Simulation of Pressure Transient Behavior for Asymmetrically Finite-Conductivity Fractured Wells in Coal Reservoirs

Transcription

1 Transp Porous Med (2013) 97: DOI /s z Simulation of Pressure Transient Behavior for Asymmetrically Finite-Conductivity Fractured Wells in 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, a new semi-analytical model considering the effects of boundary conditions was presented to investigate pressure transient behavior for asymmetrically fractured wells in coal reservoirs. The new model is more accurate than previous model proposed by Anbarci and Ertekin, SPE annual technical conference and exhibition, New Orleans, Sept 1998 because new model is expressed in the form of integral expressions and is validated well through numerical 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 behavior were discussed in detail. (2) Type curves were established to analyze both transient pressure behavior and flow characteristics in CBM reservoir. According to the characteristics of dimensionless pseudo pressure derivative curves, the process of the flow for fractured CBM wells was divided into six sub-stages. (3) This paper showed the comparison of transient steady state and pseudo steady state models. (4) The effects of parameters including transfer coefficient, wellbore storage coefficient, storage coefficient of cleat, fracture conductivity, fracture asymmetry factor, and rate coefficient on the shape of type curves were also discussed in detail, indicating that it is necessary to keep a bigger fracture conductivity and fracture symmetry for enhancing well production and reducing pressure depletion during the hydraulic 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. Wang School of Energy Resources, China University of Geosciences, Beijing100083, People s Republic of China wanglei1986sp@foxmail.com

2 354 L. Wang et al. Nomenclature Dimensionless Variables: Real Domain C fd Dimensionless fracture conductivity t D Dimensionless time C D Dimensionless wellbore storage coefficient p wd Dimensionless well bottom pressure p D Dimensionless pseudo pressure dp D Dimensionless pseudo pressure derivative p fd Dimensionless pseudo fracture pressure S k Skin factor x D j Midpoint of the j segment θ Fracture asymmetry factor λ Tranfer coefficient of CBM from matrix to cleat ω Storage coefficient of cleat Rate constant Dimensionless Variables: Laplace Domain s p D p wd p fd q(u) q fd Time variable in Laplace domain, dimensionless The dimensionless pseudo pressure p D in Laplace domain Bottom pressure p wd in Laplace domain Dimensionless pseudo fracture pressure p fd in Laplace domain Fracture rate q(x, t) in Laplace domain Dimensionless fracture rate q fd in Laplace domain Field variables c t Total compressibility, 1/psi k Effective permeability, md p Bottomhole pressure, psi p ic Initial formation pressure, psi q Rate of per unit fracture length from formation, MMscf/d μ Fluid viscosity, cp h Formation thickness, ft φ Porosity, fraction r Reservoir radius, ft r e Equivalent drainage radius, ft t Time variable, h h Formation thickness, ft τ Temperature, o R Z Gas compressibility factor, fraction x f Fracture half length, ft w Width of the fracture, ft x Integral variable C Volumetric gas concentration in the micropores, scf/ft 3 D Diffusion coefficient, ft 2 /h T Sorption time constant, h V L Total sorption capacity, scf/ft 3

