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A novel semi-analytical model for finite-conductivity multiple fractured horizontal wells in shale gas reservoirs
Journal of Natural Gas Science and Engineering 24 (2015) 35e51
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
A novel semi-analytical model for finite-conductivity multiple fractured horizontal wells in shale gas reservoirs Junjie Ren*, Ping Guo State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, Sichuan, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 December 2014 Received in revised form 7 March 2015 Accepted 9 March 2015 Available online
The use of multiple fractured horizontal wells (MFHWs) is considered an effective means for developing shale gas reservoirs, and a variety of models have recently been introduced to study the pressure behaviors of MFHWs in shale gas reservoirs. However, most of the existing models are based on some ideal assumptions that may not always be true in practice. This paper presents a novel semi-analytical model for finite-conductivity MFHWs in shale gas reservoirs with consideration of more actual conditions, such as finite-conductivity fractures, different fracture lengths, different fracture intervals, various angles between the fractures and the horizontal well, fracture asymmetry about the horizontal well, and partially penetrating fractures. Laplace transform, source function, numerical discrete method, and Gaussian elimination method are applied to solve the present model. The solution is presented in Laplace space so that the effects of the wellbore storage and skin can be easily incorporated by Duhamel's principle. Type curves of the pressure responses are plotted, possible flow regimes are identified, and a detailed analysis of the pressure characteristics is presented. The present model is much closer to the conditions of an actual reservoir, and thus it can be applied to an accurate interpretation of the pressure data of an MFHW in a shale gas reservoir. © 2015 Elsevier B.V. All rights reserved.
Keywords: Shale gas Multiple fractured horizontal wells Finite-conductivity fractures Desorption Diffusion Mathematical model
1. Introduction Shale gas reservoirs have attracted significant attention in recent years because they show great potential to offset the reduction of conventional gas production. In general, shale gas reservoirs mainly include organic matrix and inorganic natural fracture system. The nanoscale pores of organic matrix have a sufficient surface area, and thus shale gas can be stored by adsorption on the surfaces of these nanoscale pores. Hill and Nelson (2000) have pointed out that about 20e85% of the gas reserves in shale gas reservoirs are likely to come from adsorbed gas. Because the adsorption/desorption processes occur during gas production and affect the pressure and production performances of the gas wells, it is necessary to consider the effects of adsorption and desorption for simulating gas flows in shale gas reservoirs. Furthermore, owing to the very low permeability, there is usually no natural production capacity in shale gas reservoirs, and thus to obtain the economic production capacity, multiple fractured
* Corresponding author. E-mail addresses: [email protected] (J. Ren), [email protected] (P. Guo). http://dx.doi.org/10.1016/j.jngse.2015.03.015 1875-5100/© 2015 Elsevier B.V. All rights reserved.
horizontal wells (MFHWs) have been widely used for the development of shale gas reservoirs. Recently, interest in the development of models for studying the pressure behaviors in such reservoirs has grown strikingly. By incorporating gas slippage and desorption into their model, Kucuk and Sawyer (1980) studied the pressure behaviors of a vertical well in a shale gas reservoir. To study the pressure responses of a vertical well in a shale gas reservoir, Bumb and McKee (1988) presented an interesting model in which the Langmuir isotherm was introduced to describe the desorption processes. Ozkan et al. (2010) studied the pressure behaviors of a vertical well in a shale gas reservoir by introducing the idea of dual porosity. However, they ignored the effect of desorption in their model, which may not reflect the pressure characteristics under actual conditions. Guo et al. (2012) established a well test model of a horizontal well with multiple infinite-conductivity hydraulic fractures in a shale gas reservoir, which can consider the effects of diffusion and desorption on the pressure behaviors. However, the hydraulic fractures in their model were assumed to have infinite conductivity, to be perpendicular to the horizontal well, and to penetrate the formation completely, which may not always be the case under actual conditions. Zhao et al. (2013) established a tri-porosity model to study the
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J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Nomenclature
Variables C wellbore storage coefficient, m3/Pa CD dimensionless wellbore storage coefficient, dimensionless cg gas compressibility, Pa1 CfD dimensionless fracture conductivity coefficient, dimensionless D diffusion factor, m2/s h reservoir thickness, m hwi height of the ith fracture, m hwDi dimensionless height of the ith fracture, dimensionless kf permeability of hydraulic fracture, m2 kh horizontal permeability of natural fracture system, m2 kv vertical permeability of natural fracture system, m2 K0(x) second-kind modified Bessel function, zero order LfLi length of the left wing of the ith fracture, m LfLDi dimensionless length of the left wing of the ith fracture, dimensionless DLfLDi dimensionless length of discrete segment in the left wing of the ith fracture, dimensionless LfRi length of the right wing of the ith fracture, m LfRDi dimensionless length of the right wing of the ith fracture, dimensionless DLfRDi dimensionless length of discrete segment in the right wing of the ith fracture, dimensionless m number of hydraulic fractures, integer p pressure of natural fracture system, Pa p0 reference pressure, Pa pf pressure of hydraulic fracture, Pa psc pressure at standard condition, Pa q flow rate from point source, m3/s qD dimensionless flow rate from point source, dimensionless qf flow rate per unit length, m2/s qfD dimensionless flow rate per unit dimensionless length, dimensionless qfDi,j dimensionless flow rate per unit dimensionless length of the jth segment in the ith fracture, dimensionless qfi,j flow rate per unit length of the jth segment in the ith fracture, m2/s QLi production rate from the left wing of the ith fracture, m3/s QDLi dimensionless production rate from the left wing of the ith fracture, dimensionless QRi production rate from the right wing of the ith fracture, m3/s QDRi dimensionless production rate from the right wing of the ith fracture, dimensionless Qsc total production rate conditions, m3/s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisurface ffi punder r radial distance, r ¼ x2 þ y2 , m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rD dimensionless radial distance, rD ¼ x2D þ y2D , dimensionless R radius of shale matrix block, m s Laplace transform variable, dimensionless Sf skin factor, dimensionless t time, s tD dimensionless time, dimensionless T temperature of shale gas reservoir, K
Tsc V VD VE VED
temperature at standard condition, K gas concentration, m3/m3 dimensionless gas concentration, dimensionless gas concentration at equilibrium, m3/m3 dimensionless gas concentration at equilibrium, dimensionless Vi initial gas concentration, m3/m3 VL Langmuir volume, m3/m3 wf width of hydraulic fracture, m x,y,z space coordinates in x,y,z Cartesian coordinate system, m xD,yD,zD dimensionless space coordinates in xD,yD,zD Cartesian coordinate system, dimensionless xDi,yDi dimensionless space coordinates in xDi,yDi Cartesian coordinate system, dimensionless xi,yi space coordinates in xi,yi Cartesian coordinate system, m xwi,ywi,zwi space coordinates of the intersection of the ith fracture and the horizontal wellbore, m Dyi difference between ywi and ywi1, Dyi ¼ ywi ywi1, m xi,j,yi,j space coordinates of the jth end point in the ith fracture, m xDi,j,yDi,j dimensionless space coordinates of the jth end point in the ith fracture, dimensionless xmi,j,ymi,j space coordinates of the center of the jth segment in the ith fracture, m xmDi,j,ymDi,j dimensionless space coordinates of the center of the jth segment in the ith fracture, dimensionless xwDi,ywDi,zwDi dimensionless space coordinates of the intersection of the ith fracture and the horizontal wellbore, dimensionless Z Z-factor of shale gas, dimensionless m gas viscosity, Pa s f porosity, fraction b anisotropy coefficient, dimensionless u storativity ratio, dimensionless l inter-porosity flow coefficient, dimensionless s adsorption index, dimensionless qi angle between the ith fracture and the horizontal well, degree j pseudo-pressure of natural fracture system, Pa/s jD dimensionless pseudo-pressure of natural fracture system, dimensionless jf pseudo-pressure of hydraulic fracture, Pa/s jfD dimensionless pseudo-pressure of hydraulic fracture, dimensionless ji initial pseudo-pressure, Pa/s jL Langmuir pseudo-pressure, Pa/s Djs additional pseudo-pressure drop, Pa/s jwD dimensionless pseudo-pressure in horizontal wellbore considering wellbore storage and skin factor, dimensionless jwDH dimensionless pseudo-pressure in horizontal wellbore without considering wellbore storage and skin factor, dimensionless Superscript Laplace space Subscript D dimensionless
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Fig. 1. The schematic illustration of a physical model for a shale gas reservoir with an MFHW.
pressure behaviors of a multiple fractured horizontal well (MFHW) in a shale gas reservoir; however, their model also does not consider the effects of finite-conductivity fractures, partially penetrating fractures, and various angles between the fractures and the horizontal well. Wang (2014) presented a model for an MFHW in a shale gas reservoir with the effects of desorption, diffusion, viscous flow and stress-sensitivity permeability. However, the hydraulic fractures in this model were also assumed to have infinite conductivity and a complete penetration, which may not be always true in practice. Models for MFHWs with finite-conductivity hydraulic fractures in shale gas reservoirs have recently attracted significant attention, and some studies on these models have been conducted. However, most of them, if not all, are only based on the linear flow assumption (Brohi et al., 2011; Ozkan et al., 2011; Xu et al., 2013). Although these models can successfully consider the effect of fracture conductivity, they can only calculate the pressure behaviors in certain regimes (e.g., linear flow regime), and cannot reflect complete flow regimes, such as pseudo-radial flow regime and interference between fractures. In addition, these models must rely on the assumption of linear flow in shale gas reservoirs, and thus they cannot be applied to MFHWs under actual conditions, such as different fracture lengths, different fracture intervals, various angles between the fractures and the horizontal well, and partially penetrating fractures. That is because, under these conditions, the assumption of linear flow in a shale gas reservoir is invalid. Recently, Guo et al. (2014) proposed an interesting model for MFHWs with finite-conductivity fractures in an infinitely large shale gas reservoir. However, the hydraulic
37
Fig. 3. Profile view of the schematic physical model in the x z plane.
fractures considered in their paper were also perpendicular to the horizontal well and penetrated the formation completely, which may not always be true in reality. Most recently, Liu et al. (2015) presented a model for MFHWs in a shale gas reservoir that considers the effects of desorption, stress sensitivity, and fracture conductivity. However, the simulation results presented by Liu et al. (2015) show that the linear flow regime occurs before the bilinear flow regime, which is in disagreement with previous studies (Chen and Raghavan, 1997; Guo et al., 2014; Wang et al., 2014). By carefully investigating their model, we found that, even for the same hydraulic fracture, the flow rate from the reservoir into the hydraulic fracture is also assumed to be equivalent and independent of the coordinates, which may be unreasonable and lead to inaccurate simulation results. This paper proposes a novel semi-analytical model for finiteconductivity MFHWs in shale gas reservoirs which can consider more actual conditions, such as finite-conductivity fractures, different fracture lengths, different fracture intervals, various angles between the fractures and the horizontal well, fracture asymmetry about the horizontal well, and partially penetrating fractures. The present semi-analytical model is much closer to the conditions of an actual reservoir and can be applied to accurately interpret the pressure data of an MFHW in practice. 2. Physical model Fig. 1 shows an MFHW with partially (or completely) penetrating fractures located in a shale gas reservoir. The shale gas
Fig. 2. Top view of the schematic physical model in the x y plane.
