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A composite model to analyze the decline performance of a multiple fractured horizontal well in shale reservoirs
Journal of Natural Gas Science and Engineering 26 (2015) 999e1010
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A composite model to analyze the decline performance of a multiple fractured horizontal well in shale reservoirs Deliang Zhang*, Liehui Zhang, Yulong Zhao, JingJing Guo State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 February 2015 Received in revised form 21 July 2015 Accepted 22 July 2015 Available online 26 July 2015
In this work, we present a composite model that considers the effect of gas desorption to describe the fluid flow performance of a hydraulic multistage fractured horizontal (MFH) well with stimulated reservoir volume (SRV) in shale. Based on the Langmuir adsorption isotherm, Fick's law and dualporosity idealization, this MFH well model for shale gas reservoirs is tailored to our problem conditions and solved using discrete numerical methods. Then, Stehfest's numerical algorithm and the Gauss elimination method are used to obtain production decline and pressure transient type curves. The main flow regimes of shale gas MFH wells are identified with the following characteristics: wellbore storage, linear flow in the SRV region, diffusion flow and later pseudo-radial flow periods. Sensitivity analyses show that the transition flow regime and the production performance of an MFH well in shale are mainly affected by the Langmuir volume, radius and permeability of the SRV region. This study provides some insights into the mechanisms of shale gas flow and assists in understanding the production decline dynamics in shale reservoirs. © 2015 Elsevier B.V. All rights reserved.
Keywords: Shale gas reservoir Pseudo-steady & transient diffusion Multi-stage fractured horizontal (MFH) well Stimulated reservoir volume (SRV)
1. Introduction Natural gas trapped in shale formations is generally defined as a shale reservoir which is characterized by self-generating and selfpreserving hydrocarbon-rich depositions with clusters of discontinuous micro-fractures and extremely tight matrix. Since the beginning of this century, shale gas has played an increasingly important role in the energy revolution of the United States, and interest in shale has quickly spread worldwide. In contrast to conventional reservoirs that contain free gas, the majority of shale gas is adsorbed on the surfaces of rock grains (either as kerogen or minerals) in shale reservoirs. As reported, more than 85% of shale gas is adsorbed on the surfaces of the organic material and matrix particles, while the other 15% is held as free gas in natural fractures and pore spaces (Hill and Nelson, 2000). Owing to the extremely low permeability of shale matrix, the flow mechanisms at nano-Darcy scale must be described using both Fick's law and the Darcy formula, defined as a dualmechanism (Javadpour et al., 2007, 2009; Ozkan and Raghavan, 2010). Considered significant proportion of absorbed gas and
* Corresponding author. E-mail address: [email protected] (D. Zhang). http://dx.doi.org/10.1016/j.jngse.2015.07.034 1875-5100/© 2015 Elsevier B.V. All rights reserved.
diffusion flow in shale, Zhang et al. (2015) presented a new apparent permeability formulation. As shown in Fig. 1, the gas transportation in shale is summarized as: desorption in the matrix core, free gas flow in the matrix and free gas flow in the fractures. Field practices indicate that shale gas reservoirs are always extracted through multi-stage fractured horizontal (MFH) wells (Fig. 2). In shale gas reservoirs, hydraulic fracturing not only creates high-conductivity flow paths, but also connects discrete natural fractures (Clarkson, 2013). Scholars have presented many models to describe the flow dynamics and mechanisms of MFH wells. For instance, Larsen and Hegre (1991, 1994) determined transient flow regimes by plotting logelog type curves. Guo et al. (1994) presented an analytical method to predict the performance of MFH wells. However, the interference between fractures is neglected in these models, leading to inaccuracies in production and pressure calculations. To overcome this deficiency, Horne and Temeng (1995) accounted for the interference through the superposition of influence functions. Based on the same idea, Raghavan et al. (1997) determined the characteristic response of MFH wells and discerned the effects of the number, location, and fracture orientations. As extensions of previous works, Wan and Aziz (2002) and Crosby et al. (2002) used semi-analytical methods to obtain the transient pressure behavior of an MFH well considering hydraulic fractures at arbitrary angles.
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Fig. 1. Illustration of shale gas transport mechanisms (Zhao et al., 2013).
Fig. 2. MFH well in a shale gas reservoir.
Nevertheless, in these models, the flow rate of each fracture wing is assumed to be uniform. This assumption is inconsistent with actual field behavior. By dividing the fracture wings into several segments with different flow rates, Zerzar et al. (2003) obtained a comprehensive solution for MFH wells by using the boundary element method to solve partial different equations in the Laplace domain. Because the mechanisms describing the fracture network system are still immature, the effects of the stimulated region are not reflected in the model. Although previous works have presented the basic methods for solving the flow problem and are important for understanding the flow mechanisms of MFH wells, the characteristics of shale gas reservoirs (diffusion, desorption, SRV region) have not been completely considered. With the development of unconventional reservoirs, researchers have tried to extend previous models to describe gas transportation in shale by combining them with the features of unconventional reservoirs. Kucuk and Sawyer (1980) first developed a shale gas productivity model using an analytical method without considering the effect of desorption and diffusion. Carlson and Mercer (1991) studied the pressure behavior of a vertical well in shale by introducing the effects of desorption and diffusion into the classical dual-porosity model. Ozkan and Raghavan (2010) joined diffusion, stress-sensitivity and Darcy flow in shale gas to obtain a dual-porosity, dual-mechanism vertical well pressure transient model. However, the flow pattern in the model is assumed to be linear, which is not always consistent with field behavior. With the help of a numerical simulator, Freeman (2010) and Cheng (2011) examined the effects of gas desorption and discerned the typical flow regimes of an MFH well. Imad et al. (2011) developed a composite model that combined an outer
zone of single porosity with an inner zone of dual-porosity but ignored the diffusion and desorption in shale. In addition, onedirection linear flow is assumed in both the inner and outer regions, which is acceptable for certain reservoir scales. To simulate the effect the of the SRV region, Brown et al. (2009), Ozkan et al. (2011) utilized the concept posited by Lee and Brockenbrough (1986), which used a tri-linear model with inner zones of natural fractures to represent MFH well performance in unconventional reservoirs. However, without considering the desorbed gas, the model is more suitable for low permeability gas reservoirs. By incorporating inter-porosity flow from matrix to fractures, Brown et al. (2011) divided the drainage area into three linear flow regions and assumed the SRV region to be a dual-porosity medium to simulate the production dynamic and pressure behavior of an MFH well. Zhao et al. (2013) presented a tri-porosity analytical method to solve the drainage flow in an MFH well. The formation was assumed to be a tri-porosity medium over the entire flow region. Actually, the unstimulated zone is more appropriate for treatment as dual-porosity medium. Wang (2013) established a pressure transient model for an MFH well in shale, which considered the diffusive flow, Langmuir desorption, viscous flow and stress sensitivity of the reservoir permeability. Because the model failed to divide reservoir into the stimulated and unfractured zones, the effects of SRV on shale gas transport were not discussed. Zhao et al. (2014a) extended a composite MFH well model to describe the SRV in shale gas reservoirs but did not consider the effects of the Langmuir volume and Knudsen flow. Huang et al. (2015) considered gas desorption and diffusion flow to build an analytical model to research the transient pressure behavior of a fractured well in shale gas reservoirs, however, the
D. Zhang et al. / Journal of Natural Gas Science and Engineering 26 (2015) 999e1010
assumptions of homogeneous dual-porosity formation could not reflect the effects of the SRV system. Historical works have examined the transient pressure behavior of multiple fractured horizontal wells in shale gas and unconventional reservoirs through various models. However, diffusion flow, desorption gas and stimulated reservoir volume have not been considered. Additionally, previous works simplified the fracture network into a dual-porosity medium reservoir or homogeneous medium composite reservoir. Both of these simplifications cannot completely reflect the objective practice of shale hydraulic fracturing. Because natural micro-fractures and induced macrofractures exist in the rock, and to simplify the fractured and unfractured zones, a flow region divided into two radial zones for a dual-porosity medium is more appropriate for shale gas reservoirs, which is termed the radial composite model (Fig. 3). Moreover, historical studies have concentrated mainly on the pressure response and flow regime identification rather than the production decline performance. In this study, we propose a composite dual-porosity model to describe the fluid flow in a multiple fractured horizontal well in shale reservoirs. This model considers diffusion flow, desorption gas and stimulated reservoir volume. Other than identifying the flow regime, this model is mainly used to analyze the production decline performance. Source function and Gauss elimination method are used to solve the model in the Laplace domain. Then, with the assistance of the Stehfest (1970) numerical algorithm, the production decline and pressure transient type curves are obtained in the real domain. The main flow regimes of an MFH well in shale are identified as follows: wellbore storage, linear flow in the SRV region, diffusion flow and later pseudo-radial flow periods. Sensitivity analyses show that the Langmuir volume, radius and
1001
permeability in the SRV region have a great impact on the transition flow regime and production performance. 2. Mathematical model 2.1. Model assumption To facilitate the derivation of the analytical solution to the problem, some simplifications are assumed: 1) The model is for isothermal single-phase shale gas flow. 2) Natural fractures are the main flow paths and are full of free gas. Additionally, the absorbed gas is trapped on the particle surfaces of the shale matrix. 3) The inter-porosity transfer from the matrix to the net fracture system can be described by Fick's first and second laws of diffusion, which correspond to the pseudo-steady and transient diffusion mechanisms, respectively. 4) A dual-porosity composite model is used to simulate the performance of the SRV in shale. The primary flow path both in the inner and outer regions of this model is the fracture system, but the flow capacity in the inner region is much greater than that in the outer region as a result of the SRV. 5) Infinite-conductivity hydraulic fractures are assumed to be distributed along the horizontal well at equal spacing.