3 Simulation of Pressure Transient Behavior 355 Special functions K 0 (x) K 1 (x) I 0 (x) I 1 (x) Modified Bessel function (2nd kind, zero order) Modified Bessel function (2nd kind, first order) Modified Bessel function (1st kind, zero order) Modified Bessel function (1st kind, first order) Special Subscripts a i f D g sc ic w Macropore property Micropore property Fracture property Dimensionless Gas property Standard condition Initial condition 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 alternative energy of conventional petroleum-gas resources, coalbed methane (CBM) has been studied globally. Particularly in China, a numerous CBM resources of m 3 within the coal reservoirs of <2,000 m burial depth has been found which nearly equals to conventional natural 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-defined macropore and micropore structures. The macropore system is occupied by the butt and face cleats while the micropore structure is made up of rock matrix. The face cleat in the macropore system is continuous throughout the whole coal seam while the butt cleat in many cases is discontinuous, ending at an intersection with the face cleat (Ertekin and Sung 1989). Generally, the face and butt cleats intersect at right angles (Guo et al. 2003). For the sake of the CBM mainly absorbing on the coal grains surface, the transport mechanism in CBM reservoirs is different from that in conventional gas reservoirs. Two types of models can be often used to describe the process of CBM sorption: equilibrium sorption model and non-equilibrium sorption model. CBM sorption in equilibrium model is assumed to be only pressure-dependent, while in non-equilibrium model it is assumed to be both pressure- and time-dependent (Guo et al. 2003; Ertekin and Sung 1989). Langmuir adsorption isotherm is often used to describe the process of equilibrium sorption, while Fick s laws of diffusion can be used to describe the model of non-equilibrium sorption/diffusion. Therefore, based on the sorption phenomenon above, mathematical model of fractured wells in CBM reservoirs will be more complex than those in conventional reservoirs. Massive hydraulic fracturing (MHF) in coal seams is an effective technique for productivity enhancement of CBM wells. Therefore, evaluating coal seams properties of CBM fractured wells is an important task. During the past decades, there has been a continuously increasing interest in the determination of conventional formation properties from transient pressure test or flow rate data analysis (Mcguire and Sikora 1960; Prats 1961; Raghavan

4 356 L. Wang et al. et al. 1972). Gringarten et al. (1975) had made an extraordinary contribution to both the development of transient pressure analysis and type-curves analysis of fractured wells. In his work, three basic solutions were presented: the infinite fracture conductivity solution, the uniform flux solution for vertical fractures, and the uniform flux solution for horizontal fractures. Since fracture conductivity could not be ignored, a semi-analytical solution and 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 were quite significant to the later analysis of productivity and well test data for fractured wells (Tiab 1995; 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) developed a coal seam degasification model and gave application of this model. Anbarci and Ertekin (1991) presented a simplified analysis technique to determine the desorption characteristics of coal seams using pressure transient analysis. Anbarci and Ertekin (1990) developed some specialized solutions for pressure transient analysis with sorption phenomenon for singlephase gas flow in coal seams. Aminian and Ameri (2009) developed a numerical CBM model to predict production behavior of CBM reservoirs. Clarkson et al. (2009) made production data analysis of fractured and horizontal CBM wells. Nie et al. (2012) presented an analytic solution and modeled transient flow behavior of a horizontal CBM well in a coal seam. King et al. (1986) developed a numerical method for a finite or infinite conductivity vertical fracture in coal seams to simulate the transient behavior of coal seal degasification wells. Anbarci and Ertekin (1992) established a mathematical model of infinite conductivity vertical fracture in CBM reservoirs and discussed the pressure transient behavior using the model. Most of the above literatures are modeling the CBM transport characteristics of vertical well and horizontal wells (Sung et al. 1986; Anbarci and Ertekin 1990, 1991, 1992; Aminian and Ameri 2009; Clarkson et al. 2009; Nie et al. 2012), while few reports are related to CBM fractured wells (Anbarci and Ertekin 1992; Clarkson et al. 2009). At present, there is no literature about complete analytical or semi-analytical model for a finite conductivity CBM fractured well. Objectives of this paper are to establish a complete semi-analytical model for a finite conductivity CBM fractured well and discuss the transient pressure behavior for CBM fractured wells. There are several advantages in this semi-analytical model. First, the new model is in Laplace space, so it is unnecessary to scatter time, thereby reducing the amount of computation and improving the computational efficiency. Second, it is convenient to add wellbore storage effects into the constant rate solution. Third, it is generally known that the reports above about MHF technology are all based on the assumption that the well is at the center of the fracture. However, the actual fractures obtained from MHF are asymmetric to the 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 underlying impermeable strata (see Fig. 1). (2) The CBM reservoir has uniform thickness h, permeability k, and porosity φ, which do not change along with pressure. (3) The CBM production is assumed to be an isothermal process. Flow in coal matrix obeys Fick s law. Darcy flow is considered in the cleats (butt and face cleats) and hydraulic fracture(seefigs.2 and 3). We also assume that a linear flow occurs in the fracture and there is no fluid through two endpoints of the fracture (see Fig. 3).