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J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
reservoir considered here is infinitely large in the horizontal direction, and is bounded by the top and bottom parallel impermeable planes in the vertical direction. Fig. 2 shows a top view of Fig. 1 in the x y plane, and Fig. 3 shows a profile view of Fig. 1 in the x z plane. Some other assumptions of the present mathematical model are given as follows: (1) Shale gas reservoirs are dual-porosity medium including organic matrix and inorganic natural fracture system. Gas flow in natural fracture system follows Darcy's law, desorption in matrix can be described by Langmuir isotherm, and diffusion in matrix follows Fick's first law. (2) The shale gas reservoir is anisotropic (i.e., kh s kv), and the initial pressure in the shale gas reservoir is uniform. (3) Hydraulic fractures (where m is the number of hydraulic fractures) may diagonally intersect with the horizontal well, asymmetrically distribute about the horizontal well, and partially penetrate the formation. The hydraulic fractures are assumed to have finite conductivity. Gas flow within the hydraulic fractures is considered to be linear flow, and the compressibility of the hydraulic fractures can be neglected. (4) The total production rate of the MFHW is assumed to be constant with a value of Qsc, whereas the flow rate of each hydraulic fracture is distinct, and the flow rate in the same hydraulic fracture is a function of the coordinates. (5) The effects of gravity and capillary force are negligible.
by the Langmuir isotherm, we can get that
VE ¼
VL j ; jL þ j
(7)
Vi ¼
VL ji : jL þ ji
(8)
With the aid of Eqs. (7) and (8), we can obtain
VED ¼ VE Vi ¼ sjD ;
(9)
where s is the adsorption index defined in Table 1. It is difficult to directly obtain a semi-analytical model for an MFHW because of the complexity of the inner boundary condition. In general, researching the point source model is always the first step in studying a semi-analytical model for an MFHW. The inner boundary conditions for a point source with flow rate q(r ¼ 0,z ¼ zw,t) can be expressed by:
3
2
8 7 > psc qðr ¼ 0; z ¼ zw ; tÞT 6 Z < 6 ; jz zw j ε=2 vj 7 7 6 pkh hTsc lim 6 lim r dz7 ¼ : vr 7 > ε/06 r/0 : 5 4 ε zw 2 0; jz zw j > ε=2 zw þ2ε
(10) The outer boundary conditions for a shale gas reservoir that is infinitely large in the horizontal direction and bounded by impermeable planes in the vertical direction can be expressed as
3. Mathematical model 3.1. Reservoir model The hydraulic fractures of the MFHW may be penetrated partially in a shale gas reservoir, and thus the gas flow in this reservoir can be described by the following equation:
fcg p vp psc T vV 1 v p vp v p vp rkh þ kv ¼ þ : r vr mZ vr vz mZ vz Tsc vt Z vt
(1)
The definition of pseudo-pressure can be given as follows:
Zp jðpÞ ¼ p0
2p dp: mZ
jðr/∞; z; tÞ ¼ ji ;
Table 1 Dimensionless definitions of variables. Dimensionless pseudo-pressure Dimensionless time
hTsc ðji jÞ sc ðji jf Þ jD ¼ pkh Q ; jfD ¼ pkh hT Qsc psc T sc psc T
Dimensionless distance
r x y z x rD ¼ ; xD ¼ ; yD ¼ ; zD ¼ ; xDi ¼ i ; h h h h h y x y z yDi ¼ i ; xwDi ¼ wi ; ywDi ¼ wi ; zwDi ¼ wi ; h h h h L L h LfLDi ¼ fLi ; LfRDi ¼ fRi ; hwDi ¼ wi h h h
(2)
According to the definition of pseudo-pressure, Eq. (1) can be rewritten as follows:
1 v vj v vj vj 2psc T vV rkh þ kv ¼ fmcg þ : r vr vr vz vz vt Tsc vt
Storativity ratio
(3)
Based on Fick's first law, the diffusive flow in organic matrix can be expressed as
vV 6p2 D ¼ ðVE VÞ: vt R2
(4)
According to the dimensionless variables defined in Table 1, Eqs. (3) and (4) can be expressed in a dimensionless form as:
1 v vj 1 v 2 jD vj vV rD D þ 2 ¼ u D ð1 uÞ D ; rD vrD vrD vtD vtD b vz2D vVD 1 ¼ ðVED VD Þ: l vtD
(5)
(6)
Because the adsorption/desorption processes can be described
(11)
kh t tD ¼ Lh 2
u¼
fmcg L
hh where L ¼ fmcg þ 2pk Qsc
hh where L ¼ fmcg þ 2pk Qsc
R2 hh where L ¼ fmcg þ 2pk Qsc ; t ¼ 6p2 D
Inter-porosity flow coefficient
kh t l ¼ Lh 2
Adsorption index
psc Qsc T s ¼ pk hT
sc
h
VL jL ðjL þjÞðjL þji Þ
Concentration difference Anisotropy coefficient
VD ¼ V Vi ; VED ¼ VE Vi
Dimensionless flow rate from point source Dimensionless flow rate per unit length Dimensionless production rate from left and right wings of each fracture Dimensionless fracture conductivity coefficient Dimensionless wellbore storage coefficient Skin factor
qD ðrD ; zD ; tD Þ ¼ qðr;z;tÞ Qsc
b¼
qffiffiffiffi kh kv
qfD ðxD ; yD ; tD Þ ¼ hqf Qðx;y;tÞ sc QDLi ðtD Þ ¼ QQLiscðtÞ; QDRi ðtD Þ ¼ QQRiscðtÞ CfD ¼ kkf whf h
CD ¼
mC 2pLh3
hh where L ¼ fmcg þ 2pk Qsc
h hTsc Sf ¼ pk psc Qsc T Djs
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
vj vj ¼ ¼ 0: vz z¼0 vz z¼h
39
(12)
Owing to the uniform pressure at the initial time, the initial condition is:
jðr; z; t ¼ 0Þ ¼ ji :
(13)
According to the definitions in Table 1, the dimensionless forms of Eqs. (10)e(13) can be given as follows:
2
3
6 7 Z 6 7 vj 6 7 lim 6 lim rD D dzD 7 7 vrD εD /06 rD /0 4 5 ε z D ε zwD þ 2D
wD
¼
Fig. 4. Schematic of the relationship between different coordinate systems.
(14)
2
qD ðrD ¼ 0; zD ¼ zwD ; tD Þ; 0;
jD ðrD /∞; zD ; tD Þ ¼ 0; vjD vz
D zD ¼0
vj ¼ D vz
If the point source is located in (xwi,ywi,zwi), Eq. (18) can be rewritten as:
jzD zwD j εD =2 ; jzD zwD j > εD =2
"
jD ¼ qD ðxwDi ; ywDi ; zwDi ; sÞ K0 ðε0 rD Þ þ 2 (15)
n¼∞ X
# cosðnpzD ÞcosðnpzwDi ÞK0 ðεn rD Þ ;
D zD ¼1
¼ 0;
(16) where
jD ðrD ; zD ; tD ¼ 0Þ ¼ 0:
(17)
The seepage model for shale gas reservoirs consists of Eqs. (5), (6), (9), (14)e(17). Applying Fourier and Laplace transforms, the point source solution in Laplace space can be obtained as follows (Ozkan, 1988):
rD ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxD xwDi Þ2 þ ðyD ywDi Þ2 :
" jD ¼ qD ð0; zwD ; sÞ K0 ðε0 rD Þ þ 2
# cosðnpzD ÞcosðnpzwD ÞK0 ðεn rD Þ ;
(18)
n¼1
(22)
Based on the source function and principle of superposition in Laplace space (Ozkan, 1988), the dimensionless pseudo-pressure responses caused by a vertical line source solution can be expressed as
8 9 hwDi > > > > sin np zwDi hwDi n¼∞ < = 2 X sin np zwDi þ 2 jD ¼ qfD ðxwDi ; ywDi ; sÞ$ K0 ðε0 rD Þ þ 2 cosðnpzD ÞK0 ðεn rD Þ : > > nphwDi > > n¼1 : ;
n¼∞ X
(21)
n¼1
(23)
It should be noted that, as shown in Eq. (23), the dimensionless pseudo-pressure is a function of the dimensionless variable zD. Considering that the linear flow occurs along the hydraulic fracture direction, we can ignore the zD-axis directional pseudo-pressure drop within the hydraulic fracture. Therefore, the zD-axis directional average pseudo-pressure within the hydraulic fracture can be used to substitute the pseudo-pressure within the hydraulic fracture as follows:
where
εn ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 p2 f ðsÞ þ 2 ; b
f ðsÞ ¼ us þ
(19)
sð1 uÞs : 1 þ ls
(20)
jD ðrD ; sÞ ¼
1 hwDi
hwDi 2
zwDi Zþ
jD ðrD ; zD ; sÞdzD :
According to Eq. (24), Eq. (23) can be rewritten as follows:
32 2 8 9 hwDi sin np zwDi hwDi n¼∞ < = 2 X 6sin np zwDi þ 2 7 6 7 K0 ðεn rD Þ jD ¼ qfD ðxwDi ; ywDi ; sÞ$ K0 ðε0 rD Þ þ 2 4 5 : ; nphwDi n¼1
(24)
h zwDi wDi 2
(25)
40
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Following the source function and principle of superposition in Laplace space (Ozkan, 1988), one can obtain the dimensionless pseudo-pressure responses of an MFHW with m hydraulic fractures.
jD ðxD ; yD ; sÞ ¼
LfRDi Z m X i¼1
LfLDi
The relationship between the different coordinate systems, i.e., (x,y) and (xi,yi), is shown in Fig. 4. The solutions of Eqs. (28)e(32) in Laplace space can be given as follows
32 2 8 hwDi sin np zwDi hwDi n¼∞ < 2 X 6sin np zwDi þ 2 7 6 7 qfD ða sin qi ; ywDi a cos qi ; sÞ$ K0 ðε0 RD Þ þ 2 4 5 : nphwDi n¼1
9 = K0 ðεn RD Þ da; ;
2
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RD ðxD ; yD ; aÞ ¼ ðxD a sin qi Þ2 þ ðyD ywDi þ a cos qi Þ2 :
(27)
jwDH ðsÞ jfD ðxDi ; sÞ ¼
2p 6 4x CfD hwDi Di ZxDi Zx0
3.2. Fracture model
0
MFHWs produce at a rate of Qsc through m finite-conductivity fractures. In practice, the gas flow within the hydraulic fractures can be considered an approximately linear flow, and the compressibility of the fracture can be neglected for practical purposes because the fracture volume is very small (Cinco-Ley et al., 1978). The flow equation for the ith fracture in the coordinate system (xDi,yDi) can then be written as (see Appendix A) 2
v jfD ðxDi ; tD Þ 2p q ðx ; t Þ ¼ 0; CfD hwDi fD Di D vx2Di
LfLDi < xDi < LfRDi ;
LfRDi Z
qfD ðxDi ; sÞdxDi 0
3
00 00 7 qfD x ; s dx dx0 5;
Di
xDi /0þ
¼
2p Q ðt Þ; CfD hwDi DRi D
vjfD ðxDi ; tD Þ 2p ¼ Q ðt Þ; vxDi CfD hwDi DLi D xDi /0
(29)
(30)
LfRDi Z
QDRi ðtD Þ ¼
qfD ðxDi ; tD ÞdxDi ;
(31)
0
Z0 QDLi ðtD Þ ¼
qfD ðxDi ; tD ÞdxDi ;
(32)
LfLDi m X ½QDLi ðtD Þ þ QDRi ðtD Þ ¼ 1: i¼1
(33)
0 < xDi < LfRDi ;
0
(34) 2 jwDH ðsÞ jfD ðxDi ; sÞ ¼
2p 6 4 xDi CfD hwDi ZxDi Zx0 0
Z0 qfD ðxDi ; sÞdxDi LfLDi
3
7 qfD x ; s dx dx0 5; 00
00
0
LfLDi < xDi < 0 :
(28) vjfD ðxDi ; tD Þ vx
(26)
(35)
3.3. Semi-analytical solutions in Laplace space The reservoir/fracture models presented above, including Eqs. (26), (27), (34) and (35), can be applied to study pseudo-pressure behaviors of an MFHW. However, because of the complexity of the reservoir/fracture models, it is difficult to directly obtain the analytical solutions of the present models. Therefore, a semianalytical method will be used to solve these models. The left and right wings of the ith fracture are discretized into Ni segments with equal length, respectively (as shown in Fig. 5), and a P total of 2 m i¼1 Ni discrete segments can be obtained. The flow rate of a discrete segment qfi,j is considered as constant in the same time step. Note that the present reservoir and fracture models are based on different coordinate systems, i.e., (x,y) and (xi,yi), respectively. Therefore, in the following, semi-analytical solutions will be presented in a uniform coordinate system (x,y). The coordinates of the center of the jth segment in the ith fracture, (xmi,j,ymi,j), are
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
41
Fig. 5. Schematic of discretion of an MFHW.