2.2. Continuous linear source function of composite model The basic solution to the governing flow equations of the composite dual-porosity model was presented in detail by Zhao et al.
Fig. 3. Hydraulic multiple fractured horizontal well with SRV and its presentation models (Zhao et al., 2014a).
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(2014b) (Fig. 4). To solve the pressure transient analysis model for a vertical fractured well in coal seam reservoirs, the previous work derived the continuous line source function, which considered diffusion and desorption in detail using Fick's law and the Langmuir theory. Based on Zhao's work, the fully penetrating continuous line source function can be written in the Laplace domain as
Dj1f
psc T 1 ½K ðg r Þ þ AC I0 ðg1 rD Þ ¼ qscL Tsc 86:4pkf1 h 0 1 D
(1)
M g K ðg r ÞK ðg r Þ g2 K0 ðg1 rmD ÞK1 ðg2 rmD Þ where AC ¼ 12 1 1 1 mD 0 2 mD M12 g1 I1 ðg1 rmD ÞK0 ðg2 rmD Þ þ g2 I0 ðg1 rmD ÞK1 ðg2 rmD Þ qffiffiffiffiffiffiffiffiffiffi g1 ¼ f1 ðsÞ;
f1 mgi ct1 f ; u1 ¼ L 8 kf2 t > > > < 2 LL l¼ > k t > > : f2 2 6LL 8 2 Rm > > > < D t¼ > R2 > > : m p2 D
qffiffiffiffiffiffiffiffiffiffi g2 ¼ f2 ðsÞ
8 pffiffiffiffiffi i u1 s b1 s hpffiffiffiffiffi > > ls coth ls 1 > < M þ lM 12 12 f1 ðsÞ ¼ > u s b ss > 1 > þ 1 : M12 M12 ls þ 1 8 pffiffiffiffiffi i b2 s hpffiffiffiffiffi > > ls coth ls 1 < u2 s þ l f2 ðsÞ ¼ > ss > : u2 s þ b2 ls þ 1
b1 ¼ ð1 u1 u2 Þ 1 f1f ;
f2 mgi ct2 f u2 ¼ L
for transient diffusion for pseudo steady diffusion
for transient diffusion for pseudo steady diffusion
for transient flow for pseudo steady diffusion
for transient flow for pseudo steady diffusion
b2 ¼ ð1 u1 u2 Þ 1 f2f
VL jL pf aqsc psc T ih i s¼h jL pf þ j pf jL pf þ jðpi Þ kf2 hTsc
L¼
8 > > m c f > gi f tf < > > > : ff mgi ctf
1þ2
1þ2
þ
6kf2 h aqsc
for transient diffusion
þ
2kf2 h aqsc
for pseudo steady diffusion
Notably, the effects of diffusion and desorption are reflected by sand L, respectively. To describe the advantages of the SRV, the
Fig. 4. A schematic of a continuous line source in a radial composite reservoir (Zhao et al., 2014b).
D. Zhang et al. / Journal of Natural Gas Science and Engineering 26 (2015) 999e1010
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having infinite conductivity in low permeability gas reservoirs). To obtain the well bottom pressure solution, we employed the source function method and divided each half fracture wing into MM sections of length DLfi, to yield a total of MM 2 N discrete sections. For the given segment and time, the flux density is ~fi , may be assumed to be constant. Therefore, the flux density, q regarded as uniform in each segment. The total production will naturally be the summation of the flow rates from each section: MM*2N X
~fi DLfi ¼ qsc q
(2)
i¼1
The dimensionless discrete fracture rate is defined as
qDi ¼ Fig. 5. Discretion of the intercepting fractures.
~fi DLfi q q ¼ fi qsc qsc
(3)
Taking the Laplace transform, Eq. (3) can be written as reservoir parameters for the inner zone are set to be much better than those in the outer zone.
MM*2N X i¼1
2.3. Model solution for an MFH well Owing to the complexity and nonlinearity of the model, the existing methods are ill suited to obtain an accurate analytical so-
Djwi;j
qfj psc T qsc ¼ DL fj T 86:4pq sc sc kf 1 h s L
þ AC I0
DLfDj =2
Z
" K0
DLfDj =2
Z
sDLfDj M12 þ A C I0
Based on the source function, the pseudo-pressure response Djwi;j at element i caused by the production from element j can be expressed as
DLfDj =2
ffi!# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 xDi xwDj a þ yDi ywD j da g1
1
" K0
(4)
ffi! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 xDi xwDj a þ yDi ywD j g1
lution. Therefore, according to the conservation of mass and connectivity equations, a semi-analytical method is used to discretize the linear source along the fractures before solving the coupled conditions. The MFH well model assumes that the horizontal wellbore is intersected by N fractures, each with distinct properties and the ability to be distributed at any position along the wellbore. Additionally, all fractures are oriented transverse to the well and fully penetrate to the formation (Fig. 5). To simplify the calculation, we set the fractures to be uniform and equally spaced, which corresponds to the majority of field practices. In addition, we make the assumption that the flow from the reservoir to the horizontal well is insignificant compared with the flow in the hydraulic fracture
j0wDi;j ¼
qDi 1 ¼ s s
(5)
The dimensionless pressure is defined by
jD ¼
kf2 hTsc Dj 3:684 103 qscL psc T
(6)
Combining Eq. (5) with Eq. (6), we obtain the following:
jwDi;j ¼ qfDj j0wDi;j
(7)
where
ffi! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 xDi xwDj a þ yDi ywD j g1
DLfDj =2
ffi!# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 xDi xwDj a þ yDi ywD j da g1
plane. The fractures are assumed to be produced at a constant well bottom pressure (the horizontal well may always be treated as
Using the method of superposition, the pseudo-pressure response Djwi at element i can be written as
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MM*2N X
jwDi ¼
jwDi;j ¼
MM*2N X
j¼1
0 qDf j jwDi;j
(8)
j¼1
Because the flow capacity in hydraulic fractures and horizontal wellbore is much higher than in the formation, which is extremely obvious in shale reservoirs, we can treat them as having infinite conductivity. Therefore,
jwD ¼ jwDi¼1/MM*2N
(9)
By applying Eq. (8) at each element and combining it with the flux conservation condition described in Eq. (4), we obtain a 2N MMþ1 linear equation system, which can solve the 2N MMþ1 unknowns of jwD qD1 qD2 ,//qDM2N1 ,qDM2N . The matrix for this can be expressed as
2
0
jwD1;1 :: 0 jwDk;1 ::
6 6 6 6 6 6 0 4j
wD2NM;1
1
0
: jwD1;k : :: 0 : jwDk;k : :: 0 : jwD2NM;k : 1
productivity because they reflect the flow characteristics and reservoir properties, such as permeability, control radius, skin factor and fracture length. In the next section, with the help of Stehfest's algorithm, the production decline and type curves are plotted in the real domain. Sensitivity analyses demonstrate that the transition flow regime and production performance of the MFH wells in shale are mainly affected by the Langmuir volume, radius and permeability of the SRV region.