5 Simulation of Pressure Transient Behavior 357 Fig. 1 A finite conductivity vertical fracture in a bounded slap reservoir Fig. 2 Darcy flow in the cleat and pseudo-steady diffusion in matrix Fig. 3 Darcy flow in the hydraulic fracture (4) The CBM reservoir contains a slightly compressible fluid with constant compressibility c 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 x f, width w, permeability k f, and porosity φ f. (6) The CBM well with asymmetric two wings produces in a condition of a constant-volume production (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.

6 358 L. Wang et al. Fig. 4 The CBM well with asymmetric two wings produces in a condition of a constant-volume production 3 Mathematical Model 3.1 CBM Fracture Model In this section, based on the assumptions above, a two-dimensional fracture flow model is established. The fracture is saturated with a single phase gas (CBM). The CBM well is produced under constant laminar flow rate condition. The equation describing two-dimensional fracture flow model is stated as: ( k f p f p f x x ) ( + k f p f p f y y ) + p sc ZTμq sc T sc Z sc wh δ(x x w) = 0 (1) where, x w 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 ( p f x, w ) ( 2, t = p f x, w ) 2, t = p(x, 0, t) (2) and k f p f y y= w 2 = k p y y=0 + = k p y y=0 = k f p f y y= w 2 (3) Taking the integral average of pressure across the fracture width (in the y direction), and combining Eqs. (2)and(3), Eq. (1) could be replaced by ( ) p p f f x + 1 ( 2k p p ) p sc ZTμq sc x w k f y y=0 k f T sc Z sc wh δ(x x w) = 0 (4) where p f (x, t) = 1 w w 2 p f (x, y, t)dy (5) w 2 p f is the integral average of pressure in the y direction. On the surface of the fracture, it must have the following flux relationship q f 2 = kht sc Z sc p sc TZμ p p (6) y Outer boundary conditions in the fracture can be given ( ) pf = 0 (7) x x= x ( ) f pf = 0 (8) x x=x f

7 Simulation of Pressure Transient Behavior 359 Now we introduce dimensionless groups given as θ = x w x f P fd = P2 ic P2 f Pic 2q D q fd = 2x fq f q sc x D = x x f y D = x x f βt μz q D = khpic 2 q sc C fd = k fw kx f Then flow equation in the fracture can be written in dimensionless as follows: Flux condition is 2 p fd x 2 D + 2 p D + 2π δ(x D θ) = 0 (9) C fd y D yd =0 C fd p D = π y D yd =0 2 q fd (10) Outer boundary conditions are ( pfd x D ) ( pfd x D x D = 1 ) x D =1 = 0 (11) = 0 (12) We give the Eqs. (9) (12) a Laplace domain solution by means of Green s functions and Laplace transform methods, which is written as p fd (x D, s) = [ p fd (s)] avg + π 1 C fd 1 N(x, x D ) q fd (x, s) 2π C fd s N(θ, x D) (13) In the Eq. (13), [ p fd (s)] avg is the average pressure for any time in the fracture, x is the integral variable, and N(x, x D ) is the second Green s function, which is a piecewise function and is defined as N(x, x D ) = 1 4 N(x, x D ) = 1 4 [ (x + 1) 2 + (x D 1) 2 4 ] 3 1 x < x D (14) [ (x 1) 2 + (x D + 1) 2 4 ] x D < x 1 (15) CBM Reservoir Model Anbarci and Ertekin (1990) had already obtained a series of line source solutions which described the pressure transient behavior in coal seams for various combinations of outer and inner 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 theory in approximating the equilibrium isotherm to describe the model. We obtained the solution of both unsteady state and pseudo-steady state sorption/diffusion models. Spherical system elements can be used in the unsteady state model (see Fig. 5a).