8 2j 1 > > > < xmi;j ¼ 2N LfRi sin qi i ; 1 j Ni ; > 2j 1 > > : ymi;j ¼ ywi L cos qi 2Ni fRi
jwDH ðsÞ (36)
8 2ðj Ni Þ þ 1 > > LfLi sin qi > < xmi;j ¼ 2Ni ; Ni þ 1 j 2Ni : > 2ðj Ni Þ 1 > > : ymi;j ¼ ywi þ LfLi cos qi 2Ni
(37)
2Ni h m X X
i jDi;j xmDk;v ; ymDk;v ; s
i¼1 j¼1
8 v1 i h 2p < X qfDk;j ðsÞ$ ðv jÞ$DL2fRDk þ CfD hwDk : j¼1
9 j¼N i= x Xk h DL2fRDk mDk;v $qfDk;v ðsÞ q ðsÞDLfRDk ; þ ; sin qk j¼1 fDk;j 8 ð1 k m; 1 v Nk Þ; (41)
The coordinates of the jth end point in the ith fracture, (xi,j,yi,j), are
8 j1 > > > < xi;j ¼ N LfRi sin qi i ; 1 j Ni þ 1; > j1 > > : yi;j ¼ ywi LfRi cos qi Ni
(38)
8 ðj Ni Þ þ 1 > > LfLi sin qi > xi;j ¼ < Ni ; Ni þ 2 j 2Ni þ 1: > j Ni 1 > > : yi;j ¼ ywi þ LfLi cos qi Ni
2Ni h m X X
i jDi;j xmDk;v ; ymDk;v ; s
i¼1 j¼1
8 v1 i h 2p < X qfDk;j ðsÞ$ ðv jÞ$DL2fLDk þ CfD hwDk : j¼N þ1 i
(39)
9 j¼2N i= x Xk h DL2fLDk mDk;v $qfDk;v ðsÞ q ðsÞDLfLDk ; þ ; sin qk j¼N þ1 fDk;j 8 k
ð1 k m; Nk þ 1 v 2Nk Þ; (42)
The fracture intervals are given as follows:
Dyi ¼ ywi ywi1 :
(40)
Discretizing Eqs. (26), (27), (34) and (35) in Laplace space, we can obtain the following semi-analytical model:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jDi;j ðxD ; yD ; sÞ ¼ 1 þ ctg2 ðqi ÞqfDi;j ðsÞ
jwDH ðsÞ
where DLfLDk ¼ LfLDk/Nk, DLfRDk ¼ LfRDk/Nk, and for 1 i m,1 j Ni ,
32 2 9 hwDi sin np zwDi hwDi n¼∞ = < 2 X 6sin np zwDi þ 2 7 6 7 K0 ðεn RD Þ da K0 ðε0 RD Þ þ 2 4 5 ; : nphwDi
8 2xmDi;j Z xDi;j xDi;j
n¼1
(43)
42
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
for 1 i m,Niþ1 j 2Ni,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jDi;j ðxD ; yD ; sÞ ¼ 1 þ ctg2 ðqi ÞqfDi;j ðsÞ
32 2 9 hwDi hwDi sin np z sin np z þ n¼∞ = < wDi wDi 2 2 X6 7 6 7 K0 ðεn RD Þ da K0 ðε0 RD Þ þ 2 4 5 ; : nphwDi
8 2xmDi;j ZxDi;jþ1 xDi;jþ1
n¼1
(44)
4. Type curves and discussions
In Eqs. (43) and (44), RD can be expressed as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RD ðxD ; yD ; aÞ ¼ ðxD aÞ2 þ ðyD ywDi þ actgqi Þ2 :
(45)
In addition, the discrete forms of Eqs. (31)e(33) in Laplace space can be obtained as follows:
8 Ni h m
:
qfDi;j ðsÞDLfRDi
i
9 2Ni h i= 1 X þ qfDi;j ðsÞDLfLDi ¼ : ; s
(46)
j¼Ni þ1
j¼1
P Eqs. (41)e(46) compose a ð2 m N þ 1Þ-order system of linear Pm i¼1 i algebraic equations with ð2 i¼1 Ni þ 1Þ unknowns, which are jwDH ðsÞ and qfDk;v ðsÞ (1 v 2Nk,1 k m). Applying Gaussian elimination method, one can obtain the pressure responses of a finite-conductivity MFHW in a shale gas reservoir. It should be noted that the jwDH ðsÞ in Eqs. (41) and (42) does not consider the effects of the wellbore storage and skin. Owing to the solution presented in Laplace space, these effects can be easily considered by Duhamel's principle:
jwD ðsÞ ¼
sjwDH ðsÞ þ Sf h i: sþ D sjwDH ðsÞ þ Sf s2 C
(47)
Finally, jwD ðsÞ is inverted into the real space by the numerical inversion method (Stehfest, 1970).
In this section, we apply the present model to calculate the pressure and its derivative of a finite-conductivity MFHW in a shale gas reservoir. The possible flow regimes are identified according to the type curves, and a detailed analysis of the pressure characteristics is presented. Fig. 6 shows the pressure characteristics of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that there are ten possible flow regimes in the type curve: (1) Pure wellbore storage period (PWSP). In this period, the gas flow is only affected by the wellbore storage, and both the pressure and its derivative curves align in a unit slope trend. (2) Transition flow period after the PWSP. In this period, the gas flow is mainly affected by both the wellbore storage and skin. (3) Bilinear flow period (BFP), in which the pressure derivative curve appears as a line with a 1/4 slope. During this period, two linear flows occur simultaneously: one is perpendicular to the hydraulic fracture surfaces, and the other is within the hydraulic fractures (Fig. 7a). (4) Early-time linear flow period (ETLFP), in which the pressure derivative curve appears with a 1/2 slope. During this period, the gas flow mainly appears as a linear flow perpendicular to the hydraulic fracture surfaces (Fig. 7b). In addition, each hydraulic fracture produces independently in this period.
Fig. 6. Type curve of a finite-conductivity MFHW in a shale gas reservoir.
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
43
Fig. 7. Flow regimes of a finite-conductivity MFHW in a shale gas reservoir.
(5) Transition flow period after the ETLFP. In this period, the fracture height and anisotropy coefficient have an important effect on the type curves, which will be discussed in detail in the following. (6) Intermediate-time pseudo-radial flow period (ITPRFP), in which the pressure derivative is a constant value of 1/(2 m). During this period, a pseudo-radial flow occurs around each fracture (Fig. 7c), and interference between adjacent fractures is not felt. (7) Intermediate-time linear flow period (ITLFP), in which the pressure derivative curve appears as a line with a 1/2 slope. In this period, interference between adjacent fractures occurs, and a linear flow takes place in the shale gas reservoir (Fig. 7d). (8) Transition flow period after the ITLFP. In this period, the duration mainly depends on the occurrence time of the gas diffusion from organic matrix to natural fracture system. (9) Diffusive flow period (DFP), in which the pressure derivative curve appears as a “dip”. In this period, gas adsorbed in the organic matrix begins desorbing from the matrix surface,
and then diffuses into the natural fracture system. Because of the complement of the adsorbed gas, the wellbore pressure drops slowly, and a “dip” in the pressure derivative curve appears. (10) Late-time pseudo-radial flow period (LTPRFP), in which the pressure derivative keeps a 0.5 value and a pseudo-radial flow occurs, as shown in Fig. 7e. Note that, owing to the differences in the system properties, the flow regimes described above may not be present in all MFHWs in shale gas reservoirs. In fact, the flow regimes become more complex under the actual conditions. In the following, we study in detail the effects of the relevant parameters on the pressure behaviors. Fig. 8 shows the effect of the fracture height, hwi, on the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that the fracture height has a significant effect on the early-time pressure behaviors. A smaller fracture height leads to higher pressure derivative curves during the BFP and ETLFP. It should be noted that, as the fracture height decreases, the end of
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J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Fig. 8. Effect of fracture height, hwi, on the pressure behaviors of a finite-conductivity MFHW (hw1 ¼ hw2 ¼ hw3).
the ETLFP occurs earlier, and the duration of the transition flow period after the ETLFP becomes longer. Fig. 9 shows the effect of the anisotropy coefficient, b, on the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that the anisotropy coefficient only affects the ETLFP and ITPRFP. The larger the anisotropy coefficient is, the higher the pressure derivative curves in the ETLFP and ITPRFP become. In addition, with the increase of the anisotropy coefficient, the durations of the ETLFP and ITPRFP become shorter. Fig. 10 shows the effect of the length of the fracture wings, Lfi, on
the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It is clear that as the length of the fracture wings increases, the positions of the pressure derivative curves during the ETLFP are lower, the duration of the ITPRFP is shorter, and the duration of the BFP becomes longer. Fig. 11 shows the effect of the fracture asymmetry, LfLi/LfRi, on the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that the fracture asymmetry only affects the BFP and ETLFP. As the fracture asymmetry increases, the ETLFP occurs later and the duration of the BFP becomes shorter. Note that
Fig. 9. Effect of anisotropy coefficient, b, on the pressure behaviors of a finite-conductivity MFHW.
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
45
Fig. 10. Effect of the length of the fracture wings, Lfi, on the pressure behaviors of a finite-conductivity MFHW (Lfi ¼ LfLi ¼ LfRi, and Lf1 ¼ Lf2 ¼ Lf3).
a transitional flow period, which is caused by the fracture asymmetry, exists between the BFP and ETLFP in type curves. Increasing the value of LfLi/LfRi makes the transitional flow period appear more obviously because the transitional flow period results from the interference between the left and right wings of each fracture. When the flow regime of the shorter wing of each fracture is located in the ETLFP, the flow regime of the longer wing of the fracture still remains in the BFP, and thus a transitional flow period will appear under this condition. Fig. 12 shows the effect of the fracture interval, Dyi, on the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that the fracture interval mainly affects the duration of the ITPRFP. With the decrease of the fracture interval, the duration of the ITPRFP becomes shorter.
Fig. 13 shows the effect of the number of hydraulic fractures, m, on the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that the number of hydraulic fractures mainly affects the size of the early- and intermediate-time pressures and their derivatives. With a larger value of m, the positions of the pressure and its derivative curves in the early- and intermediate-time periods are lower. Fig. 14 shows the effect of the dimensionless fracture conductivity coefficient, CfD, on the pressure behaviors of a finiteconductivity MFHW in a shale gas reservoir. It can be seen that the BFP is affected by CfD. Increasing the value of CfD will decrease the duration of the BFP and the magnitude of the pressure derivative in the BFP. Fig. 15 shows the effect of the storativity ratio, u, on the pressure
Fig. 11. Effect of fracture asymmetry, LfLi/LfRi, on the pressure behaviors of a finite-conductivity MFHW (LfL1 ¼ LfL2 ¼ LfL3, and LfR1 ¼ LfR2 ¼ LfR3).