0
: jwD1;2NM : :: 0 : jwDk;2NM : :: 0 : jwD2NM;2NM : 1
32 3 2 3 qfD1 0 1 6 7 6 7 1 7 76 qfD2 7 6 0 7 6 7 6 :: 7 , 1 7 76 7¼6 7 6 7 6 :: 7 , 1 7 76 7 6 7 1 54 qfDM2N 5 4 0 5 1 jwD 0
3. Type curves and production decline curves The dimensionless bottom-hole pseudo-pressure (jwD) and its derivative (djwD/dtD) can be obtained by using a numerical algorithm (Stehfest, 1970) to invert jwD back into real space. Therefore, we can plot the standard logelog type curves of jwD and j’wD* tD/CD vs. tD/CD (Fig. 6). The standard rate decline curves of qD vs. tD can be expressed as
qD ¼
(10)
Skin s
(11)
where jSD is the pseudo-pressure response at the MFH well bottom with the skin effect defined in Laplace space and Skin is the dimensionless skin coefficient. Ultimately, the wellbore storage effect can be easily considered by the identity (Everdingen and Hurst, 1949):
jwD ¼
jSD 1 þ CD s2 jSD
(13)
where
When considering the effect of the skin factor (Skin),
jSD ¼ jwD þ
1 s2 jwD
(12)
where jwD is the pseudo-pressure response at the MFH well bottom with the storage effect defined in Laplace space and CD ¼ 2pf Cc L2 is f 1 tf1 the dimensionless wellbore storage coefficient. The production decline and type curves are extremely important for engineers to recognize the flow regimes and predict
qD ¼
aqsc psc T kf2 hTsc jfi jwf
After this procedure, the Blasingame type curves, which are a powerful instrument for rate decline analyses (Blasingame et al., 1991; Blasingame and Lee, 1994), can be obtained using the following expressions: Dimensionless decline time:
tDd ¼
t 2 D 0:5 reD 1 ½lnðreD Þ 0:5
(14)
Dimensionless decline rate function:
qDd ¼ qD ½lnðreD Þ 0:5 Dimensionless decline rate integral:
Fig. 6. Comparison of the pseudo-steady and transient diffusion models on well test type curves.
(15)
D. Zhang et al. / Journal of Natural Gas Science and Engineering 26 (2015) 999e1010
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Table 1 Synthetic data used for the discussion of the results. Initial reservoir pressure, pi, MPa Formation thickness, h, m Gas specific gravity, gg, fraction Radius of inner region, rm Skin factor, Skin Langmuir pressure, PL(MPa) Inner region Permeability, kf1, mD Porosity, ∅f2 Total compressibility, Ctf1, MPa-1
Z qDdi ¼
tDa 0
15 50 0.65 1000 0.01 1.5
Reservoir temperature, T, K Half fracture length, xf, m Well production rate, qsc, 104 m3/d Wellbore storage coefficient, CD The sorption time constant, t, h Langmuir Volume, VL (m3/m3) Outer region Permeability, kf2, mD Porosity, ∅f2 Total compressibility, Ctf2, MPa-1
0.1 0.02 1.0E-02
qDd ðtÞdt (16)
tDa
Dimensionless decline rate integral derivative:
q0Ddi ¼
dqDdi dlnðtDa Þ
(17)
3.1. Type curves for an MFH well in shale In this section, we will analyze the pressure response from the mathematical model derived in this paper. The basic synthesis data are given in Table 1. The results are shown in Fig. 6. Fig. 6 shows the type curves of pseudo-pressure and the derivative type curves of pseudo-pressure for an MFH well with the induced SRV in a shale gas reservoir, which may be used to analyze the transient pressure to recognize the fluid flow characteristics in the reservoir. The curves may be cataloged into the following flow periods. Stage 1: Early wellbore storage period; in this stage, the curve exhibits a straight line with a unit slope for both the pseudopressure and pseudo-pressure derivative curves. Stage 2: A transition period occurs from the wellbore storage to the linear flow period in the SRV, which is characterized by a hump in the pseudo-pressure derivative curves. Stage 3: Linear flow exists in the SRV region, which is characterized by a slope of 1/2 in both the pseudo-pressure and pseudo-pressure derivative curves on a logelog scale. Because
Stage 6: A later pseudo radial flow period; during this period, the fluid transfer between the matrix and fractures reaches a dynamically balanced state. This is characterized by a horizontal line with a value of 0.5 on the derivative curves. It is noted that the inter-porosity flow period in Fig. 6 characterized by a “valley” on the derivative curve is masked by the transition period. To prove the existence of this flow period and analyze its characteristics, we plot another type curve with an SRV radius assumption of 3000 m (shown as the dashed line in Fig. 6). In
Transient diffusion
Production, qsc (104m3/d)
Pseudo-steady diffusion Conventional reservoir
10
1
1.0E+0
0.02 0.01 1.0E-02
the duration is very short, it is sometimes hidden by the large wellbore storage coefficient CD. Fig. 6 shows the pseudo-steady diffusion and transient diffusion curves. By comparing the two diffusion mechanisms in this stage, we see that transient diffusion causes a higher pressure drop when produced at a constant production rate, meaning that more energy is consumed in the liner flow duration. We note that linear flow is the optimal flow mode because it has the lowest flow resistance among all flow regimes. This reason can also explain why MFH well technology is a powerful tool for the exploitation of unconventional reservoirs. Stage 4: A later pseudo-radial flow period in the SRV region; this period is extremely sensitive to the fracture length and the SRV area radius and is characterized by a horizontal line with the value of 1/(2 M12). Most of the time, it is theoretically existed for the ratio of the half fracture length to the SRV radius is not large enough. In this paper, the ratio is set to 0.075, which ensures that the duration is easily observed. Stage 5: A transition period occurs from the pseudo-radial flow period in the SRV region to the later pseudo-radial flow period in the outer region.
100
0 1.0E-1
330 75 5 105 2 105 40
1.0E+1 1.0E+2 Time, t (Day)
1.0E+3
1.0E+4
Fig. 7. Comparison of the pseudo-steady and transient diffusion models on the production decline.