8 360 L. Wang et al. Fig. 5 Unsteady state and pseudo-steady state sorption/diffusion models: a spherical system elements. b Cube system elements Macropore transport equation in the cleat can be obtained by performing a mass balance on a radial elemental volume. It is similar to the one proposed by Anbarci and Ertekin (1990) 1 p 2 (r a ) = φμc g p 2 + 2p sct μz V (16) r a r a r a αk t αkt sc t Based on the description above, the unsteady state sorption equation for coal matrix can be written as V t = 3D R Substituting Eq. (17) into Eq. (16), we obtain C r i ri =R (17) 1 r a p 2 (r a ) = φμc g p 2 + 6p sct μzd C r a r a αk t αkt sc R r i ri =R (18) Initial condition is given as p(r a, 0) = p ic (19) Inner boundary condition is given as p βt μz r a (r w, t) rw 0 = r a 2kh q sc (20) Outer boundary conditions for infinite, constant and closed can be given separately as p(r a, t) = p ic (21) p(r e, t) = p ic (22) p(r e, t) = 0 r a (23) We introduce the dimensionless groups, which are defined as follows ω = φμc g p fd = p2 ic p2 f p 2 ic q D C D = C C ic r Da = r a x f τ = R2 D r Di = r i R q βt μz D = t D = αkt x 2 f khpic 2 q sc = φμc g + 6p sct μz T sc q D p 2 ic λ = αkτ x 2 f p Da = p2 ic p2 p 2 ic q D

9 Simulation of Pressure Transient Behavior 361 So, the macropore equations can be given in the dimensionless form as follows 1 p Da (r Da ) = ω p Da (1 ω) C D r Da r Da r Da t D λ r Di rdi =1 Initial condition is Inner boundary condition is (24) p Da (r Da, 0) = 0 (25) p Da r Da = 1 (26) r Da rda 0 Outer boundary conditions are given separately as follows p Da (r Da, t) = 0 (27) p Da (r Da = r ed, t) = 0 (28) p Da (r ed, t) = 0 r Da (29) Using Fick s law of diffusion, the micropore diffusivity equation describing the transport of CBM in the coal matrix can be written as ( 1 ri 2 ri 2 r ) = C i r i t (30) Initial condition in the coal matrix elements is given as: C(r i, t = 0) = C ic (31) Using the existing symmetry condition, center of the element could be treated as a no flow boundary: C (r i = 0, t) = 0 (32) r i Concentration of the CBM on the external surface of the matrix elements is estimated at the CBM pressure in the cleats: Letting C(r i = R, t) = C p(ra,t) (33) C D = C C ic r Di = r i R = φμc g = 6p sct μz T sc q D p 2 ic τ = R2 D r Da = r a x f λ = αkτ x 2 f Then the micropore equation is written as ( ) 1 C D r Di r Di r Di r Di Initial condition is Inner condition becomes t D = αkt x 2 f = λ C D t D (34) C D (r Di, t D ) = 0 (35) C D (r Di = 0, t D ) r Di = 0 (36)