46
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Fig. 12. Effect of fracture interval, Dyi, on the pressure behaviors of a finite-conductivity MFHW (Dy1 ¼ Dy2 ¼ Dy3).
behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that the storativity ratio has an important effect on the BFP, ETLFP, ITPRFP, ITLFP, and DFP. With the increase of the storativity ratio, the pressure derivative curve during the BFP, ETLFP, and ITLFP is lower, and the “dip” in the pressure derivative curve is shallower and narrower. Fig. 16 shows the effect of the inter-porosity flow coefficient, l, on the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that l influences the occurrence time of the DFP. A smaller value of l will lead to an earlier occurrence of the DFP. Fig. 17 shows the effect of the adsorption index, s, on the pressure behaviors of a finite-conductivity MFHW in a shale gas
reservoir. It is clear that the adsorption index mainly affects the DFP. The smaller the value of the adsorption index is, the shallower and narrower the “dip” in the pressure derivative curve becomes because the adsorption index is proportional to the amount of adsorbed gas in organic matrix. With the increase of the adsorption index, more of the adsorbed gas begins desorbing from the matrix surface, and then diffuses into the natural fracture system. Therefore, the wellbore pressure drops slowly and the pressure derivative curve shows a “dip”. 5. Simplified model and validation In the case of u ¼ 1, Eq. (20) is simplified into f(s) ¼ s, and thus
Fig. 13. Effect of the number of hydraulic fractures, m, on the pressure behaviors of a finite-conductivity MFHW.
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
47
Fig. 14. Effect of dimensionless fracture conductivity coefficient, CfD, on the pressure behaviors of a finite-conductivity MFHW.
the model presented above is reduced to a well-test model for MFHWs in conventional gas reservoirs which does not take into consideration the effects of desorption and diffusion in organic matrix. To validate the present model, the theoretical pressure data for MFHWs in conventional gas reservoirs are calculated using the present model with u ¼ 1, and are in comparison with the real pressure data. To show the advantages of the present model, a comparison between the present model and the previous model (Guo et al., 2014) is conducted as follows. Table 2 shows the basic data of an MFHW in a conventional gas reservoir. Fig. 18 shows the matching results of the real pressure
data and the type curves proposed by Guo et al. (2014), and Fig. 19 shows the matching results of the real pressure data and the type curves proposed by the model presented in this paper. The main matching parameters of Guo et al.'s model and the present model are summarized in Table 3, where ‘N/A’ indicates not applicable. The comparison between the two models shown in Table 3 indicates that the fracture asymmetry was evaluated by the present model, whereas Guo et al.'s model was unable to obtain information on the fracture asymmetry. Compared with Guo et al.'s model, the present model can accommodate hydraulic fractures with more realistic configurations; therefore, the present model can make a better agreement with the real pressure data (see Figs. 18 and 19).
Fig. 15. Effect of storativity ratio, u, on the pressure behaviors of a finite-conductivity MFHW.
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J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Fig. 16. Effect of inter-porosity flow coefficient, l, on the pressure behaviors of a finite-conductivity MFHW.
6. Conclusions (1) A novel semi-analytical model for finite-conductivity MFHWs in shale gas reservoirs was presented. Compared with the existing models, the present model can take into account more actual conditions, such as finite-conductivity fractures, different fracture lengths, different fracture intervals, various angles between the fractures and the horizontal well, fracture asymmetry about the horizontal well, and partially penetrating fractures. (2) The solution of the present model was obtained based on the source function method and numerical discrete method in
Table 2 Basic data of an MFHW in a conventional gas reservoir. Name of parameters
Value
Estimated reservoir thickness, h Porosity, f Reservoir temperature, T Gas viscosity, m Gas compressibility, cg Total production rate, Qsc
12.4 m 0.121 369.3 K 2.2 105 Pa s 4.2 108 Pa1 1.31 m3/s
Fig. 17. Effect of adsorption index, s, on the pressure behaviors of a finite-conductivity MFHW.
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
49
Fig. 18. Matching results of real pressure data and type curves proposed by Guo et al.
Table 3 Summary of matching parameters.
Laplace space. Because the present model does not require reservoir-space and time discretions, it can improve the computational efficiency. (3) Based on the present model, type curves for finiteconductivity MFHWs in shale gas reservoirs were plotted, the possible flow regimes were identified, and a detailed analysis of the pressure characteristics was presented. (4) The present semi-analytical model is much closer to the conditions of an actual reservoir, and thus it can be applied to an accurate interpretation of the pressure data of an MFHW.
Name of parameters
The present model
Guo et al.'s model
Number of hydraulic fractures, m Dimensionless fracture conductivity coefficient, CfD Dimensionless wellbore storage coefficient, CD Skin factor, Sf Fracture height, hwi Fracture half-length, Lfi Reservoir permeability, kh Fracture interval, Dyi Fracture asymmetry, LfLi/LfRi Length of the left wing, LfLi Length of the right wing, LfRi
4 10
4 10
104
104
0.001 12.4 m 21.8 m 5 1016 m2 150 m 7.13 38.24 m 5.36 m
0.001 12.4 m 21.8 m 5 1016 m2 150 m N/A N/A N/A
Fig. 19. Matching results of real pressure data and type curves proposed by the present model.
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J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Acknowledgments
Z0
The authors acknowledge the support by the National Basic Research Program of China (2014CB239205).
QLi ðtÞ ¼
qf ðxi ; tÞdxi ;
(A8)
LfLi m X ½QLi ðtÞ þ QRi ðtÞ ¼ Qsc :
Appendix A
(A9)
i¼1
To study the flow characteristics within a hydraulic fracture, a new coordinate system (xi,yi) for the ith fracture was established (as shown in Fig. 4). Because the gas flow within hydraulic fractures can be considered as an approximately linear flow and the compressibility of the hydraulic fractures can be neglected for practical purposes, the governing equation for the ith fracture in the coordinate system (xi,yi) can be given as
v pf vpf ðxi ; yi ; tÞ v pf vpf ðxi ; yi ; tÞ þ ¼ 0; vxi mZ vxi vyi mZ vyi w w LfLi < xi < LfRi ; f < yi < f : 2 2
(A1)
p Z f 2p dp: jf pf ¼ mZ
(A2)
p0
According to Eq. (A2), Eq. (A1) can be rewritten as
v vjf ðxi ; yi ; tÞ v vjf ðxi ; yi ; tÞ þ ¼ 0; vxi vxi vyi vyi
2
(A3)
The flow rate per unit length in the ith fracture can be expressed as
kf Tsc vjf ðxi ; yi ; tÞ kh Tsc vjðxi ; yi ; tÞ ¼ : w w vyi vyi 2psc T 2psc T yi ¼ f yi ¼ f
2
(A4)
Because the fracture width is negligibly small compared with the discharge area of an MFHW, the yi -axis directional pseudopressure drop within the fractures can be ignored. Therefore, the yDi -axis directional average pseudo-pressure within the ith fracture can be used to substitute the pseudo-pressure within the fracture as follows: wf
1 jf ðxi ; tÞ ¼ wf
Z2
jf ðxi ; yi ; tÞdyi ;
LfLi < xi < LfRi :
wf
hwi w
2f
wf 2
Z hwi
w
2f
kf Tsc vjf ðxi ; yi ; tÞ dyi vxi 2psc T
(A12)
w
2f
" # v2 jf ðxi ; tÞ kh vjðxi ; yi ; tÞ vjðxi ; yi ; tÞ þ wf w vyi vyi wf kf vx2i yi ¼ 2 yi ¼ 2f
¼ 0; LfLi < xi < LfRi :
(A13)
v2 jf ðxi ; tÞ 2psc T þ qf ðxi ; tÞ ¼ 0; LfLi < xi < LfRi : 2 w k h T vxi f f wi sc (A14)
The wellbore conditions are
Z2
(A11)
2
According to Eqs. (A4) and (A13), we can obtain
# " k Tsc vjðxi ; yi ; tÞ vjðxi ; yi ; tÞ qf ðxi ; tÞ ¼ hwi h : wf w vyi vyi 2psc T yi ¼ yi ¼ f
kf Tsc vjf ðxi ; yi ; tÞ dyi vxi 2psc T
(A10)
2
According to Eqs. (A10)e(A12), Eq. (A3) can be rewritten as follows:
w w LfLi < xi < LfRi ; f < yi < f : 2 2
2
kf Tsc vjf ðxi ; yi ; tÞ k Tsc vjðxi ; yi ; tÞ ¼ h wf ; w vyi vyi 2psc T 2psc T yi ¼ f yi ¼
2
The definition of the fracture pseudo-pressure is given as follows
According to the mass conservation principle, the fracture surface condition can be written as
According to Eq. (A12), Eqs. (A5) and (A6) can also be rewritten, respectively, as
¼ QRi ðtÞ;
(A5)
hwi wf
kf Tsc vjf ðxi ; tÞ ¼ QRi ðtÞ; vxi xi /0þ 2psc T
(A15)
hwi wf
kf Tsc vjf ðxi ; tÞ ¼ QLi ðtÞ: vxi xi /0 2psc T
(A16)
xi /0þ
¼ QLi ðtÞ;
(A6)
With the dimensionless variables defined in Table 1, Eqs. (A14)e(A16), and (A7)e(A9) can be given in a dimensionless form:
v2 jfD ðxDi ; tD Þ 2p qfD ðxDi ; tD Þ ¼ 0; LfLDi < xDi < LfRDi ; 2 CfD hwDi vxDi
xi /0
where
(A17) ZLfRi
QRi ðtÞ ¼
qf ðxi ; tÞdxi ; 0
(A7)
vjfD ðxDi ; tD Þ 2p ¼ Q ðt Þ; vxDi CfD hwDi DRi D xDi /0þ
(A18)
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
vjfD ðxDi ; tD Þ 2p ¼ Q ðt Þ; vxDi CfD hwDi DLi D xDi /0
(A19)
LfRDi Z
QDRi ðtD Þ ¼
qfD ðxDi ; tD ÞdxDi ;
(A20)
0
Z0 QDLi ðtD Þ ¼
qfD ðxDi ; tD ÞdxDi ;
(A21)
LfLDi m X ½QDLi ðtD Þ þ QDRi ðtD Þ ¼ 1:
(A22)
i¼1
References Brohi, I.G., Pooladi-Darvish, M., Aguilera, R., 2011. Modeling fractured horizontal wells as dual porosity composite reservoirs-application to tight gas, shale gas and tight oil cases. In: SPE Paper 144057 Presented at the SPE Western North American Region Meeting. Anchorage, Alaska, USA. Bumb, A.C., McKee, C.R., 1988. Gas-well testing in the presence of desorption for coalbed methane and Devonian shale. SPE Form. Eval. 3 (1), 179e185. Chen, C.-C., Raghavan, R., 1997. A multiply-fractured horizontal well in a rectangular drainage region. SPE J. 2 (4), 455e465. Cinco-Ley, H., Samaniego, F., Dominguez, N., 1978. Transient pressure behavior for a well with a finite-conductivity vertical fracture. SPE J. 18 (4), 253e264. Guo, J.J., Zhang, L.H., Wang, H.T., Feng, G.Q., 2012. Pressure transient analysis for
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multi-stage fractured horizontal wells in shale gas reservoirs. Trans. Porous Media 93 (3), 635e653. Guo, J.J., Zhang, L.H., Wang, H.T., 2014. Pressure transient characteristics of multistage fractured horizontal wells in shale gas reservoirs with consideration of multiple mechanisms. In: Proceedings of the 5th International Conference on Porous Media and its Applications in Science and Engineering. Kona, Hawaii, USA. Hill, D.G., Nelson, C.R., 2000. Gas productive fractured shales: an overview and update. Gas. Tips 6 (3), 4e13. Kucuk, F., Sawyer, W.K., 1980. Transient flow in naturally fractured reservoirs and its application to Devonian gas shales. In: SPE Paper 9397 Presented at the SPE Annual Technical Conference and Exhibition. Dallas, Texas. Liu, M.J., Xiao, C., Wang, Y.C., Li, Z., Zhang, Y.Y., Chen, S., Wang, G.D., 2015. Sensitivity analysis of geometry for multi-stage fractured horizontal wells with consideration of finite-conductivity fractures in shale gas reservoirs. J. Nat. Gas. Sci. Eng. 22, 182e195. Ozkan, E., 1988. Performance of Horizontal Wells (Ph.D. dissertation). University of Tulsa, Tulsa, Oklahoma, USA. Ozkan, E., Raghavan, R., Apaydin, O.G., 2010. Modeling of fluid transfer from shale matrix to fracture network. In: SPE Paper 134830 Presented at the SPE Annual Technical Conference and Exhibition. Florence, Italy. Ozkan, E., Brown, M., Raghavan, R., Kazemi, H., 2011. Comparison of fracturedhorizontal-well performance in tight sand and shale reservoirs. SPE Reser. Eval. Eng. 14 (2), 248e259. Stehfest, H., 1970. Numerical inversion of Laplace transforms. Commun. ACM 13 (1), 47e49. Wang, H.T., 2014. Performance of multiple fractured horizontal wells in shale gas reservoirs with consideration of multiple mechanisms. J. Hydrol. 510, 299e312. Wang, X.D., Luo, W.J., Hou, X.C., Wang, J.L., 2014. Pressure transient analysis of multi-stage fractured horizontal wells in boxed reservoirs. Pet. Explor. Dev. 41 (1), 82e87. Xu, B.X., Haghighi, M., Li, X.F., Cooke, D., 2013. Development of new type curves for production analysis in naturally fractured shale gas/tight gas reservoirs. J. Pet. Sci. Eng. 105, 107e115. Zhao, Y.L., Zhang, L.H., Zhao, J.Z., Luo, J.X., Zhang, B.N., 2013. “Triple porosity” modeling of transient well test and rate decline analysis for multi-fractured horizontal well in shale gas reservoirs. J. Pet. Sci. Eng. 110, 253e262.