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addition, the “valley” is easily observed under the pseudo-steady diffusion model but is not obvious for transient diffusion model. 3.2. Production decline curves for an MFH well in shale Fig. 7 shows the production decline curves of an MFH well with SRV in a shale gas reservoir at constant bottom-hole pressure, and compares this to the difference between the transient diffusion and pseudo-steady diffusion in a conventional reservoir. The synthesis data are listed in Table 1. The lines with markers are obtained by the model presented in Sec. 2. The dashed line is obtained using the homogeneous model without the desorbed gas and SRV area. Based on Fig. 7, we may safely draw the conclusion that the production of pseudo-steady diffusion is higher than that of transient diffusion, which is consistent with the analysis of the type curves. In the pseudo-steady diffusion model, the gas adsorbed on the particle surfaces can desorb and diffuse into the fracture system instantaneously. However, for the transient diffusion model, gas diffusion requires some time to complete. Therefore, compared with the transient diffusion model, the pressure drop associated with pseudo-steady state diffusion is relatively small, which leads to higher production when the inner boundary condition is set to be constant pressure. In the later period, the transient diffusion reaches the level of pseudo-steady diffusion, and their production gradually becomes identical. Additionally, because the desorbed gas can provide a continuous supplement, the difference between the conventional and unconventional models is much more obvious in the later period. 3.3. Sensitivity analyses To understand the effect of the parameters on well performance and obtain more accurate results during well testing, we discuss the sensitivity of some important properties in the model. The basic synthetic data are given in Table 1. To clearly display the differences, all production decline curves are drawn with dimensional variables in the real domain. 3.3.1. Langmuir volume (VL) The Langmuir volume is defined as the maximum amount of gas adsorbed by a unit volume shale core at infinite pressure. Figs. 8 and 9 show the effect of the Langmuir volume (VL) on the well testing type curves and production decline performance,
respectively, in an infinite reservoir. It is clearly observed that VL mainly influences the curves in the middle and later flow periods, which includes the inter-porosity flow period. The larger VL is, the greater production rate for a constant pressure drop, or less pressure drop occurs for a constant production rate. Fig. 8 shows in detail that the production curve with less VL declines quickly after 1e2 years. Additionally, if we set the economic limit production as 104 m3/d, when VL is 40 m3/m3, the production duration will last approximately 2000 days. However, when VL is 0 m3/m3, the duration is reduced by half. 3.3.2. SRV region radius (rm) The variable rm is defined as the inner region radius, which is used as the SRV region radius. Figs. 10 and 11 show the influence of SRV radius (rm) on the well pressure and production decline performance. The SRV, which always improves the well performance dramatically, can be defined as a reservoir region containing the effectively stimulated fracture network and the main hydraulic fractures. This concept was first proposed for an unconventional reservoir and has an important impact on pressure and production performance. For the well testing type curves, rm mainly affects the pseudo-radial flow period of the SRV region. As rm increases from 1000 m to 2000 m, the duration of the pseudo-radial flow will be longer, which also means less pressure consumed when produced at a constant flow rate. When the well produces at a constant bottom-hole pressure, a larger rm results in a better stimulation and larger flow surface which definitely leads to greater production. 3.3.3. Permeability in SRV region (kf1) The variable kf1 is the permeability in the SRV region. Figs. 12 and 13 show the pressure and production performance for different permeability in the SRV region (kf1). The larger kf1 is, the smaller the pressure drop and lower the derivative curves are on the logelog plot. In other words, better fracture effects will result in a lower energy loss from seepage. When produced at a constant bottom-hole pressure, the large value of kf1 will lead to a large production rate. The effect of kf1 is more evident at early times. Notably, we can gain more insight from the detail in Fig. 15. As kf1 increases from 0.5 mD to 2 mD, the cumulative stimulated production improves only by approximately 7% (the yellow (in the web version) region in Fig. 13), which is much less than the stimulated production when kf1 increases from 0.1 mD to 0.5 mD (the green region in Fig. 13). This interesting phenomenon implies that there is
1.0E+0
ψWD , ψ’WD tD/CD
Dimensionless pseudo-pressure and derivative
1.0E+1
1.0E-1
1.0E-2 VL=0 VL=20
1.0E-3 1.0E-3
VL=40
1.0E-1
1.0E+1 1.0E+3 Dimensionless time, tD/CD
1.0E+5
Fig. 8. Effect of Langmuir volume (VL) on well testing type curves.
1.0E+7
D. Zhang et al. / Journal of Natural Gas Science and Engineering 26 (2015) 999e1010
100
1007
VL=0
Production, qsc (104m3/d)
VL=20 VL=40
10
1
0 1.0E+0
1.0E+1
1.0E+2
1.0E+3
1.0E+4
Time, t (Day) Fig. 9. Effect of Langmuir volume (VL) on well production performance curves.
rm=1000m rm=1500m rm=2000m
1.0E+0
ψWD , ψ’WD tD/CD
Dimensionless pseudo-pressure and derivative
1.0E+1
1.0E-1
1.0E-2
1.0E-3 1.0E-3
1.0E-1
1.0E+1 1.0E+3 Dimensionless time, tD/CD
1.0E+5
1.0E+7
Fig. 10. Effect of SRV radius (rm) on the pseudo-pressure and its derivative curves.
100
rm=1000m
Production, qsc (104m3/d)
rm=1500m rm=2000m
10
1
0 1.0E+0
1.0E+1
1.0E+2
1.0E+3
Time, t (Day) Fig. 11. Effect of SRV radius (rm) on the well production performance.
1.0E+4
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kf1=0.1mD kf1=0.5mD
1.0E+0
ψWD , ψ’WD tD/CD
Dimensionless pseudo-pressure and derivative
1.0E+1
kf1=2mD
1.0E-1
1.0E-2
1.0E-3
1.0E-4 1.0E-3
1.0E-1
1.0E+1 1.0E+3 Dimensionless time, tD/CD
1.0E+5
1.0E+7
Fig. 12. Effect of induced fracture network permeability in the SRV on pseudo-pressure and its derivative curves.
Production, qsc (104m3/d)
100
10
1 kf1=0.1mD kf1=0.5mD kf1=2mD
0 1.0E-1
1.0E+0
1.0E+1
1.0E+2
1.0E+3
Time, t (Day) Fig. 13. Effect of kf1 on the well production performance.
Fig. 14. Comparison of well testing type curves in homogeneous and composite reservoirs.
1.0E+4
D. Zhang et al. / Journal of Natural Gas Science and Engineering 26 (2015) 999e1010
1009
Fig. 15. Comparison of production performances in homogeneous and composite reservoirs.
an optimal SRV permeability in shale gas extraction, and the blind pursuit of higher fracture conductivity is not affordable or reasonable. We should therefore try our best to find a balance between obtaining better benefits and pursuing higher conductivity (McGuire and Sikora, 1960).
3.4. Comparison of homogeneous and composite reservoirs Figs. 14 and 15 compare the type curves and production performance of an MFH well in homogeneous and composite reservoirs. In general, the existence of the SRV can obviously improve the flow capacity, which leads to a less pressure drop and pressure derivative on the logelog scale for a constant rate and higher production under the constant pressure well condition. For example, with the consideration of SRV effects (kf1 ¼ 5 kf2 ¼ 0.1 mD), after 100 days, the production rate is nearly 3.2 104 m3/d, however, for a well without SRV (kf1 ¼ kf2 ¼ 0.02 mD), the rate is only 0.8 104 m3/d.
(3) The Langmuir volume (VL) mainly influences the well productivity in the later flow period. In the inter-porosity flow period especially, a large VL means that more gas will be desorbed at the same reservoir temperature and pressure, which will result in a higher production rate with a constant pressure drop. Acknowledgments This work is supported by the National Science Fund for Distinguished Young Scholars of China (Grant No. 51125019), the National Natural Science Foundation of China (Grant No. 51404206), the National Basic Research Program of China (Grant No. 2014CB239205) and the Scientific Research Fund of Sichuan Provincial Education Department (Grant No. 14ZA0038). Nomenclature cg ctf1
4. Conclusions ctf2 This paper investigates the transient pressure response and performance of a multistage fractured horizontal (MFH) well in a composite shale gas reservoir and analyzes the effects of the stimulated reservoir volume (SRV) and its characteristic properties. The main conclusions of this paper are as follows: (1) By dividing the flow region into two radial composite zones to reflect the effect of the SRV region and using the continuous linear source function to consider desorption and diffusion, a mathematical model was established to describe the flow behavior of MFH wells in shale, which is observed as an extension of the conventional MFH well model. (2) The fracture network may contribute significantly to the well production performance in shale gas reservoirs. A higher SRV region permeability (kf1) represents not only a higher flow rate but also an earlier decline. In a shale gas reservoir, there is an optimal SRV permeability. The blind pursuit of higher fracture conductivity is not affordable or reasonable. We should try to find a balance between obtaining better benefits and pursuing higher conductivity.