10 362 L. Wang et al. Table 1 Coefficients for different outer boundary conditions Outer boundary Coefficient A Coefficient B Infinite 1 0 Constant pressure 1 K 0 (r ed s 1/2 )/I 0 (r ed s 1/2 ) Closed 1 K 1 (r ed s 1/2 )/I 1 (r ed s 1/2 ) Outer condition becomes C D (r Di = 1, t D ) = C D pda (r Da,t D ) (37) By employing the Lapalace transform, the solution of Eqs. (34) (37) can be obtained C D rdi rdi=1 = p Da( λs coth λs 1) (38) Now substituting the Eq. (38) into Lapalace transform equation of Eq. (24) and combining Eqs. (25) (29), we can get the point source solution where p Da = AK 0 {r Da Ɣ}+BI 0 {r Da Ɣ} (39) Ɣ = ws + (1 w) ( λs coth λs 1) (40) λ If we would like to gain PSS model (see Fig. 5b), Eq. (28) would be only replaced by Ɣ = ws + (1 w) λ s s + 1/λ Through integral of Eq.(39), uniform solution of the fracture can be obtained p fd (x D, s) = (41) q fd (x, s)[ak 0 {[(x D x ) 2 ] 1/2 Ɣ}+BI 0 {[(x D x ) 2 ] 1/2 Ɣ}]dx (42) In Eqs. (39) and(42), the coefficients A and B are listed in Table 1 for different boundary combinations. Substituting the Eqs. (42) into Eq. (13) will yield π q fd (x, s)[ak 0 {[(x D x ) 2 ] 1/2 Ɣ}+BI 0 {[(x D x ) 2 ] 1/2 Ɣ}]dx = [ p fd (s)] avg 1 C fd 1 N(x, x D ) q fd (x, s) 2π C fd s N(θ, x D) (43) Eq. (43) is a new semi-analytical model for an asymmetric fracture in CBM reservoirs, and we need to discrete the Eq. (43) to get its solution.

11 Simulation of Pressure Transient Behavior 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, q fd (x, s)[ak 0 {[(x D x ) 2 ] 1/2 Ɣ}+BI 0 {[(x D x ) 2 ] 1/2 Ɣ}]dx = 1 2 A 2N i=n B 2N 1 i=n i=n+1 q fdi q fdi q fdi xdi+1 xdi x Di+1 x Di K 0 [ x Di+1 xdi xd j x Ɣ]dx A I 0 [ xd j x Ɣ]dx B 1 i=n q fdi x Di+1 x Di K 0 [ xd j + x Ɣ]dx I 0 [ x D j + x Ɣ]dx (44) Integral of the right side in the Eq. (43) would be transformed as, π 1 C fd 1 N(x, x D j ) q fd (x, s)dx = π 2N q fdi C fd i=1 x Di+1 x Di N(x, x D j )dx (45) where x D j is the midpoint of the j segment. With the descriptions above about steady flow, we can know 1 2N 2 x q fdi (s) = 1 (46) s i=1 The unknowns q fdi (s) and [ p fd (s)] avg can be obtained from Eqs. (44) (46). Then take q fdi (s) and [ p fd (s)] avg back into Eq. (42)andmakex D = θ to get the bottom-hole pressure solution in Laplace domains. Then, by Stehfest numerical algorithm, p fd (θ, s) q fd (t D ) and p fd (t D ) can be figured out for any given t D. To obtain the solution including wellbore storage and skin effect, we need the relationship given as 1 p wd = s 2 (47) C D + s/[s p fd (θ, s) + S k ] If substituting the solved p fd (θ, s) into Eq. (47), we can obtain the solution including wellbore storage 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) were selected, as listed in Table 2. In our model, C fd = 1, 000 and θ = 0 were proposed considering the neglect of fracture conductivity and fracture asymmetry in Anbarci s model.

12 364 L. Wang et al. Table 2 Basic data used in this paper τ = h x f = 100 ft φ = 0.01 μ = cp c t = psi 1 K = 26 md T = 530 R Z = h = 6ft q sc = 0.2 MMscf/d V L = scf/ft 3 p L = psi p ic = 447.7psi Fig. 6 Comparison of our results against the results in the literature As shown in Fig. 6, there is a better agreement between the solution obtained in this work and the results of numerical simulation. Anbarci s results are relatively lower than that from numerical simulation. It is also verified that Green s Function method which is used to solve the 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 under the same condition presented in Table 2. Figure 7a shows the effects of the fracture conductivity C fd of 0.5, 5, and 50 on CBM pressure. 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 C fd on pressure depend upon C fd values. The fracture conductivity C fd < 5 shows stronger influences than those when C fd > 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 larger the porosity is, the higher the bottomhole pressure will be. Moreover, a smaller porosity value (ϕ <0.1) will lead to a lower location of the pressure curve, which means the pressure depletion will be faster for a smaller porosity CBM reservoir. As shown in Fig. 7c, from the effects of fracture asymmetry factor θ ranging from 0 to 1 on bottomhole pressure it is found that the bottomhole pressure changes slightly with the θ<0.5, while overall lower values for the CBM well will appear with the θ>0.5, which means off-centered well in the fracture will lead a bigger pressure depletion. Therefore, well location in the fracture is a significant factor to affect bottomhole pressure. Fig. 7d shows the effects of CBM reservoir permeability k on bottomhole pressure. It can be seen that, at the same time point, the smaller the permeability