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
A novel semi-analytical model for finite-conductivity multiple fractured horizontal wells in shale gas reservoirs Junjie Ren*, Ping Guo State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, Sichuan, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 December 2014 Received in revised form 7 March 2015 Accepted 9 March 2015 Available online
The use of multiple fractured horizontal wells (MFHWs) is considered an effective means for developing shale gas reservoirs, and a variety of models have recently been introduced to study the pressure behaviors of MFHWs in shale gas reservoirs. However, most of the existing models are based on some ideal assumptions that may not always be true in practice. This paper presents a novel semi-analytical model for finite-conductivity MFHWs in shale gas reservoirs with consideration of more actual conditions, such as finite-conductivity fractures, different fracture lengths, different fracture intervals, various angles between the fractures and the horizontal well, fracture asymmetry about the horizontal well, and partially penetrating fractures. Laplace transform, source function, numerical discrete method, and Gaussian elimination method are applied to solve the present model. The solution is presented in Laplace space so that the effects of the wellbore storage and skin can be easily incorporated by Duhamel's principle. Type curves of the pressure responses are plotted, possible flow regimes are identified, and a detailed analysis of the pressure characteristics is presented. The present model is much closer to the conditions of an actual reservoir, and thus it can be applied to an accurate interpretation of the pressure data of an MFHW in a shale gas reservoir. © 2015 Elsevier B.V. All rights reserved.
Keywords: Shale gas Multiple fractured horizontal wells Finite-conductivity fractures Desorption Diffusion Mathematical model
1. Introduction Shale gas reservoirs have attracted significant attention in recent years because they show great potential to offset the reduction of conventional gas production. In general, shale gas reservoirs mainly include organic matrix and inorganic natural fracture system. The nanoscale pores of organic matrix have a sufficient surface area, and thus shale gas can be stored by adsorption on the surfaces of these nanoscale pores. Hill and Nelson (2000) have pointed out that about 20e85% of the gas reserves in shale gas reservoirs are likely to come from adsorbed gas. Because the adsorption/desorption processes occur during gas production and affect the pressure and production performances of the gas wells, it is necessary to consider the effects of adsorption and desorption for simulating gas flows in shale gas reservoirs. Furthermore, owing to the very low permeability, there is usually no natural production capacity in shale gas reservoirs, and thus to obtain the economic production capacity, multiple fractured
* Corresponding author. E-mail addresses: [email protected] (J. Ren), [email protected] (P. Guo). http://dx.doi.org/10.1016/j.jngse.2015.03.015 1875-5100/© 2015 Elsevier B.V. All rights reserved.
horizontal wells (MFHWs) have been widely used for the development of shale gas reservoirs. Recently, interest in the development of models for studying the pressure behaviors in such reservoirs has grown strikingly. By incorporating gas slippage and desorption into their model, Kucuk and Sawyer (1980) studied the pressure behaviors of a vertical well in a shale gas reservoir. To study the pressure responses of a vertical well in a shale gas reservoir, Bumb and McKee (1988) presented an interesting model in which the Langmuir isotherm was introduced to describe the desorption processes. Ozkan et al. (2010) studied the pressure behaviors of a vertical well in a shale gas reservoir by introducing the idea of dual porosity. However, they ignored the effect of desorption in their model, which may not reflect the pressure characteristics under actual conditions. Guo et al. (2012) established a well test model of a horizontal well with multiple infinite-conductivity hydraulic fractures in a shale gas reservoir, which can consider the effects of diffusion and desorption on the pressure behaviors. However, the hydraulic fractures in their model were assumed to have infinite conductivity, to be perpendicular to the horizontal well, and to penetrate the formation completely, which may not always be the case under actual conditions. Zhao et al. (2013) established a tri-porosity model to study the
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J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Nomenclature
Variables C wellbore storage coefficient, m3/Pa CD dimensionless wellbore storage coefficient, dimensionless cg gas compressibility, Pa1 CfD dimensionless fracture conductivity coefficient, dimensionless D diffusion factor, m2/s h reservoir thickness, m hwi height of the ith fracture, m hwDi dimensionless height of the ith fracture, dimensionless kf permeability of hydraulic fracture, m2 kh horizontal permeability of natural fracture system, m2 kv vertical permeability of natural fracture system, m2 K0(x) second-kind modified Bessel function, zero order LfLi length of the left wing of the ith fracture, m LfLDi dimensionless length of the left wing of the ith fracture, dimensionless DLfLDi dimensionless length of discrete segment in the left wing of the ith fracture, dimensionless LfRi length of the right wing of the ith fracture, m LfRDi dimensionless length of the right wing of the ith fracture, dimensionless DLfRDi dimensionless length of discrete segment in the right wing of the ith fracture, dimensionless m number of hydraulic fractures, integer p pressure of natural fracture system, Pa p0 reference pressure, Pa pf pressure of hydraulic fracture, Pa psc pressure at standard condition, Pa q flow rate from point source, m3/s qD dimensionless flow rate from point source, dimensionless qf flow rate per unit length, m2/s qfD dimensionless flow rate per unit dimensionless length, dimensionless qfDi,j dimensionless flow rate per unit dimensionless length of the jth segment in the ith fracture, dimensionless qfi,j flow rate per unit length of the jth segment in the ith fracture, m2/s QLi production rate from the left wing of the ith fracture, m3/s QDLi dimensionless production rate from the left wing of the ith fracture, dimensionless QRi production rate from the right wing of the ith fracture, m3/s QDRi dimensionless production rate from the right wing of the ith fracture, dimensionless Qsc total production rate conditions, m3/s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisurface ffi punder r radial distance, r ¼ x2 þ y2 , m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rD dimensionless radial distance, rD ¼ x2D þ y2D , dimensionless R radius of shale matrix block, m s Laplace transform variable, dimensionless Sf skin factor, dimensionless t time, s tD dimensionless time, dimensionless T temperature of shale gas reservoir, K
Tsc V VD VE VED
temperature at standard condition, K gas concentration, m3/m3 dimensionless gas concentration, dimensionless gas concentration at equilibrium, m3/m3 dimensionless gas concentration at equilibrium, dimensionless Vi initial gas concentration, m3/m3 VL Langmuir volume, m3/m3 wf width of hydraulic fracture, m x,y,z space coordinates in x,y,z Cartesian coordinate system, m xD,yD,zD dimensionless space coordinates in xD,yD,zD Cartesian coordinate system, dimensionless xDi,yDi dimensionless space coordinates in xDi,yDi Cartesian coordinate system, dimensionless xi,yi space coordinates in xi,yi Cartesian coordinate system, m xwi,ywi,zwi space coordinates of the intersection of the ith fracture and the horizontal wellbore, m Dyi difference between ywi and ywi1, Dyi ¼ ywi ywi1, m xi,j,yi,j space coordinates of the jth end point in the ith fracture, m xDi,j,yDi,j dimensionless space coordinates of the jth end point in the ith fracture, dimensionless xmi,j,ymi,j space coordinates of the center of the jth segment in the ith fracture, m xmDi,j,ymDi,j dimensionless space coordinates of the center of the jth segment in the ith fracture, dimensionless xwDi,ywDi,zwDi dimensionless space coordinates of the intersection of the ith fracture and the horizontal wellbore, dimensionless Z Z-factor of shale gas, dimensionless m gas viscosity, Pa s f porosity, fraction b anisotropy coefficient, dimensionless u storativity ratio, dimensionless l inter-porosity flow coefficient, dimensionless s adsorption index, dimensionless qi angle between the ith fracture and the horizontal well, degree j pseudo-pressure of natural fracture system, Pa/s jD dimensionless pseudo-pressure of natural fracture system, dimensionless jf pseudo-pressure of hydraulic fracture, Pa/s jfD dimensionless pseudo-pressure of hydraulic fracture, dimensionless ji initial pseudo-pressure, Pa/s jL Langmuir pseudo-pressure, Pa/s Djs additional pseudo-pressure drop, Pa/s jwD dimensionless pseudo-pressure in horizontal wellbore considering wellbore storage and skin factor, dimensionless jwDH dimensionless pseudo-pressure in horizontal wellbore without considering wellbore storage and skin factor, dimensionless Superscript Laplace space Subscript D dimensionless
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Fig. 1. The schematic illustration of a physical model for a shale gas reservoir with an MFHW.
pressure behaviors of a multiple fractured horizontal well (MFHW) in a shale gas reservoir; however, their model also does not consider the effects of finite-conductivity fractures, partially penetrating fractures, and various angles between the fractures and the horizontal well. Wang (2014) presented a model for an MFHW in a shale gas reservoir with the effects of desorption, diffusion, viscous flow and stress-sensitivity permeability. However, the hydraulic fractures in this model were also assumed to have infinite conductivity and a complete penetration, which may not be always true in practice. Models for MFHWs with finite-conductivity hydraulic fractures in shale gas reservoirs have recently attracted significant attention, and some studies on these models have been conducted. However, most of them, if not all, are only based on the linear flow assumption (Brohi et al., 2011; Ozkan et al., 2011; Xu et al., 2013). Although these models can successfully consider the effect of fracture conductivity, they can only calculate the pressure behaviors in certain regimes (e.g., linear flow regime), and cannot reflect complete flow regimes, such as pseudo-radial flow regime and interference between fractures. In addition, these models must rely on the assumption of linear flow in shale gas reservoirs, and thus they cannot be applied to MFHWs under actual conditions, such as different fracture lengths, different fracture intervals, various angles between the fractures and the horizontal well, and partially penetrating fractures. That is because, under these conditions, the assumption of linear flow in a shale gas reservoir is invalid. Recently, Guo et al. (2014) proposed an interesting model for MFHWs with finite-conductivity fractures in an infinitely large shale gas reservoir. However, the hydraulic
37
Fig. 3. Profile view of the schematic physical model in the x z plane.
fractures considered in their paper were also perpendicular to the horizontal well and penetrated the formation completely, which may not always be true in reality. Most recently, Liu et al. (2015) presented a model for MFHWs in a shale gas reservoir that considers the effects of desorption, stress sensitivity, and fracture conductivity. However, the simulation results presented by Liu et al. (2015) show that the linear flow regime occurs before the bilinear flow regime, which is in disagreement with previous studies (Chen and Raghavan, 1997; Guo et al., 2014; Wang et al., 2014). By carefully investigating their model, we found that, even for the same hydraulic fracture, the flow rate from the reservoir into the hydraulic fracture is also assumed to be equivalent and independent of the coordinates, which may be unreasonable and lead to inaccurate simulation results. This paper proposes a novel semi-analytical model for finiteconductivity MFHWs in shale gas reservoirs which can consider more actual conditions, such as finite-conductivity fractures, different fracture lengths, different fracture intervals, various angles between the fractures and the horizontal well, fracture asymmetry about the horizontal well, and partially penetrating fractures. The present semi-analytical model is much closer to the conditions of an actual reservoir and can be applied to accurately interpret the pressure data of an MFHW in practice. 2. Physical model Fig. 1 shows an MFHW with partially (or completely) penetrating fractures located in a shale gas reservoir. The shale gas
Fig. 2. Top view of the schematic physical model in the x y plane.