C h kf1 kf2 DLfi L М M12 pf1 pf2 psc ~fi q qfi qsc rm r rm R s
gas compressibility, MPa1 total compressibility in the natural fracture system of the inner region, MPa1 total compressibility in the natural fracture system of the outer region (for composite dual porosity model), MPa1 wellbore storage coefficient, m3/MPa formation thickness, m fracture network permeability in SRV region, D micro-fracture permeability in outer region, D length of the i-th discrete fracture, m half horizontal well length, m molar mass of natural gas, mol/kg mobility ratio, dimensionless fracture network system pressure in SRV, MPa micro-fracture system pressure in outer region, MPa pressure at standard condition, psc ¼ 0.10325 MPa continuous line source flow rate, m3/h production rate of the i-th discrete fracture, m3/d well production rate, m3/d radial distance of the SRV, m radius, m SRV region radius, m universal gas constant, R ¼ 0.008314 (MPa m3)/(K∙kmol) Laplace variable
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Skin t T Tsc VL zg
skin factor, dimensionless time, h temperature, K temperature at standard condition, Tsc ¼ 293 K Langmuir volume, dimensionless gas deviation factor, sm3/m3 j pseudo-pressure, MPa2/cp f formation porosity, fraction ∅f1 fracture network porosity in SRV, fraction ∅f2 micro-fracture porosity, fraction m gas viscosity, cp mgi gas viscosity at the initial reservoir condition, cp u1 storability coefficient in SRV, dimensionless u2 storability coefficient in outer region, dimensionless l dimensionless transfer constants, dimensionless s ad- and desorption coefficient, dimensionless g1, g 2, f1(s), f2(s) variables groups
Subscript D dimensionless Superscript e Laplace transform References Blasingame, T.A., Lee, W.J., 1994. In: . The Variable-rate Reservoir Limits Testing of Gas Wells. Paper SPE 17708-MS presented at SPE Gas Technology Symposium, 13e15 June, Dallas, Texas. Blasingame, T.A., McCray, T.L., Lee, W.J., 1991. In: . Decline Curve Analysis for Variable Pressure Drop/variable Flow Rate Systems. Paper SPE-21513-MS presented at SPE Gas Technology Symposium, 22e24 January, Houston, Texas. Brown, M., Ozkan, E., Raghavan, R., Kazemi, H., 2009. In: . Practical Solutions for Pressure Transient Responses of Fractured Horizontal Wells in Unconventional Reservoirs. http://dx.doi.org/10.2118/125043-MS. Paper SPE 125043 presented at the SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana. Brown, M., Ozkan, E., Raghavan, R., etal, 2011. Practical solutions for pressuretransient responses of fractured horizontal wells in unconventional reservoirs. SPE Reserv. Eval. Eng. 14, 663e676. Carlson, E.S., Mercer, J.C., 1991. Devonian shale gas production: mechanisms and simple models. J. Pet. Technol. 43 (4), 476e482. Cheng, Y.M., 2011. In: . Pressure Transient Characteristics of Hydraulically Fractured Horizontal Shale Gas Wells. Paper SPE149311 presented at the SPE Eastern Regional Meeting. Columbus, Ohio, USA. Clarkson, C.R., 2013. Production data analysis of unconventional gas wells: review of theory and best practices. Int. J. Coal Geol. 109e110, 101e146. Crosby, D.G., Rahman, M.M., Rahman, M.K., et al., 2002. Single and multiple transverse fracture initiation from horizontal wells. J. Pet. Sci. Eng. 35, 191e204. Everdingen, V., Hust, A.F., 1949. The application of the Laplace transformation to flow problems in reservoirs. J. Pet. Technol. 1 (12), 305e324. Freeman, C.M., 2010. In: . A Numerical Study of Microscale Flow Behavior in Tight Gas and Shale Gas. SPE Paper 141125 presented at the SPE Annual Technical Conference and Exhibition, Florence, Italy. Guo, G.L., Evans, R.D., Chang, M.M., 1994. In: . Pressure-transient Behavior for a Horizontal Well Intersecting Multiple Random Discrete Fractures. SPE Paper 28390 presented at SPE Annual Technical Conference and Exhibition, New Orleans, USA.
Hill, David G., Nelson, C.R., 2000. Gas productive fractured shales: an overview and update. Gas. TIPS 6, 4e13. Horne, R.N., Temeng, K.O., 1995. In: . Relative Productivities and Pressure Transient Modeling of Horizontal Wells with Multiple Fractures. SPE Paper 29891 presented at SPE Middle East Oil Show, Bahrain, Bahrain. Huang, Ting, Guo, Xiao, Chen, Feifei, 2015. Modeling transient pressure behavior of a fractured well for shale gas reservoirs based on the properties of nanopores. J. Nat. Gas Sci. Eng. 23, 387e398. Imad, B., Mehran, P.D., Roberto, A., 2011. In: . Modeling Fractured Horizontal Wells as Dual Porosity Composite Reservoirs Application to Tight Gas, Shale Gas and Tight Oil Cases. http://dx.doi.org/10.2118/144057-MS. Paper SPE 144057 presented at the SPE Western North American Regional Meeting, Anchorage, Alaska. Javadpour, F., 2009. Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone). J. Can. Pet. Technol. 48 (8), 16e21. Javadpour, F., Fisher, D., Unsworth, M., 2007. Nanoscale gas flow in shale gas sediments. J. Can. Pet. Technol. 46 (10), 55e61. Kucuk, F., Sawyer, W.K., 1980. In: . Transient Flow in Naturally Fractured Reservoirs and its Application to Devonian Gas Shales. SPE Paper 9397 presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas. Larsen, L., Hegre, R., 1991. In: . Pressure Transient Behavior of Horizontal Wells with Finite Conductivity Vertical Fractures. http://dx.doi.org/10.2118/22076-MS. Paper SPE 22076 presented at the International Arctic Technology Conference, Anchorage, Alaska. Larsen, L., Hegre, R., 1994. In: . Pressure Transient Analysis of Multi-fractured Horizontal Wells. http://dx.doi.org/10.2118/28389-MS. Paper SPE 28389 presented at the SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana. Lee, S.T., Brockenbrough, J.R., 1986. A new approximate analytic solution for finite conductivity vertical fractures. SPE Eval. 1 (1), 75e88. McGuire, W.J., Sikora, V.J., 1960. The effect of vertical fractures on well productivity. J. Pet. Technol. 10 (12), 72e74. Ozkan, E., Raghavan, R., 2010. In: . Modeling of Fluid Transfer from Shale Matrix to Fracture Network. http://dx.doi.org/10.2118/134830-MS. Paper SPE 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 RE.&E. 14 (2), 248e259. http://dx.doi.org/10.2118/121290-PA. Raghavan, R., Chen, Chin-Cheng, Agarwal, Bijin, 1997. An analysis of horizontal wells intercepted by multiple fractures. SPE J. 2 (3), 235e245. Stehfest, H., 1970. Algorithm 368: numerical inversion of Laplace transforms. Commun. ACM 13 (1), 47e49. http://dx.doi.org/10.1145/361953.361969. Wan, J., Aziz, K., 2002. Semi-analytical well model of horizontal wells with multiple hydraulic fractures. SPE J. 437e445. Wang, H.-T., 2013. Performance of multiple fractured horizontal wells in shale gas reservoirs with consideration of multiple mechanisms. J. Hydrol. 510, 299e312. http://dx.doi.org/10.1016/j.jhydrol.2013.12.019. Zerzar, A., Bettam, Y., Tiab, D., 2003. In: . Interpretation of Multiple Hydraulically Fractured Horizontal Wells in Closed Systems. SPE Paper 84888 presented at SPE International Improved Oil Recovery Conference in Asia Pacific, Kuala Lumpur, Malaysia. Zhang, Longjun, Li, Daolun, Lu, Detang, et al., 2015. A new formulation of apparent permeability for gas transport in shale. J. Nat. Gas Sci. Eng. 23, 221e226. Zhao, Y.L., Zhang, L.H., Zhao, J.Z., et al., 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. Zhao, Yu-Long, Zhang, Lie-Hui, Luo, Jian-Xin, et al., 2014a. Performance of fractured horizontal well with stimulated reservoir volume in unconventional gas reservoir. J. Hydrol. 512, 447e456. Zhao, Yu-long, Zhang, Lie-Hui, Feng, Guo-Qing, et al., 2014b. Performance analysis of fractured wells with stimulated reservoir volume in coal seam reservoirs. Oil Gas. Sci. Technol. http://dx.doi.org/10.2516/ogst/2014026.