13 Simulation of Pressure Transient Behavior 365 Fig. 7 The effects of parameters on bottomhole pressure behavior were analyzed in detail: a the effect of the fracture conductivity C fd on CBM pressure. b The effects of the different porosity φ values on bottomhole pressure. c The effects of fracture asymmetry factor θ ranging from 0 to 1 on bottomhole pressure. d The effects of CBM reservoir permeability k on bottomhole pressure. e The effects of sorption time constant τ on bottomhole pressure. f The effects of fracture half length x f on bottomhole pressure. g The effect of CBM viscosity μ on bottomhole pressure. h The effects of Langmuir volume V L on bottomhole pressure. i The effects of total compressibility c t on bottomhole pressure

14 366 L. Wang et al. Fig. 8 Type curves used for the analysis of transient pressure behavior effected by external boundary conditions (r ed = 5). Stage 1: wellbore storage effect. Stage 2: wellbore storage transition region. Stage 3: linear flow region. Stage 4: diffusion/desorption region. Stage 5: pseudo-radial region. Stage 6: boundary dominated region k is, the lower the bottomhole pressure will be. Moreover, the pressure depletion with smaller permeability is much larger than the one with higher permeability values. According to the definition of the sorption time constant above, the sorption time constant τ is negatively related 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 between different pressure curves are remarkable. Fig. 7f shows the effects of fracture half-length x f on bottomhole pressure p. At the same time point, the smaller the fracture x f is, the lower the bottomhole pressure will be. It follows that a shorter fracture will lead to a bigger pressure depletion. Moreover, the slopes of the pressure curves of CBM wells with different fracture half-length vary slightly, which implies that the rate of the pressure drop at different x f values is approximate. CBM viscosity μ is referred to fluid property instead of fracture and CBM reservoir. From its effects on bottomhole pressure p, as shown in Fig. 7g, it was found that the bigger μ is, the lower the bottomhole pressure will be. Moreover, the bottomhole pressure of CBM well with a high viscosity value will reduce rapidly with depletion. The Langmuir volume V L mainly affects the change of bottomhole pressure with the CBM production after 1 day(fig. 7h). The lower bottomhole pressure occurs in CBM well in the coal reservoir with asmallv L. Usually, a weaker sorption capacity will cause rapid pressure depletion. Fig. 7i shows the effects of total compressibility c t on bottomhole pressure p. At the same time point, the smaller c t is, the lower the bottomhole pressure will be. It follows that a smaller total compressibility will lead to a bigger pressure depletion. However, as the result of time increasing, pressure values for different c t values are almost same, which implies that the effect of c t 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 be used to analyze transient pressure and rate decline so as to recognize the flow characteristics of fluids in CBM reservoir. In addition, by Type curves matching, some reservoir property parameters, such as permeability, skin factor, gas in place, fracture half-length, and gas reservoir drainage area, can be obtained (Wang et al. 2012; Nie et al. 2012). Figures 8, 9,and 10 show that the Type curves of pressure and derivative pressure analysis for asymmetrically fractured CBM well model reflect transient and pseudo steady state in coalbed.