38
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
reservoir considered here is infinitely large in the horizontal direction, and is bounded by the top and bottom parallel impermeable planes in the vertical direction. Fig. 2 shows a top view of Fig. 1 in the x y plane, and Fig. 3 shows a profile view of Fig. 1 in the x z plane. Some other assumptions of the present mathematical model are given as follows: (1) Shale gas reservoirs are dual-porosity medium including organic matrix and inorganic natural fracture system. Gas flow in natural fracture system follows Darcy's law, desorption in matrix can be described by Langmuir isotherm, and diffusion in matrix follows Fick's first law. (2) The shale gas reservoir is anisotropic (i.e., kh s kv), and the initial pressure in the shale gas reservoir is uniform. (3) Hydraulic fractures (where m is the number of hydraulic fractures) may diagonally intersect with the horizontal well, asymmetrically distribute about the horizontal well, and partially penetrate the formation. The hydraulic fractures are assumed to have finite conductivity. Gas flow within the hydraulic fractures is considered to be linear flow, and the compressibility of the hydraulic fractures can be neglected. (4) The total production rate of the MFHW is assumed to be constant with a value of Qsc, whereas the flow rate of each hydraulic fracture is distinct, and the flow rate in the same hydraulic fracture is a function of the coordinates. (5) The effects of gravity and capillary force are negligible.
by the Langmuir isotherm, we can get that
VE ¼
VL j ; jL þ j
(7)
Vi ¼
VL ji : jL þ ji
(8)
With the aid of Eqs. (7) and (8), we can obtain
VED ¼ VE Vi ¼ sjD ;
(9)
where s is the adsorption index defined in Table 1. It is difficult to directly obtain a semi-analytical model for an MFHW because of the complexity of the inner boundary condition. In general, researching the point source model is always the first step in studying a semi-analytical model for an MFHW. The inner boundary conditions for a point source with flow rate q(r ¼ 0,z ¼ zw,t) can be expressed by:
3
2
8 7 > psc qðr ¼ 0; z ¼ zw ; tÞT 6 Z < 6 ; jz zw j ε=2 vj 7 7 6 pkh hTsc lim 6 lim r dz7 ¼ : vr 7 > ε/06 r/0 : 5 4 ε zw 2 0; jz zw j > ε=2 zw þ2ε
(10) The outer boundary conditions for a shale gas reservoir that is infinitely large in the horizontal direction and bounded by impermeable planes in the vertical direction can be expressed as
3. Mathematical model 3.1. Reservoir model The hydraulic fractures of the MFHW may be penetrated partially in a shale gas reservoir, and thus the gas flow in this reservoir can be described by the following equation:
fcg p vp psc T vV 1 v p vp v p vp rkh þ kv ¼ þ : r vr mZ vr vz mZ vz Tsc vt Z vt
(1)
The definition of pseudo-pressure can be given as follows:
Zp jðpÞ ¼ p0
2p dp: mZ
jðr/∞; z; tÞ ¼ ji ;
Table 1 Dimensionless definitions of variables. Dimensionless pseudo-pressure Dimensionless time
hTsc ðji jÞ sc ðji jf Þ jD ¼ pkh Q ; jfD ¼ pkh hT Qsc psc T sc psc T
Dimensionless distance
r x y z x rD ¼ ; xD ¼ ; yD ¼ ; zD ¼ ; xDi ¼ i ; h h h h h y x y z yDi ¼ i ; xwDi ¼ wi ; ywDi ¼ wi ; zwDi ¼ wi ; h h h h L L h LfLDi ¼ fLi ; LfRDi ¼ fRi ; hwDi ¼ wi h h h
(2)
According to the definition of pseudo-pressure, Eq. (1) can be rewritten as follows:
1 v vj v vj vj 2psc T vV rkh þ kv ¼ fmcg þ : r vr vr vz vz vt Tsc vt
Storativity ratio
(3)
Based on Fick's first law, the diffusive flow in organic matrix can be expressed as
vV 6p2 D ¼ ðVE VÞ: vt R2
(4)
According to the dimensionless variables defined in Table 1, Eqs. (3) and (4) can be expressed in a dimensionless form as:
1 v vj 1 v 2 jD vj vV rD D þ 2 ¼ u D ð1 uÞ D ; rD vrD vrD vtD vtD b vz2D vVD 1 ¼ ðVED VD Þ: l vtD
(5)
(6)
Because the adsorption/desorption processes can be described
(11)
kh t tD ¼ Lh 2
u¼
fmcg L
hh where L ¼ fmcg þ 2pk Qsc
hh where L ¼ fmcg þ 2pk Qsc
R2 hh where L ¼ fmcg þ 2pk Qsc ; t ¼ 6p2 D
Inter-porosity flow coefficient
kh t l ¼ Lh 2
Adsorption index
psc Qsc T s ¼ pk hT
sc
h
VL jL ðjL þjÞðjL þji Þ
Concentration difference Anisotropy coefficient
VD ¼ V Vi ; VED ¼ VE Vi
Dimensionless flow rate from point source Dimensionless flow rate per unit length Dimensionless production rate from left and right wings of each fracture Dimensionless fracture conductivity coefficient Dimensionless wellbore storage coefficient Skin factor
qD ðrD ; zD ; tD Þ ¼ qðr;z;tÞ Qsc
b¼
qffiffiffiffi kh kv
qfD ðxD ; yD ; tD Þ ¼ hqf Qðx;y;tÞ sc QDLi ðtD Þ ¼ QQLiscðtÞ; QDRi ðtD Þ ¼ QQRiscðtÞ CfD ¼ kkf whf h
CD ¼
mC 2pLh3
hh where L ¼ fmcg þ 2pk Qsc
h hTsc Sf ¼ pk psc Qsc T Djs
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
vj vj ¼ ¼ 0: vz z¼0 vz z¼h
39
(12)
Owing to the uniform pressure at the initial time, the initial condition is:
jðr; z; t ¼ 0Þ ¼ ji :
(13)
According to the definitions in Table 1, the dimensionless forms of Eqs. (10)e(13) can be given as follows:
2
3
6 7 Z 6 7 vj 6 7 lim 6 lim rD D dzD 7 7 vrD εD /06 rD /0 4 5 ε z D ε zwD þ 2D
wD
¼
Fig. 4. Schematic of the relationship between different coordinate systems.
(14)
2
qD ðrD ¼ 0; zD ¼ zwD ; tD Þ; 0;
jD ðrD /∞; zD ; tD Þ ¼ 0; vjD vz
D zD ¼0
vj ¼ D vz
If the point source is located in (xwi,ywi,zwi), Eq. (18) can be rewritten as:
jzD zwD j εD =2 ; jzD zwD j > εD =2
"
jD ¼ qD ðxwDi ; ywDi ; zwDi ; sÞ K0 ðε0 rD Þ þ 2 (15)
n¼∞ X
# cosðnpzD ÞcosðnpzwDi ÞK0 ðεn rD Þ ;
D zD ¼1
¼ 0;
(16) where
jD ðrD ; zD ; tD ¼ 0Þ ¼ 0:
(17)
The seepage model for shale gas reservoirs consists of Eqs. (5), (6), (9), (14)e(17). Applying Fourier and Laplace transforms, the point source solution in Laplace space can be obtained as follows (Ozkan, 1988):
rD ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxD xwDi Þ2 þ ðyD ywDi Þ2 :
" jD ¼ qD ð0; zwD ; sÞ K0 ðε0 rD Þ þ 2
# cosðnpzD ÞcosðnpzwD ÞK0 ðεn rD Þ ;
(18)
n¼1
(22)
Based on the source function and principle of superposition in Laplace space (Ozkan, 1988), the dimensionless pseudo-pressure responses caused by a vertical line source solution can be expressed as
8 9 hwDi > > > > sin np zwDi hwDi n¼∞ < = 2 X sin np zwDi þ 2 jD ¼ qfD ðxwDi ; ywDi ; sÞ$ K0 ðε0 rD Þ þ 2 cosðnpzD ÞK0 ðεn rD Þ : > > nphwDi > > n¼1 : ;
n¼∞ X
(21)
n¼1
(23)
It should be noted that, as shown in Eq. (23), the dimensionless pseudo-pressure is a function of the dimensionless variable zD. Considering that the linear flow occurs along the hydraulic fracture direction, we can ignore the zD-axis directional pseudo-pressure drop within the hydraulic fracture. Therefore, the zD-axis directional average pseudo-pressure within the hydraulic fracture can be used to substitute the pseudo-pressure within the hydraulic fracture as follows:
where
εn ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 p2 f ðsÞ þ 2 ; b
f ðsÞ ¼ us þ
(19)
sð1 uÞs : 1 þ ls
(20)
jD ðrD ; sÞ ¼
1 hwDi
hwDi 2
zwDi Zþ
jD ðrD ; zD ; sÞdzD :
According to Eq. (24), Eq. (23) can be rewritten as follows:
32 2 8 9 hwDi sin np zwDi hwDi n¼∞ < = 2 X 6sin np zwDi þ 2 7 6 7 K0 ðεn rD Þ jD ¼ qfD ðxwDi ; ywDi ; sÞ$ K0 ðε0 rD Þ þ 2 4 5 : ; nphwDi n¼1
(24)
h zwDi wDi 2
(25)
40
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Following the source function and principle of superposition in Laplace space (Ozkan, 1988), one can obtain the dimensionless pseudo-pressure responses of an MFHW with m hydraulic fractures.