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A composite model to analyze the decline performance of a multiple fractured horizontal well in shale reservoirs Deliang Zhang*, Liehui Zhang, Yulong Zhao, JingJing Guo State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 February 2015 Received in revised form 21 July 2015 Accepted 22 July 2015 Available online 26 July 2015
In this work, we present a composite model that considers the effect of gas desorption to describe the fluid flow performance of a hydraulic multistage fractured horizontal (MFH) well with stimulated reservoir volume (SRV) in shale. Based on the Langmuir adsorption isotherm, Fick's law and dualporosity idealization, this MFH well model for shale gas reservoirs is tailored to our problem conditions and solved using discrete numerical methods. Then, Stehfest's numerical algorithm and the Gauss elimination method are used to obtain production decline and pressure transient type curves. The main flow regimes of shale gas MFH wells are identified with the following characteristics: wellbore storage, linear flow in the SRV region, diffusion flow and later pseudo-radial flow periods. Sensitivity analyses show that the transition flow regime and the production performance of an MFH well in shale are mainly affected by the Langmuir volume, radius and permeability of the SRV region. This study provides some insights into the mechanisms of shale gas flow and assists in understanding the production decline dynamics in shale reservoirs. © 2015 Elsevier B.V. All rights reserved.
Keywords: Shale gas reservoir Pseudo-steady & transient diffusion Multi-stage fractured horizontal (MFH) well Stimulated reservoir volume (SRV)
1. Introduction Natural gas trapped in shale formations is generally defined as a shale reservoir which is characterized by self-generating and selfpreserving hydrocarbon-rich depositions with clusters of discontinuous micro-fractures and extremely tight matrix. Since the beginning of this century, shale gas has played an increasingly important role in the energy revolution of the United States, and interest in shale has quickly spread worldwide. In contrast to conventional reservoirs that contain free gas, the majority of shale gas is adsorbed on the surfaces of rock grains (either as kerogen or minerals) in shale reservoirs. As reported, more than 85% of shale gas is adsorbed on the surfaces of the organic material and matrix particles, while the other 15% is held as free gas in natural fractures and pore spaces (Hill and Nelson, 2000). Owing to the extremely low permeability of shale matrix, the flow mechanisms at nano-Darcy scale must be described using both Fick's law and the Darcy formula, defined as a dualmechanism (Javadpour et al., 2007, 2009; Ozkan and Raghavan, 2010). Considered significant proportion of absorbed gas and
* Corresponding author. E-mail address: [email protected] (D. Zhang). http://dx.doi.org/10.1016/j.jngse.2015.07.034 1875-5100/© 2015 Elsevier B.V. All rights reserved.
diffusion flow in shale, Zhang et al. (2015) presented a new apparent permeability formulation. As shown in Fig. 1, the gas transportation in shale is summarized as: desorption in the matrix core, free gas flow in the matrix and free gas flow in the fractures. Field practices indicate that shale gas reservoirs are always extracted through multi-stage fractured horizontal (MFH) wells (Fig. 2). In shale gas reservoirs, hydraulic fracturing not only creates high-conductivity flow paths, but also connects discrete natural fractures (Clarkson, 2013). Scholars have presented many models to describe the flow dynamics and mechanisms of MFH wells. For instance, Larsen and Hegre (1991, 1994) determined transient flow regimes by plotting logelog type curves. Guo et al. (1994) presented an analytical method to predict the performance of MFH wells. However, the interference between fractures is neglected in these models, leading to inaccuracies in production and pressure calculations. To overcome this deficiency, Horne and Temeng (1995) accounted for the interference through the superposition of influence functions. Based on the same idea, Raghavan et al. (1997) determined the characteristic response of MFH wells and discerned the effects of the number, location, and fracture orientations. As extensions of previous works, Wan and Aziz (2002) and Crosby et al. (2002) used semi-analytical methods to obtain the transient pressure behavior of an MFH well considering hydraulic fractures at arbitrary angles.
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Fig. 1. Illustration of shale gas transport mechanisms (Zhao et al., 2013).
Fig. 2. MFH well in a shale gas reservoir.
Nevertheless, in these models, the flow rate of each fracture wing is assumed to be uniform. This assumption is inconsistent with actual field behavior. By dividing the fracture wings into several segments with different flow rates, Zerzar et al. (2003) obtained a comprehensive solution for MFH wells by using the boundary element method to solve partial different equations in the Laplace domain. Because the mechanisms describing the fracture network system are still immature, the effects of the stimulated region are not reflected in the model. Although previous works have presented the basic methods for solving the flow problem and are important for understanding the flow mechanisms of MFH wells, the characteristics of shale gas reservoirs (diffusion, desorption, SRV region) have not been completely considered. With the development of unconventional reservoirs, researchers have tried to extend previous models to describe gas transportation in shale by combining them with the features of unconventional reservoirs. Kucuk and Sawyer (1980) first developed a shale gas productivity model using an analytical method without considering the effect of desorption and diffusion. Carlson and Mercer (1991) studied the pressure behavior of a vertical well in shale by introducing the effects of desorption and diffusion into the classical dual-porosity model. Ozkan and Raghavan (2010) joined diffusion, stress-sensitivity and Darcy flow in shale gas to obtain a dual-porosity, dual-mechanism vertical well pressure transient model. However, the flow pattern in the model is assumed to be linear, which is not always consistent with field behavior. With the help of a numerical simulator, Freeman (2010) and Cheng (2011) examined the effects of gas desorption and discerned the typical flow regimes of an MFH well. Imad et al. (2011) developed a composite model that combined an outer
zone of single porosity with an inner zone of dual-porosity but ignored the diffusion and desorption in shale. In addition, onedirection linear flow is assumed in both the inner and outer regions, which is acceptable for certain reservoir scales. To simulate the effect the of the SRV region, Brown et al. (2009), Ozkan et al. (2011) utilized the concept posited by Lee and Brockenbrough (1986), which used a tri-linear model with inner zones of natural fractures to represent MFH well performance in unconventional reservoirs. However, without considering the desorbed gas, the model is more suitable for low permeability gas reservoirs. By incorporating inter-porosity flow from matrix to fractures, Brown et al. (2011) divided the drainage area into three linear flow regions and assumed the SRV region to be a dual-porosity medium to simulate the production dynamic and pressure behavior of an MFH well. Zhao et al. (2013) presented a tri-porosity analytical method to solve the drainage flow in an MFH well. The formation was assumed to be a tri-porosity medium over the entire flow region. Actually, the unstimulated zone is more appropriate for treatment as dual-porosity medium. Wang (2013) established a pressure transient model for an MFH well in shale, which considered the diffusive flow, Langmuir desorption, viscous flow and stress sensitivity of the reservoir permeability. Because the model failed to divide reservoir into the stimulated and unfractured zones, the effects of SRV on shale gas transport were not discussed. Zhao et al. (2014a) extended a composite MFH well model to describe the SRV in shale gas reservoirs but did not consider the effects of the Langmuir volume and Knudsen flow. Huang et al. (2015) considered gas desorption and diffusion flow to build an analytical model to research the transient pressure behavior of a fractured well in shale gas reservoirs, however, the
D. Zhang et al. / Journal of Natural Gas Science and Engineering 26 (2015) 999e1010
assumptions of homogeneous dual-porosity formation could not reflect the effects of the SRV system. Historical works have examined the transient pressure behavior of multiple fractured horizontal wells in shale gas and unconventional reservoirs through various models. However, diffusion flow, desorption gas and stimulated reservoir volume have not been considered. Additionally, previous works simplified the fracture network into a dual-porosity medium reservoir or homogeneous medium composite reservoir. Both of these simplifications cannot completely reflect the objective practice of shale hydraulic fracturing. Because natural micro-fractures and induced macrofractures exist in the rock, and to simplify the fractured and unfractured zones, a flow region divided into two radial zones for a dual-porosity medium is more appropriate for shale gas reservoirs, which is termed the radial composite model (Fig. 3). Moreover, historical studies have concentrated mainly on the pressure response and flow regime identification rather than the production decline performance. In this study, we propose a composite dual-porosity model to describe the fluid flow in a multiple fractured horizontal well in shale reservoirs. This model considers diffusion flow, desorption gas and stimulated reservoir volume. Other than identifying the flow regime, this model is mainly used to analyze the production decline performance. Source function and Gauss elimination method are used to solve the model in the Laplace domain. Then, with the assistance of the Stehfest (1970) numerical algorithm, the production decline and pressure transient type curves are obtained in the real domain. The main flow regimes of an MFH well in shale are identified as follows: wellbore storage, linear flow in the SRV region, diffusion flow and later pseudo-radial flow periods. Sensitivity analyses show that the Langmuir volume, radius and
1001
permeability in the SRV region have a great impact on the transition flow regime and production performance. 2. Mathematical model 2.1. Model assumption To facilitate the derivation of the analytical solution to the problem, some simplifications are assumed: 1) The model is for isothermal single-phase shale gas flow. 2) Natural fractures are the main flow paths and are full of free gas. Additionally, the absorbed gas is trapped on the particle surfaces of the shale matrix. 3) The inter-porosity transfer from the matrix to the net fracture system can be described by Fick's first and second laws of diffusion, which correspond to the pseudo-steady and transient diffusion mechanisms, respectively. 4) A dual-porosity composite model is used to simulate the performance of the SRV in shale. The primary flow path both in the inner and outer regions of this model is the fracture system, but the flow capacity in the inner region is much greater than that in the outer region as a result of the SRV. 5) Infinite-conductivity hydraulic fractures are assumed to be distributed along the horizontal well at equal spacing.
