Influencing Factors on Bottomhole Pressure

The effects of parameters on bottomhole pressure behavior, including fracture conductivity,coal reservoir porosity and permeability, fracture asymmetry factor, sorption time constant,fracture half-length, and coalbed methane (CBM) viscosity, were all analyzed in detail underthe same condition presented in Table 2.

Figure 7a shows the effects of the fracture conductivity CfD of 0.5, 5, and 50 on CBMpressure. It can be seen that, at the same time point, the larger the fracture conductivity is,the higher the bottomhole pressure is. However, the effects of CfD on pressure depend uponCfD values. The fracture conductivity CfD < 5 shows stronger influences than those whenCfD > 5 on the bottomhole pressure. The effects of the different porosity φ values (0.01, 0.1,and 0.25) on bottomhole pressure were shown in Fig. 7b. At the same time point, the largerthe porosity is, the higher the bottomhole pressure will be. Moreover, a smaller porosityvalue (ϕ < 0.1) will lead to a lower location of the pressure curve, which means the pressuredepletion will be faster for a smaller porosity CBM reservoir. As shown in Fig. 7c, from theeffects of fracture asymmetry factor θ ranging from 0 to 1 on bottomhole pressure it is foundthat the bottomhole pressure changes slightly with the θ < 0.5, while overall lower values forthe CBM well will appear with the θ > 0.5, which means off-centered well in the fracture willlead a bigger pressure depletion. Therefore, well location in the fracture is a significant factorto affect bottomhole pressure. Fig. 7d shows the effects of CBM reservoir permeability k onbottomhole pressure. It can be seen that, at the same time point, the smaller the permeability

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Fig. 7 The effects of parameters on bottomhole pressure behavior were analyzed in detail: a the effect of thefracture conductivity CfD on CBM pressure. b The effects of the different porosity φ values on bottomholepressure. c The effects of fracture asymmetry factor θ ranging from 0 to 1 on bottomhole pressure. d Theeffects of CBM reservoir permeability k on bottomhole pressure. e The effects of sorption time constant τ onbottomhole pressure. f The effects of fracture half length xf on bottomhole pressure. g The effect of CBMviscosity μ on bottomhole pressure. h The effects of Langmuir volume VL on bottomhole pressure. i Theeffects of total compressibility ct on bottomhole pressure

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Fig. 8 Type curves used for theanalysis of transient pressurebehavior effected by externalboundary conditions (reD = 5).Stage 1: wellbore storage effect.Stage 2: wellbore storagetransition region. Stage 3: linearflow region. Stage 4:diffusion/desorption region.Stage 5: pseudo-radial region.Stage 6: boundary dominatedregion

k is, the lower the bottomhole pressure will be. Moreover, the pressure depletion with smallerpermeability is much larger than the one with higher permeability values. According to thedefinition of the sorption time constant above, the sorption time constant τ is negativelyrelated to the diffusion coefficient D. Therefore, τ reflects the diffusion ability from matrix.Its effects on bottomhole pressure were shown in Fig. 7e. It can be seen that the bigger τ

is, the lower the bottomhole pressure will be. Furthermore, differences of the slope betweendifferent pressure curves are remarkable. Fig. 7f shows the effects of fracture half-length xf

on bottomhole pressure p. At the same time point, the smaller the fracture xf is, the lower thebottomhole pressure will be. It follows that a shorter fracture will lead to a bigger pressuredepletion. Moreover, the slopes of the pressure curves of CBM wells with different fracturehalf-length vary slightly, which implies that the rate of the pressure drop at different xf valuesis approximate. CBM viscosity μ is referred to fluid property instead of fracture and CBMreservoir. From its effects on bottomhole pressure p, as shown in Fig. 7g, it was found thatthe bigger μ is, the lower the bottomhole pressure will be. Moreover, the bottomhole pressureof CBM well with a high viscosity value will reduce rapidly with depletion. The Langmuirvolume VL mainly affects the change of bottomhole pressure with the CBM production after1 day (Fig. 7h). The lower bottomhole pressure occurs in CBM well in the coal reservoir witha small VL. Usually, a weaker sorption capacity will cause rapid pressure depletion. Fig. 7ishows the effects of total compressibility ct on bottomhole pressure p. At the same timepoint, the smaller ct is, the lower the bottomhole pressure will be. It follows that a smallertotal compressibility will lead to a bigger pressure depletion. However, as the result of timeincreasing, pressure values for different ct values are almost same, which implies that theeffect of ct on pressure curves is weak in the late time.

4.3 Well Test Curves under Different External Boundary Conditions

The transient transport characteristics are graphically showed by Type curves, which can beused to analyze transient pressure and rate decline so as to recognize the flow characteristicsof fluids in CBM reservoir. In addition, by Type curves matching, some reservoir propertyparameters, such as permeability, skin factor, gas in place, fracture half-length, and gasreservoir drainage area, can be obtained (Wang et al. 2012; Nie et al. 2012). Figures 8, 9, and10 show that the Type curves of pressure and derivative pressure analysis for asymmetricallyfractured CBM well model reflect transient and pseudo steady state in coalbed.