jD ðxD ; yD ; sÞ ¼
LfRDi Z m X i¼1
LfLDi
The relationship between the different coordinate systems, i.e., (x,y) and (xi,yi), is shown in Fig. 4. The solutions of Eqs. (28)e(32) in Laplace space can be given as follows
32 2 8 hwDi sin np zwDi hwDi n¼∞ < 2 X 6sin np zwDi þ 2 7 6 7 qfD ða sin qi ; ywDi a cos qi ; sÞ$ K0 ðε0 RD Þ þ 2 4 5 : nphwDi n¼1
9 = K0 ðεn RD Þ da; ;
2
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RD ðxD ; yD ; aÞ ¼ ðxD a sin qi Þ2 þ ðyD ywDi þ a cos qi Þ2 :
(27)
jwDH ðsÞ jfD ðxDi ; sÞ ¼
2p 6 4x CfD hwDi Di ZxDi Zx0
3.2. Fracture model
0
MFHWs produce at a rate of Qsc through m finite-conductivity fractures. In practice, the gas flow within the hydraulic fractures can be considered an approximately linear flow, and the compressibility of the fracture can be neglected for practical purposes because the fracture volume is very small (Cinco-Ley et al., 1978). The flow equation for the ith fracture in the coordinate system (xDi,yDi) can then be written as (see Appendix A) 2
v jfD ðxDi ; tD Þ 2p q ðx ; t Þ ¼ 0; CfD hwDi fD Di D vx2Di
LfLDi < xDi < LfRDi ;
LfRDi Z
qfD ðxDi ; sÞdxDi 0
3
00 00 7 qfD x ; s dx dx0 5;
Di
xDi /0þ
¼
2p Q ðt Þ; CfD hwDi DRi D
vjfD ðxDi ; tD Þ 2p ¼ Q ðt Þ; vxDi CfD hwDi DLi D xDi /0
(29)
(30)
LfRDi Z
QDRi ðtD Þ ¼
qfD ðxDi ; tD ÞdxDi ;
(31)
0
Z0 QDLi ðtD Þ ¼
qfD ðxDi ; tD ÞdxDi ;
(32)
LfLDi m X ½QDLi ðtD Þ þ QDRi ðtD Þ ¼ 1: i¼1
(33)
0 < xDi < LfRDi ;
0
(34) 2 jwDH ðsÞ jfD ðxDi ; sÞ ¼
2p 6 4 xDi CfD hwDi ZxDi Zx0 0
Z0 qfD ðxDi ; sÞdxDi LfLDi
3
7 qfD x ; s dx dx0 5; 00
00
0
LfLDi < xDi < 0 :
(28) vjfD ðxDi ; tD Þ vx
(26)
(35)
3.3. Semi-analytical solutions in Laplace space The reservoir/fracture models presented above, including Eqs. (26), (27), (34) and (35), can be applied to study pseudo-pressure behaviors of an MFHW. However, because of the complexity of the reservoir/fracture models, it is difficult to directly obtain the analytical solutions of the present models. Therefore, a semianalytical method will be used to solve these models. The left and right wings of the ith fracture are discretized into Ni segments with equal length, respectively (as shown in Fig. 5), and a P total of 2 m i¼1 Ni discrete segments can be obtained. The flow rate of a discrete segment qfi,j is considered as constant in the same time step. Note that the present reservoir and fracture models are based on different coordinate systems, i.e., (x,y) and (xi,yi), respectively. Therefore, in the following, semi-analytical solutions will be presented in a uniform coordinate system (x,y). The coordinates of the center of the jth segment in the ith fracture, (xmi,j,ymi,j), are
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
41
Fig. 5. Schematic of discretion of an MFHW.
8 2j 1 > > > < xmi;j ¼ 2N LfRi sin qi i ; 1 j Ni ; > 2j 1 > > : ymi;j ¼ ywi L cos qi 2Ni fRi
jwDH ðsÞ (36)
8 2ðj Ni Þ þ 1 > > LfLi sin qi > < xmi;j ¼ 2Ni ; Ni þ 1 j 2Ni : > 2ðj Ni Þ 1 > > : ymi;j ¼ ywi þ LfLi cos qi 2Ni
(37)
2Ni h m X X
i jDi;j xmDk;v ; ymDk;v ; s
i¼1 j¼1
8 v1 i h 2p < X qfDk;j ðsÞ$ ðv jÞ$DL2fRDk þ CfD hwDk : j¼1
9 j¼N i= x Xk h DL2fRDk mDk;v $qfDk;v ðsÞ q ðsÞDLfRDk ; þ ; sin qk j¼1 fDk;j 8 ð1 k m; 1 v Nk Þ; (41)
The coordinates of the jth end point in the ith fracture, (xi,j,yi,j), are
8 j1 > > > < xi;j ¼ N LfRi sin qi i ; 1 j Ni þ 1; > j1 > > : yi;j ¼ ywi LfRi cos qi Ni
(38)
8 ðj Ni Þ þ 1 > > LfLi sin qi > xi;j ¼ < Ni ; Ni þ 2 j 2Ni þ 1: > j Ni 1 > > : yi;j ¼ ywi þ LfLi cos qi Ni
2Ni h m X X
i jDi;j xmDk;v ; ymDk;v ; s
i¼1 j¼1
8 v1 i h 2p < X qfDk;j ðsÞ$ ðv jÞ$DL2fLDk þ CfD hwDk : j¼N þ1 i
(39)
9 j¼2N i= x Xk h DL2fLDk mDk;v $qfDk;v ðsÞ q ðsÞDLfLDk ; þ ; sin qk j¼N þ1 fDk;j 8 k
ð1 k m; Nk þ 1 v 2Nk Þ; (42)
The fracture intervals are given as follows:
Dyi ¼ ywi ywi1 :
(40)
Discretizing Eqs. (26), (27), (34) and (35) in Laplace space, we can obtain the following semi-analytical model:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jDi;j ðxD ; yD ; sÞ ¼ 1 þ ctg2 ðqi ÞqfDi;j ðsÞ
jwDH ðsÞ
where DLfLDk ¼ LfLDk/Nk, DLfRDk ¼ LfRDk/Nk, and for 1 i m,1 j Ni ,
32 2 9 hwDi sin np zwDi hwDi n¼∞ = < 2 X 6sin np zwDi þ 2 7 6 7 K0 ðεn RD Þ da K0 ðε0 RD Þ þ 2 4 5 ; : nphwDi
8 2xmDi;j Z xDi;j xDi;j
n¼1
(43)
42
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
for 1 i m,Niþ1 j 2Ni,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jDi;j ðxD ; yD ; sÞ ¼ 1 þ ctg2 ðqi ÞqfDi;j ðsÞ
32 2 9 hwDi hwDi sin np z sin np z þ n¼∞ = < wDi wDi 2 2 X6 7 6 7 K0 ðεn RD Þ da K0 ðε0 RD Þ þ 2 4 5 ; : nphwDi
8 2xmDi;j ZxDi;jþ1 xDi;jþ1
n¼1
(44)
4. Type curves and discussions
In Eqs. (43) and (44), RD can be expressed as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RD ðxD ; yD ; aÞ ¼ ðxD aÞ2 þ ðyD ywDi þ actgqi Þ2 :
(45)
In addition, the discrete forms of Eqs. (31)e(33) in Laplace space can be obtained as follows:
8 Ni h m
:
qfDi;j ðsÞDLfRDi
i
9 2Ni h i= 1 X þ qfDi;j ðsÞDLfLDi ¼ : ; s
(46)
j¼Ni þ1
j¼1
P Eqs. (41)e(46) compose a ð2 m N þ 1Þ-order system of linear Pm i¼1 i algebraic equations with ð2 i¼1 Ni þ 1Þ unknowns, which are jwDH ðsÞ and qfDk;v ðsÞ (1 v 2Nk,1 k m). Applying Gaussian elimination method, one can obtain the pressure responses of a finite-conductivity MFHW in a shale gas reservoir. It should be noted that the jwDH ðsÞ in Eqs. (41) and (42) does not consider the effects of the wellbore storage and skin. Owing to the solution presented in Laplace space, these effects can be easily considered by Duhamel's principle:
jwD ðsÞ ¼
sjwDH ðsÞ þ Sf h i: sþ D sjwDH ðsÞ þ Sf s2 C
(47)
Finally, jwD ðsÞ is inverted into the real space by the numerical inversion method (Stehfest, 1970).
In this section, we apply the present model to calculate the pressure and its derivative of a finite-conductivity MFHW in a shale gas reservoir. The possible flow regimes are identified according to the type curves, and a detailed analysis of the pressure characteristics is presented. Fig. 6 shows the pressure characteristics of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that there are ten possible flow regimes in the type curve: (1) Pure wellbore storage period (PWSP). In this period, the gas flow is only affected by the wellbore storage, and both the pressure and its derivative curves align in a unit slope trend. (2) Transition flow period after the PWSP. In this period, the gas flow is mainly affected by both the wellbore storage and skin. (3) Bilinear flow period (BFP), in which the pressure derivative curve appears as a line with a 1/4 slope. During this period, two linear flows occur simultaneously: one is perpendicular to the hydraulic fracture surfaces, and the other is within the hydraulic fractures (Fig. 7a). (4) Early-time linear flow period (ETLFP), in which the pressure derivative curve appears with a 1/2 slope. During this period, the gas flow mainly appears as a linear flow perpendicular to the hydraulic fracture surfaces (Fig. 7b). In addition, each hydraulic fracture produces independently in this period.
Fig. 6. Type curve of a finite-conductivity MFHW in a shale gas reservoir.
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
43
Fig. 7. Flow regimes of a finite-conductivity MFHW in a shale gas reservoir.
(5) Transition flow period after the ETLFP. In this period, the fracture height and anisotropy coefficient have an important effect on the type curves, which will be discussed in detail in the following. (6) Intermediate-time pseudo-radial flow period (ITPRFP), in which the pressure derivative is a constant value of 1/(2 m). During this period, a pseudo-radial flow occurs around each fracture (Fig. 7c), and interference between adjacent fractures is not felt. (7) Intermediate-time linear flow period (ITLFP), in which the pressure derivative curve appears as a line with a 1/2 slope. In this period, interference between adjacent fractures occurs, and a linear flow takes place in the shale gas reservoir (Fig. 7d). (8) Transition flow period after the ITLFP. In this period, the duration mainly depends on the occurrence time of the gas diffusion from organic matrix to natural fracture system. (9) Diffusive flow period (DFP), in which the pressure derivative curve appears as a “dip”. In this period, gas adsorbed in the organic matrix begins desorbing from the matrix surface,
and then diffuses into the natural fracture system. Because of the complement of the adsorbed gas, the wellbore pressure drops slowly, and a “dip” in the pressure derivative curve appears. (10) Late-time pseudo-radial flow period (LTPRFP), in which the pressure derivative keeps a 0.5 value and a pseudo-radial flow occurs, as shown in Fig. 7e. Note that, owing to the differences in the system properties, the flow regimes described above may not be present in all MFHWs in shale gas reservoirs. In fact, the flow regimes become more complex under the actual conditions. In the following, we study in detail the effects of the relevant parameters on the pressure behaviors. Fig. 8 shows the effect of the fracture height, hwi, on the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that the fracture height has a significant effect on the early-time pressure behaviors. A smaller fracture height leads to higher pressure derivative curves during the BFP and ETLFP. It should be noted that, as the fracture height decreases, the end of
44
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Fig. 8. Effect of fracture height, hwi, on the pressure behaviors of a finite-conductivity MFHW (hw1 ¼ hw2 ¼ hw3).
the ETLFP occurs earlier, and the duration of the transition flow period after the ETLFP becomes longer. Fig. 9 shows the effect of the anisotropy coefficient, b, on the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that the anisotropy coefficient only affects the ETLFP and ITPRFP. The larger the anisotropy coefficient is, the higher the pressure derivative curves in the ETLFP and ITPRFP become. In addition, with the increase of the anisotropy coefficient, the durations of the ETLFP and ITPRFP become shorter. Fig. 10 shows the effect of the length of the fracture wings, Lfi, on
the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It is clear that as the length of the fracture wings increases, the positions of the pressure derivative curves during the ETLFP are lower, the duration of the ITPRFP is shorter, and the duration of the BFP becomes longer. Fig. 11 shows the effect of the fracture asymmetry, LfLi/LfRi, on the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that the fracture asymmetry only affects the BFP and ETLFP. As the fracture asymmetry increases, the ETLFP occurs later and the duration of the BFP becomes shorter. Note that
Fig. 9. Effect of anisotropy coefficient, b, on the pressure behaviors of a finite-conductivity MFHW.