15 Simulation of Pressure Transient Behavior 367 Fig. 9 Comparison of transient steady state TSS and pseudo steady state PSS models Fig. 10 The effect of parameters on the shape of type curves: a the effect of transfer coefficient λ on the shape of type curves. b The effect of wellbore storage coefficient C D on the shape of type curves. c The effect of cleat storage coefficient ω on the shape of type curves. d The effect of fracture conductivity C fd on the shape of type curves. e The effect of fracture asymmetry factor θ on the shape of type curves. f The effect of rate constant on the shape of type curves

16 368 L. Wang et al. The Type curves which are used for the analysis of transient pressure behavior affected by external boundary conditions are shown in Fig. 8. The used data were listed in Table 3.It can 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 wellbore storage effect. Stage 2: It can be seen that the curve in this stage is upward bending, for the reason of wellbore storage effects. Therefore, this region can be called as wellbore storage transition region. Stage 3: Linear flow in CBM reservoirs is observed in this stage. Both dimensionless pseudo pressure curve and its derivative curve show two parallel straight lines with the slope 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 very short while wellbore storage effect is dominated for a long time, it is hard to find out the bi-linear flow region. In this region, the flow pathway is mainly dominated by the natural cleat 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 diffusion/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 dimensionless pseudo pressure derivative curve has a V-like shape. The shape, location, and size of the type curves in this region are controlled by the wellbore storage C D, transfer coefficient λ, cleat storage coefficient ω, rate constant, fracture conductivity C fd,and fracture asymmetry factor θ(see Fig. 10). Stage 5: The diffusion rate in matrix is equal to the flow rate from the cleat to wellbore, so fluid flow in this stage reaches a dynamic balance state. It can be seen from the plot that the dimensionless pseudo pressure derivative curve nearly shows a zero slope straight line. So, this stage is named as pseudo-radial region. Stage 6: There is no difference in the shape of the curve under the different boundary conditions (from Stage 1 to Stage 5); however, we can see simply that difference when flow reaches pseudo-steady state. For infinite-acting coalbeds, the dimensionless pressure curve slowly rises with the increase in dimensionless time, but the dimensionless pressure derivative curve shows a zero slope straight line. For constant pressure case, the dimensionless pressure curve shows a zero slope straight line, but the dimensionless pressure derivative curve goes down rapidly. For closed boundary case, both the dimensionless pressure curve and its derivative curve show a unit slope straight line and go up rapidly. 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 same parameters listed in Table 3. Two models perform some differences for the same parameters. For wellbore storage effect region, these dimensionless pseudo pressure and pseudo pressure derivative curves of two models exhibit good agreement with a unit slope straight line. However, for linear flow region, dimensionless pseudo pressure and pseudo pressure derivative curves in PSS model are higher than those in TSS model, which means pressure depletion in PSS model is much bigger than that in TSS model. It is the result of the geometry of two models. For PSS model, we can simply see a diffusion region, which is not observed in TSS model still showing a 1/2 slope straight line. For pseudo radial flow region, dimensionless pseudo pressure and pseudo pressure derivative curves in PSS model are lower than those