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Fig. 9 Comparison of transientsteady state TSS and pseudosteady state PSS models

Fig. 10 The effect of parameters on the shape of type curves: a the effect of transfer coefficient λ on the shapeof type curves. b The effect of wellbore storage coefficient CD on the shape of type curves. c The effect ofcleat storage coefficient ω on the shape of type curves. d The effect of fracture conductivity CfD on the shapeof type curves. e The effect of fracture asymmetry factor θ on the shape of type curves. f The effect of rateconstant � on the shape of type curves

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The Type curves which are used for the analysis of transient pressure behavior affectedby external boundary conditions are shown in Fig. 8. The used data were listed in Table 3. Itcan be found that the curves are sub-divided into six stages as marked in the figure:

Stage 1: In this stage, the curve has a straight line with the slope of one, reflecting wellborestorage effect.Stage 2: It can be seen that the curve in this stage is upward bending, for the reason ofwellbore storage effects. Therefore, this region can be called as wellbore storage transitionregion.Stage 3: Linear flow in CBM reservoirs is observed in this stage. Both dimensionlesspseudo pressure curve and its derivative curve show two parallel straight lines with theslope of 1/2. Why is bi-linear flow reported by Cinco-Ley and Fernando Samaniego(1981) not observed in this stage? For the reason that the time of bi-linear flow is veryshort while wellbore storage effect is dominated for a long time, it is hard to find out thebi-linear flow region. In this region, the flow pathway is mainly dominated by the naturalcleat network. The flow in coal matrix is weak due to the low diffusion/desorption rate.Stage 4: When most of the fluid flow towards wellbore from the cleat network, the diffu-sion/desorption rate becomes high, indicating that the diffusion in matrix is predominant.Thus, the stage 4 is named as diffusion/desorption region. It can be seen that the dimen-sionless pseudo pressure derivative curve has a V-like shape. The shape, location, andsize of the type curves in this region are controlled by the wellbore storage CD, transfercoefficient λ, cleat storage coefficient ω, rate constant �, fracture conductivity CfD, andfracture asymmetry factor θ (see Fig. 10).Stage 5: The diffusion rate in matrix is equal to the flow rate from the cleat to wellbore, sofluid flow in this stage reaches a dynamic balance state. It can be seen from the plot thatthe dimensionless pseudo pressure derivative curve nearly shows a zero slope straightline. So, this stage is named as pseudo-radial region.Stage 6: There is no difference in the shape of the curve under the different boundaryconditions (from Stage 1 to Stage 5); however, we can see simply that difference whenflow reaches pseudo-steady state. For infinite-acting coalbeds, the dimensionless pres-sure curve slowly rises with the increase in dimensionless time, but the dimensionlesspressure derivative curve shows a zero slope straight line. For constant pressure case,the dimensionless pressure curve shows a zero slope straight line, but the dimensionlesspressure derivative curve goes down rapidly. For closed boundary case, both the dimen-sionless pressure curve and its derivative curve show a unit slope straight line and go uprapidly. Therefore, this region can be named as boundary dominated region.

4.4 Comparison of Transient Steady State and Pseudo Steady State (PSS) Models

Figure 9 shows a comparison of transient steady state (TSS) and PSS models for the sameparameters listed in Table 3. Two models perform some differences for the same parameters.For wellbore storage effect region, these dimensionless pseudo pressure and pseudo pressurederivative curves of two models exhibit good agreement with a unit slope straight line. How-ever, for linear flow region, dimensionless pseudo pressure and pseudo pressure derivativecurves in PSS model are higher than those in TSS model, which means pressure depletionin PSS model is much bigger than that in TSS model. It is the result of the geometry of twomodels. For PSS model, we can simply see a diffusion region, which is not observed in TSSmodel still showing a 1/2 slope straight line. For pseudo radial flow region, dimensionlesspseudo pressure and pseudo pressure derivative curves in PSS model are lower than those

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Table 3 Basic data used for typecurves analysis CD = 10−5 ω = 0.001 λ = 0.005

θ = 0 CfD = 1 � = 1

in TSS model, which means pressure depletion in PSS model is much smaller than that inTSS model. This is because, in a short time, CBM diffuse from matrix to cleat, which sup-plementary pressure depletion in PSS model. The difference between TSS and PSS modelsis that spherical system elements (Fig. 5a) can be assumed in the TSS model; however, cubicelements (Fig. 5b) are assumed in PSS model. Using TSS model, we obtain the Eq. (40) andusing PSS model we can get the Eq. (41). Equations (40) and (41) also reflect the differencebetween TSS and PSS models. Therefore, we can see that difference in Fig. 9.

4.5 Parameters Influence on Shape of Type Curves

Transfer coefficient λ is a function defined in the above sections, which is positively pro-portional to permeability K and sorption time constant τ but is negatively proportional tothe square of fracture half-length. According to the definition of the sorption time constantin the above sections, τ is negatively related to the diffusion coefficient D. So, a bigger λ

reflect a weaker diffusion ability. Figure 10a shows the effect of transfer coefficient λ on theshape of type curves for the same parameters listed in Table 3. A smaller λ value leads to theearly time of diffusion from matrix to cleat, which also means a bigger diffusion coefficientwill cause the early time of diffusion from matrix to cleat. Therefore, λ reflects the diffusionability in matrix and starting time of gas transfer from matrix to cleat.