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
45
Fig. 10. Effect of the length of the fracture wings, Lfi, on the pressure behaviors of a finite-conductivity MFHW (Lfi ¼ LfLi ¼ LfRi, and Lf1 ¼ Lf2 ¼ Lf3).
a transitional flow period, which is caused by the fracture asymmetry, exists between the BFP and ETLFP in type curves. Increasing the value of LfLi/LfRi makes the transitional flow period appear more obviously because the transitional flow period results from the interference between the left and right wings of each fracture. When the flow regime of the shorter wing of each fracture is located in the ETLFP, the flow regime of the longer wing of the fracture still remains in the BFP, and thus a transitional flow period will appear under this condition. Fig. 12 shows the effect of the fracture interval, Dyi, on the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that the fracture interval mainly affects the duration of the ITPRFP. With the decrease of the fracture interval, the duration of the ITPRFP becomes shorter.
Fig. 13 shows the effect of the number of hydraulic fractures, m, on the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that the number of hydraulic fractures mainly affects the size of the early- and intermediate-time pressures and their derivatives. With a larger value of m, the positions of the pressure and its derivative curves in the early- and intermediate-time periods are lower. Fig. 14 shows the effect of the dimensionless fracture conductivity coefficient, CfD, on the pressure behaviors of a finiteconductivity MFHW in a shale gas reservoir. It can be seen that the BFP is affected by CfD. Increasing the value of CfD will decrease the duration of the BFP and the magnitude of the pressure derivative in the BFP. Fig. 15 shows the effect of the storativity ratio, u, on the pressure
Fig. 11. Effect of fracture asymmetry, LfLi/LfRi, on the pressure behaviors of a finite-conductivity MFHW (LfL1 ¼ LfL2 ¼ LfL3, and LfR1 ¼ LfR2 ¼ LfR3).
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J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Fig. 12. Effect of fracture interval, Dyi, on the pressure behaviors of a finite-conductivity MFHW (Dy1 ¼ Dy2 ¼ Dy3).
behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that the storativity ratio has an important effect on the BFP, ETLFP, ITPRFP, ITLFP, and DFP. With the increase of the storativity ratio, the pressure derivative curve during the BFP, ETLFP, and ITLFP is lower, and the “dip” in the pressure derivative curve is shallower and narrower. Fig. 16 shows the effect of the inter-porosity flow coefficient, l, on the pressure behaviors of a finite-conductivity MFHW in a shale gas reservoir. It can be seen that l influences the occurrence time of the DFP. A smaller value of l will lead to an earlier occurrence of the DFP. Fig. 17 shows the effect of the adsorption index, s, on the pressure behaviors of a finite-conductivity MFHW in a shale gas
reservoir. It is clear that the adsorption index mainly affects the DFP. The smaller the value of the adsorption index is, the shallower and narrower the “dip” in the pressure derivative curve becomes because the adsorption index is proportional to the amount of adsorbed gas in organic matrix. With the increase of the adsorption index, more of the adsorbed gas begins desorbing from the matrix surface, and then diffuses into the natural fracture system. Therefore, the wellbore pressure drops slowly and the pressure derivative curve shows a “dip”. 5. Simplified model and validation In the case of u ¼ 1, Eq. (20) is simplified into f(s) ¼ s, and thus
Fig. 13. Effect of the number of hydraulic fractures, m, on the pressure behaviors of a finite-conductivity MFHW.
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
47
Fig. 14. Effect of dimensionless fracture conductivity coefficient, CfD, on the pressure behaviors of a finite-conductivity MFHW.
the model presented above is reduced to a well-test model for MFHWs in conventional gas reservoirs which does not take into consideration the effects of desorption and diffusion in organic matrix. To validate the present model, the theoretical pressure data for MFHWs in conventional gas reservoirs are calculated using the present model with u ¼ 1, and are in comparison with the real pressure data. To show the advantages of the present model, a comparison between the present model and the previous model (Guo et al., 2014) is conducted as follows. Table 2 shows the basic data of an MFHW in a conventional gas reservoir. Fig. 18 shows the matching results of the real pressure
data and the type curves proposed by Guo et al. (2014), and Fig. 19 shows the matching results of the real pressure data and the type curves proposed by the model presented in this paper. The main matching parameters of Guo et al.'s model and the present model are summarized in Table 3, where ‘N/A’ indicates not applicable. The comparison between the two models shown in Table 3 indicates that the fracture asymmetry was evaluated by the present model, whereas Guo et al.'s model was unable to obtain information on the fracture asymmetry. Compared with Guo et al.'s model, the present model can accommodate hydraulic fractures with more realistic configurations; therefore, the present model can make a better agreement with the real pressure data (see Figs. 18 and 19).
Fig. 15. Effect of storativity ratio, u, on the pressure behaviors of a finite-conductivity MFHW.
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J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Fig. 16. Effect of inter-porosity flow coefficient, l, on the pressure behaviors of a finite-conductivity MFHW.
6. Conclusions (1) A novel semi-analytical model for finite-conductivity MFHWs in shale gas reservoirs was presented. Compared with the existing models, the present model can take into account more actual conditions, such as finite-conductivity fractures, different fracture lengths, different fracture intervals, various angles between the fractures and the horizontal well, fracture asymmetry about the horizontal well, and partially penetrating fractures. (2) The solution of the present model was obtained based on the source function method and numerical discrete method in
Table 2 Basic data of an MFHW in a conventional gas reservoir. Name of parameters
Value
Estimated reservoir thickness, h Porosity, f Reservoir temperature, T Gas viscosity, m Gas compressibility, cg Total production rate, Qsc
12.4 m 0.121 369.3 K 2.2 105 Pa s 4.2 108 Pa1 1.31 m3/s
Fig. 17. Effect of adsorption index, s, on the pressure behaviors of a finite-conductivity MFHW.
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
49
Fig. 18. Matching results of real pressure data and type curves proposed by Guo et al.
Table 3 Summary of matching parameters.
Laplace space. Because the present model does not require reservoir-space and time discretions, it can improve the computational efficiency. (3) Based on the present model, type curves for finiteconductivity MFHWs in shale gas reservoirs were plotted, the possible flow regimes were identified, and a detailed analysis of the pressure characteristics was presented. (4) The present semi-analytical model is much closer to the conditions of an actual reservoir, and thus it can be applied to an accurate interpretation of the pressure data of an MFHW.
Name of parameters
The present model
Guo et al.'s model
Number of hydraulic fractures, m Dimensionless fracture conductivity coefficient, CfD Dimensionless wellbore storage coefficient, CD Skin factor, Sf Fracture height, hwi Fracture half-length, Lfi Reservoir permeability, kh Fracture interval, Dyi Fracture asymmetry, LfLi/LfRi Length of the left wing, LfLi Length of the right wing, LfRi
4 10
4 10
104
104
0.001 12.4 m 21.8 m 5 1016 m2 150 m 7.13 38.24 m 5.36 m
0.001 12.4 m 21.8 m 5 1016 m2 150 m N/A N/A N/A
Fig. 19. Matching results of real pressure data and type curves proposed by the present model.
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J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
Acknowledgments
Z0
The authors acknowledge the support by the National Basic Research Program of China (2014CB239205).
QLi ðtÞ ¼
qf ðxi ; tÞdxi ;
(A8)
LfLi m X ½QLi ðtÞ þ QRi ðtÞ ¼ Qsc :
Appendix A
(A9)
i¼1
To study the flow characteristics within a hydraulic fracture, a new coordinate system (xi,yi) for the ith fracture was established (as shown in Fig. 4). Because the gas flow within hydraulic fractures can be considered as an approximately linear flow and the compressibility of the hydraulic fractures can be neglected for practical purposes, the governing equation for the ith fracture in the coordinate system (xi,yi) can be given as
v pf vpf ðxi ; yi ; tÞ v pf vpf ðxi ; yi ; tÞ þ ¼ 0; vxi mZ vxi vyi mZ vyi w w LfLi < xi < LfRi ; f < yi < f : 2 2
(A1)
p Z f 2p dp: jf pf ¼ mZ
(A2)
p0
According to Eq. (A2), Eq. (A1) can be rewritten as
v vjf ðxi ; yi ; tÞ v vjf ðxi ; yi ; tÞ þ ¼ 0; vxi vxi vyi vyi
2
(A3)
The flow rate per unit length in the ith fracture can be expressed as
kf Tsc vjf ðxi ; yi ; tÞ kh Tsc vjðxi ; yi ; tÞ ¼ : w w vyi vyi 2psc T 2psc T yi ¼ f yi ¼ f
2
(A4)
Because the fracture width is negligibly small compared with the discharge area of an MFHW, the yi -axis directional pseudopressure drop within the fractures can be ignored. Therefore, the yDi -axis directional average pseudo-pressure within the ith fracture can be used to substitute the pseudo-pressure within the fracture as follows: wf
1 jf ðxi ; tÞ ¼ wf
Z2
jf ðxi ; yi ; tÞdyi ;
LfLi < xi < LfRi :
wf
hwi w
2f
wf 2
Z hwi
w
2f
kf Tsc vjf ðxi ; yi ; tÞ dyi vxi 2psc T
(A12)
w
2f
" # v2 jf ðxi ; tÞ kh vjðxi ; yi ; tÞ vjðxi ; yi ; tÞ þ wf w vyi vyi wf kf vx2i yi ¼ 2 yi ¼ 2f
¼ 0; LfLi < xi < LfRi :
(A13)
v2 jf ðxi ; tÞ 2psc T þ qf ðxi ; tÞ ¼ 0; LfLi < xi < LfRi : 2 w k h T vxi f f wi sc (A14)
The wellbore conditions are
Z2
(A11)
2
According to Eqs. (A4) and (A13), we can obtain
# " k Tsc vjðxi ; yi ; tÞ vjðxi ; yi ; tÞ qf ðxi ; tÞ ¼ hwi h : wf w vyi vyi 2psc T yi ¼ yi ¼ f
kf Tsc vjf ðxi ; yi ; tÞ dyi vxi 2psc T
(A10)
2
According to Eqs. (A10)e(A12), Eq. (A3) can be rewritten as follows:
w w LfLi < xi < LfRi ; f < yi < f : 2 2
2
kf Tsc vjf ðxi ; yi ; tÞ k Tsc vjðxi ; yi ; tÞ ¼ h wf ; w vyi vyi 2psc T 2psc T yi ¼ f yi ¼
2
The definition of the fracture pseudo-pressure is given as follows
According to the mass conservation principle, the fracture surface condition can be written as
According to Eq. (A12), Eqs. (A5) and (A6) can also be rewritten, respectively, as
¼ QRi ðtÞ;
(A5)
hwi wf
kf Tsc vjf ðxi ; tÞ ¼ QRi ðtÞ; vxi xi /0þ 2psc T
(A15)
hwi wf
kf Tsc vjf ðxi ; tÞ ¼ QLi ðtÞ: vxi xi /0 2psc T
(A16)
xi /0þ
¼ QLi ðtÞ;
(A6)
With the dimensionless variables defined in Table 1, Eqs. (A14)e(A16), and (A7)e(A9) can be given in a dimensionless form:
v2 jfD ðxDi ; tD Þ 2p qfD ðxDi ; tD Þ ¼ 0; LfLDi < xDi < LfRDi ; 2 CfD hwDi vxDi
xi /0
where
(A17) ZLfRi
QRi ðtÞ ¼
qf ðxi ; tÞdxi ; 0
(A7)
vjfD ðxDi ; tD Þ 2p ¼ Q ðt Þ; vxDi CfD hwDi DRi D xDi /0þ
(A18)
J. Ren, P. Guo / Journal of Natural Gas Science and Engineering 24 (2015) 35e51
vjfD ðxDi ; tD Þ 2p ¼ Q ðt Þ; vxDi CfD hwDi DLi D xDi /0
(A19)
LfRDi Z
QDRi ðtD Þ ¼
qfD ðxDi ; tD ÞdxDi ;
(A20)
0
Z0 QDLi ðtD Þ ¼
qfD ðxDi ; tD ÞdxDi ;
(A21)
LfLDi m X ½QDLi ðtD Þ þ QDRi ðtD Þ ¼ 1:
(A22)
i¼1
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