2.2. Continuous linear source function of composite model The basic solution to the governing flow equations of the composite dual-porosity model was presented in detail by Zhao et al.
Fig. 3. Hydraulic multiple fractured horizontal well with SRV and its presentation models (Zhao et al., 2014a).
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(2014b) (Fig. 4). To solve the pressure transient analysis model for a vertical fractured well in coal seam reservoirs, the previous work derived the continuous line source function, which considered diffusion and desorption in detail using Fick's law and the Langmuir theory. Based on Zhao's work, the fully penetrating continuous line source function can be written in the Laplace domain as
Dj1f
psc T 1 ½K ðg r Þ þ AC I0 ðg1 rD Þ ¼ qscL Tsc 86:4pkf1 h 0 1 D
(1)
M g K ðg r ÞK ðg r Þ g2 K0 ðg1 rmD ÞK1 ðg2 rmD Þ where AC ¼ 12 1 1 1 mD 0 2 mD M12 g1 I1 ðg1 rmD ÞK0 ðg2 rmD Þ þ g2 I0 ðg1 rmD ÞK1 ðg2 rmD Þ qffiffiffiffiffiffiffiffiffiffi g1 ¼ f1 ðsÞ;
f1 mgi ct1 f ; u1 ¼ L 8 kf2 t > > > < 2 LL l¼ > k t > > : f2 2 6LL 8 2 Rm > > > < D t¼ > R2 > > : m p2 D
qffiffiffiffiffiffiffiffiffiffi g2 ¼ f2 ðsÞ
8 pffiffiffiffiffi i u1 s b1 s hpffiffiffiffiffi > > ls coth ls 1 > < M þ lM 12 12 f1 ðsÞ ¼ > u s b ss > 1 > þ 1 : M12 M12 ls þ 1 8 pffiffiffiffiffi i b2 s hpffiffiffiffiffi > > ls coth ls 1 < u2 s þ l f2 ðsÞ ¼ > ss > : u2 s þ b2 ls þ 1
b1 ¼ ð1 u1 u2 Þ 1 f1f ;
f2 mgi ct2 f u2 ¼ L
for transient diffusion for pseudo steady diffusion
for transient diffusion for pseudo steady diffusion
for transient flow for pseudo steady diffusion
for transient flow for pseudo steady diffusion
b2 ¼ ð1 u1 u2 Þ 1 f2f
VL jL pf aqsc psc T ih i s¼h jL pf þ j pf jL pf þ jðpi Þ kf2 hTsc
L¼
8 > > m c f > gi f tf < > > > : ff mgi ctf
1þ2
1þ2
þ
6kf2 h aqsc
for transient diffusion
þ
2kf2 h aqsc
for pseudo steady diffusion
Notably, the effects of diffusion and desorption are reflected by sand L, respectively. To describe the advantages of the SRV, the
Fig. 4. A schematic of a continuous line source in a radial composite reservoir (Zhao et al., 2014b).
D. Zhang et al. / Journal of Natural Gas Science and Engineering 26 (2015) 999e1010
1003
having infinite conductivity in low permeability gas reservoirs). To obtain the well bottom pressure solution, we employed the source function method and divided each half fracture wing into MM sections of length DLfi, to yield a total of MM 2 N discrete sections. For the given segment and time, the flux density is ~fi , may be assumed to be constant. Therefore, the flux density, q regarded as uniform in each segment. The total production will naturally be the summation of the flow rates from each section: MM*2N X
~fi DLfi ¼ qsc q
(2)
i¼1
The dimensionless discrete fracture rate is defined as
qDi ¼ Fig. 5. Discretion of the intercepting fractures.
~fi DLfi q q ¼ fi qsc qsc
(3)
Taking the Laplace transform, Eq. (3) can be written as reservoir parameters for the inner zone are set to be much better than those in the outer zone.
MM*2N X i¼1
2.3. Model solution for an MFH well Owing to the complexity and nonlinearity of the model, the existing methods are ill suited to obtain an accurate analytical so-
Djwi;j
qfj psc T qsc ¼ DL fj T 86:4pq sc sc kf 1 h s L
þ AC I0
DLfDj =2
Z
" K0
DLfDj =2
Z
sDLfDj M12 þ A C I0
Based on the source function, the pseudo-pressure response Djwi;j at element i caused by the production from element j can be expressed as
DLfDj =2
ffi!# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 xDi xwDj a þ yDi ywD j da g1
1
" K0
(4)
ffi! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 xDi xwDj a þ yDi ywD j g1
lution. Therefore, according to the conservation of mass and connectivity equations, a semi-analytical method is used to discretize the linear source along the fractures before solving the coupled conditions. The MFH well model assumes that the horizontal wellbore is intersected by N fractures, each with distinct properties and the ability to be distributed at any position along the wellbore. Additionally, all fractures are oriented transverse to the well and fully penetrate to the formation (Fig. 5). To simplify the calculation, we set the fractures to be uniform and equally spaced, which corresponds to the majority of field practices. In addition, we make the assumption that the flow from the reservoir to the horizontal well is insignificant compared with the flow in the hydraulic fracture
j0wDi;j ¼
qDi 1 ¼ s s
(5)
The dimensionless pressure is defined by
jD ¼
kf2 hTsc Dj 3:684 103 qscL psc T
(6)
Combining Eq. (5) with Eq. (6), we obtain the following:
jwDi;j ¼ qfDj j0wDi;j
(7)
where
ffi! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 xDi xwDj a þ yDi ywD j g1
DLfDj =2
ffi!# sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 xDi xwDj a þ yDi ywD j da g1
plane. The fractures are assumed to be produced at a constant well bottom pressure (the horizontal well may always be treated as
Using the method of superposition, the pseudo-pressure response Djwi at element i can be written as
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D. Zhang et al. / Journal of Natural Gas Science and Engineering 26 (2015) 999e1010
MM*2N X
jwDi ¼
jwDi;j ¼
MM*2N X
j¼1
0 qDf j jwDi;j
(8)
j¼1
Because the flow capacity in hydraulic fractures and horizontal wellbore is much higher than in the formation, which is extremely obvious in shale reservoirs, we can treat them as having infinite conductivity. Therefore,
jwD ¼ jwDi¼1/MM*2N
(9)
By applying Eq. (8) at each element and combining it with the flux conservation condition described in Eq. (4), we obtain a 2N MMþ1 linear equation system, which can solve the 2N MMþ1 unknowns of jwD qD1 qD2 ,//qDM2N1 ,qDM2N . The matrix for this can be expressed as
2
0
jwD1;1 :: 0 jwDk;1 ::
6 6 6 6 6 6 0 4j
wD2NM;1
1
0
: jwD1;k : :: 0 : jwDk;k : :: 0 : jwD2NM;k : 1
productivity because they reflect the flow characteristics and reservoir properties, such as permeability, control radius, skin factor and fracture length. In the next section, with the help of Stehfest's algorithm, the production decline and type curves are plotted in the real domain. Sensitivity analyses demonstrate that the transition flow regime and production performance of the MFH wells in shale are mainly affected by the Langmuir volume, radius and permeability of the SRV region.