Figure 10b shows the effect of wellbore storage coefficient CD on the shape of type curvesfor the same parameters listed in Table 3. We can see from the curves that in the early time,a bigger CD will lead to a bigger pressure depletion, which will make the time of linear flowshorter, and make the starting time of diffusion from matrix to cleat ahead of time.

According to the definition of the storage coefficient of cleat in the above sections, ω isnegatively proportional to the porosity ϕ. Therefore, ω reflects storage ability of the CBM inthe cleat and matrix. A bigger ω means a stronger storage ability. Figure 10c shows the effectof cleat storage coefficient ω on the shape of type curves for the same parameters listed inTable 3. We can see from the curves that in the early time, a bigger ω will lead to a smallerpressure depletion, which will make the time of linear flow longer, and prolong the startingtime of diffusion from matrix to cleat and short total time of diffusion from matrix to cleat.In addition, there is no difference in the reaching of pseudo radial flow, which indicates inthe late time, their pressure depletions are same.

Figure 10d shows the effect of fracture conductivity CfD on the shape of type curvesfor the same parameters listed in Table 3. In the entire flow period, a smaller CfD willlead to bigger pressure depletion as expected, which indicates that fracture conductivityis important to the effect on production. In the hydraulic fracturing design, it is neces-sary to keep bigger fracture conductivity to enhance well production and reduce pressuredepletion.

Figure 10e shows the effect of fracture asymmetry factor θ on the shape of type curvesfor the same parameters listed in Table 3. In the entire flow period, a bigger θ will lead to abigger pressure depletion as expected, which indicates that fracture asymmetry factor is alsoimportant to the effect on production. In the hydraulic fracturing design, in order to enhancewell production and reduce pressure depletion, we should keep the fracture symmetry.

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According to the definition of the rate coefficient in the above sections, � is a functionrelated to the production qsc and is proportional to qsc. Therefore, � reflects the productivityof CBM. A bigger qsc means a stronger productivity. Figure 10f shows the effect of rateconstant � on the shape of type curves for the same parameters listed in Table 3. In theentire flow period, a bigger � will lead to a smaller pressure depletion as expected but thestarting time of diffusion from matrix to cleat occurs early, which indicates that a biggerwell production needs a smaller depletion. In a word, to enhance well production and reducepressure depletion, each parameter of CBM well in the practical production process shouldbe coordinated appropriately.

5 Conclusions

(1) We have successfully constructed comprehensive fracture flow model, formation flowmodel for asymmetrically fractured wells centered in a constant pressure boundary, aclosed boundary and an infinite boundary, circular CBM reservoir.

(2) Fracture asymmetry factor and fracture conductivity are both considered in new semi-analytical model which is different from previous models. The new model is moreaccurate than that previous model Anbarci presented because new model is gained inthe form of Bessel integral expressions and is validated well with numerical simulation.

(3) Effects of parameters on bottomhole pressure behavior are discussed in details includingfracture conductivity CfD, the CBM reservoir porosity ϕ, fracture asymmetry factor θ ,CBM reservoir permeability k, sorption time constant τ , fracture half-length xf , andCBM viscosity μ.

(4) Type curves are established to make transient pressure analysis and recognize the flowcharacteristics for a real CBM reservoir. Flow for CBM fractured wells is divided into sixstages according to characteristics of dimensionless pseudo pressure derivative curve:wellbore storage effects region showing a unit slope straight line, wellbore storage tran-sition region, linear flow region showing a 1/2 slope straight line, diffusion/desorptionregion showing V-shaped curve, pseudo-radial region approximately showing zero slopestraight line, and boundary dominated region decided by different boundary conditions.

(5) Comparison of TSS and pseudo steady state(PSS) models indicates that for PSS model,diffusion/desorption region can be simply seen, which is not observed in TSS modelstill showing a 1/2 slope straight line.

(6) Effects of parameters on shape of type curves are discussed in details including Transfercoefficient λ, wellbore storage coefficient CD, storage coefficient of cleat ω, fractureconductivity CfD, fracture asymmetry factor θ , and rate coefficient �. In the hydraulicfracturing design, to enhance well production and reduce pressure depletion, it is nec-essary for us to keep bigger fracture conductivity and fracture symmetry.

Acknowledgments This article was supported by Important National Science and Technology SpecificProjects of the twelfth five Years Plan Period (Grant No.2011ZX05013-002) and the National Basic ResearchProgram of China (Grant No.2011ZX05009-004).

6 Appendix A

α = 2.637 × 10−4 β = 1.422 × 106

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7 Appendix B

See Table 4.

Table 4 SI metric conversionfactors bbl × 0.1589874 m3

cP × 0.001 Pa s

ft × 0.3048 m

ft2 × 0.0929 m2

psi × 6.894757 kPa

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