17 Simulation of Pressure Transient Behavior 369 Table 3 Basic data used for type curves analysis C D = 10 5 ω = λ = θ = 0 C fd = 1 = 1 in TSS model, which means pressure depletion in PSS model is much smaller than that in TSS model. This is because, in a short time, CBM diffuse from matrix to cleat, which supplementary pressure depletion in PSS model. The difference between TSS and PSS models is that spherical system elements (Fig. 5a) canbeassumed inthe TSSmodel; however, cubic elements (Fig. 5b) are assumed in PSS model. Using TSS model, we obtain the Eq. (40)and using PSS model we canget the Eq. (41). Equations (40) and(41) also reflect the difference between TSS and PSS models. Therefore, we can see that difference in Fig Parameters Influence on Shape of Type Curves Transfer coefficient λ is a function defined in the above sections, which is positively proportional to permeability K and sorption time constant τ but is negatively proportional to the square of fracture half-length. According to the definition of the sorption time constant in 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 the shape of type curves for the same parameters listed in Table 3. A smaller λ value leads to the early time of diffusion from matrix to cleat, which also means a bigger diffusion coefficient will cause the early time of diffusion from matrix to cleat. Therefore, λ reflects the diffusion ability in matrix and starting time of gas transfer from matrix to cleat. Figure 10b shows the effect of wellbore storage coefficient C D on the shape of type curves for the same parameters listed in Table 3. We can see from the curves that in the early time, a bigger C D will lead to a bigger pressure depletion, which will make the time of linear flow shorter, 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 is negatively proportional to the porosity ϕ. Therefore, ω reflects storage ability of the CBM in the cleat and matrix. A bigger ω means a stronger storage ability. Figure 10c shows the effect of cleat storage coefficient ω on the shape of type curves for the same parameters listed in Table 3. We can see from the curves that in the early time, a bigger ω will lead to a smaller pressure depletion, which will make the time of linear flow longer, and prolong the starting time 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 in the late time, their pressure depletions are same. Figure 10d shows the effect of fracture conductivity C fd on the shape of type curves for the same parameters listed in Table 3. In the entire flow period, a smaller C fd will lead to bigger pressure depletion as expected, which indicates that fracture conductivity is important to the effect on production. In the hydraulic fracturing design, it is necessary to keep bigger fracture conductivity to enhance well production and reduce pressure depletion. Figure 10e shows the effect of fracture asymmetry factor θ on the shape of type curves for the same parameters listed in Table 3. In the entire flow period, a bigger θ will lead to a bigger pressure depletion as expected, which indicates that fracture asymmetry factor is also important to the effect on production. In the hydraulic fracturing design, in order to enhance well production and reduce pressure depletion, we should keep the fracture symmetry.

18 370 L. Wang et al. According to the definition of the rate coefficient in the above sections, is a function related to the production q sc and is proportional to q sc. Therefore, reflects the productivity of CBM. A bigger q sc means a stronger productivity. Figure 10f shows the effect of rate constant on the shape of type curves for the same parameters listed in Table 3. Inthe entire flow period, a bigger will lead to a smaller pressure depletion as expected but the starting time of diffusion from matrix to cleat occurs early, which indicates that a bigger well production needs a smaller depletion. In a word, to enhance well production and reduce pressure depletion, each parameter of CBM well in the practical production process should be coordinated appropriately. 5 Conclusions (1) We have successfully constructed comprehensive fracture flow model, formation flow model for asymmetrically fractured wells centered in a constant pressure boundary, a closed boundary and an infinite boundary, circular CBM reservoir. (2) Fracture asymmetry factor and fracture conductivity are both considered in new semianalytical model which is different from previous models. The new model is more accurate than that previous model Anbarci presented because new model is gained in the form of Bessel integral expressions and is validated well with numerical simulation. (3) Effects of parameters on bottomhole pressure behavior are discussed in details including fracture conductivity C fd, the CBM reservoir porosity ϕ, fracture asymmetry factor θ, CBM reservoir permeability k, sorption time constant τ, fracture half-length x f,and CBM viscosity μ. (4) Type curves are established to make transient pressure analysis and recognize the flow characteristics for a real CBM reservoir. Flow for CBM fractured wells is divided into six stages according to characteristics of dimensionless pseudo pressure derivative curve: wellbore storage effects region showing a unit slope straight line, wellbore storage transition region, linear flow region showing a 1/2 slope straight line, diffusion/desorption region showing V-shaped curve, pseudo-radial region approximately showing zero slope straight 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 model still showing a 1/2 slope straight line. (6) Effects of parameters on shape of type curves are discussed in details including Transfer coefficient λ, wellbore storage coefficient C D, storage coefficient of cleat ω, fracture conductivity C fd, fracture asymmetry factor θ, and rate coefficient. In the hydraulic fracturing design, to enhance well production and reduce pressure depletion, it is necessary for us to keep bigger fracture conductivity and fracture symmetry. Acknowledgments This article was supported by Important National Science and Technology Specific Projects of the twelfth five Years Plan Period (Grant No.2011ZX ) and the National Basic Research Program of China (Grant No.2011ZX ). 6 Appendix A α = β =

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