0
: jwD1;2NM : :: 0 : jwDk;2NM : :: 0 : jwD2NM;2NM : 1
32 3 2 3 qfD1 0 1 6 7 6 7 1 7 76 qfD2 7 6 0 7 6 7 6 :: 7 , 1 7 76 7¼6 7 6 7 6 :: 7 , 1 7 76 7 6 7 1 54 qfDM2N 5 4 0 5 1 jwD 0
3. Type curves and production decline curves The dimensionless bottom-hole pseudo-pressure (jwD) and its derivative (djwD/dtD) can be obtained by using a numerical algorithm (Stehfest, 1970) to invert jwD back into real space. Therefore, we can plot the standard logelog type curves of jwD and j’wD* tD/CD vs. tD/CD (Fig. 6). The standard rate decline curves of qD vs. tD can be expressed as
qD ¼
(10)
Skin s
(11)
where jSD is the pseudo-pressure response at the MFH well bottom with the skin effect defined in Laplace space and Skin is the dimensionless skin coefficient. Ultimately, the wellbore storage effect can be easily considered by the identity (Everdingen and Hurst, 1949):
jwD ¼
jSD 1 þ CD s2 jSD
(13)
where
When considering the effect of the skin factor (Skin),
jSD ¼ jwD þ
1 s2 jwD
(12)
where jwD is the pseudo-pressure response at the MFH well bottom with the storage effect defined in Laplace space and CD ¼ 2pf Cc L2 is f 1 tf1 the dimensionless wellbore storage coefficient. The production decline and type curves are extremely important for engineers to recognize the flow regimes and predict
qD ¼
aqsc psc T kf2 hTsc jfi jwf
After this procedure, the Blasingame type curves, which are a powerful instrument for rate decline analyses (Blasingame et al., 1991; Blasingame and Lee, 1994), can be obtained using the following expressions: Dimensionless decline time:
tDd ¼
t 2 D 0:5 reD 1 ½lnðreD Þ 0:5
(14)
Dimensionless decline rate function:
qDd ¼ qD ½lnðreD Þ 0:5 Dimensionless decline rate integral:
Fig. 6. Comparison of the pseudo-steady and transient diffusion models on well test type curves.
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1005
Table 1 Synthetic data used for the discussion of the results. Initial reservoir pressure, pi, MPa Formation thickness, h, m Gas specific gravity, gg, fraction Radius of inner region, rm Skin factor, Skin Langmuir pressure, PL(MPa) Inner region Permeability, kf1, mD Porosity, ∅f2 Total compressibility, Ctf1, MPa-1
Z qDdi ¼
tDa 0
15 50 0.65 1000 0.01 1.5
Reservoir temperature, T, K Half fracture length, xf, m Well production rate, qsc, 104 m3/d Wellbore storage coefficient, CD The sorption time constant, t, h Langmuir Volume, VL (m3/m3) Outer region Permeability, kf2, mD Porosity, ∅f2 Total compressibility, Ctf2, MPa-1
0.1 0.02 1.0E-02
qDd ðtÞdt (16)
tDa
Dimensionless decline rate integral derivative:
q0Ddi ¼
dqDdi dlnðtDa Þ
(17)
3.1. Type curves for an MFH well in shale In this section, we will analyze the pressure response from the mathematical model derived in this paper. The basic synthesis data are given in Table 1. The results are shown in Fig. 6. Fig. 6 shows the type curves of pseudo-pressure and the derivative type curves of pseudo-pressure for an MFH well with the induced SRV in a shale gas reservoir, which may be used to analyze the transient pressure to recognize the fluid flow characteristics in the reservoir. The curves may be cataloged into the following flow periods. Stage 1: Early wellbore storage period; in this stage, the curve exhibits a straight line with a unit slope for both the pseudopressure and pseudo-pressure derivative curves. Stage 2: A transition period occurs from the wellbore storage to the linear flow period in the SRV, which is characterized by a hump in the pseudo-pressure derivative curves. Stage 3: Linear flow exists in the SRV region, which is characterized by a slope of 1/2 in both the pseudo-pressure and pseudo-pressure derivative curves on a logelog scale. Because
Stage 6: A later pseudo radial flow period; during this period, the fluid transfer between the matrix and fractures reaches a dynamically balanced state. This is characterized by a horizontal line with a value of 0.5 on the derivative curves. It is noted that the inter-porosity flow period in Fig. 6 characterized by a “valley” on the derivative curve is masked by the transition period. To prove the existence of this flow period and analyze its characteristics, we plot another type curve with an SRV radius assumption of 3000 m (shown as the dashed line in Fig. 6). In
Transient diffusion
Production, qsc (104m3/d)
Pseudo-steady diffusion Conventional reservoir
10
1
1.0E+0
0.02 0.01 1.0E-02
the duration is very short, it is sometimes hidden by the large wellbore storage coefficient CD. Fig. 6 shows the pseudo-steady diffusion and transient diffusion curves. By comparing the two diffusion mechanisms in this stage, we see that transient diffusion causes a higher pressure drop when produced at a constant production rate, meaning that more energy is consumed in the liner flow duration. We note that linear flow is the optimal flow mode because it has the lowest flow resistance among all flow regimes. This reason can also explain why MFH well technology is a powerful tool for the exploitation of unconventional reservoirs. Stage 4: A later pseudo-radial flow period in the SRV region; this period is extremely sensitive to the fracture length and the SRV area radius and is characterized by a horizontal line with the value of 1/(2 M12). Most of the time, it is theoretically existed for the ratio of the half fracture length to the SRV radius is not large enough. In this paper, the ratio is set to 0.075, which ensures that the duration is easily observed. Stage 5: A transition period occurs from the pseudo-radial flow period in the SRV region to the later pseudo-radial flow period in the outer region.
100
0 1.0E-1
330 75 5 105 2 105 40
1.0E+1 1.0E+2 Time, t (Day)
1.0E+3
1.0E+4
Fig. 7. Comparison of the pseudo-steady and transient diffusion models on the production decline.
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addition, the “valley” is easily observed under the pseudo-steady diffusion model but is not obvious for transient diffusion model. 3.2. Production decline curves for an MFH well in shale Fig. 7 shows the production decline curves of an MFH well with SRV in a shale gas reservoir at constant bottom-hole pressure, and compares this to the difference between the transient diffusion and pseudo-steady diffusion in a conventional reservoir. The synthesis data are listed in Table 1. The lines with markers are obtained by the model presented in Sec. 2. The dashed line is obtained using the homogeneous model without the desorbed gas and SRV area. Based on Fig. 7, we may safely draw the conclusion that the production of pseudo-steady diffusion is higher than that of transient diffusion, which is consistent with the analysis of the type curves. In the pseudo-steady diffusion model, the gas adsorbed on the particle surfaces can desorb and diffuse into the fracture system instantaneously. However, for the transient diffusion model, gas diffusion requires some time to complete. Therefore, compared with the transient diffusion model, the pressure drop associated with pseudo-steady state diffusion is relatively small, which leads to higher production when the inner boundary condition is set to be constant pressure. In the later period, the transient diffusion reaches the level of pseudo-steady diffusion, and their production gradually becomes identical. Additionally, because the desorbed gas can provide a continuous supplement, the difference between the conventional and unconventional models is much more obvious in the later period. 3.3. Sensitivity analyses To understand the effect of the parameters on well performance and obtain more accurate results during well testing, we discuss the sensitivity of some important properties in the model. The basic synthetic data are given in Table 1. To clearly display the differences, all production decline curves are drawn with dimensional variables in the real domain. 3.3.1. Langmuir volume (VL) The Langmuir volume is defined as the maximum amount of gas adsorbed by a unit volume shale core at infinite pressure. Figs. 8 and 9 show the effect of the Langmuir volume (VL) on the well testing type curves and production decline performance,
respectively, in an infinite reservoir. It is clearly observed that VL mainly influences the curves in the middle and later flow periods, which includes the inter-porosity flow period. The larger VL is, the greater production rate for a constant pressure drop, or less pressure drop occurs for a constant production rate. Fig. 8 shows in detail that the production curve with less VL declines quickly after 1e2 years. Additionally, if we set the economic limit production as 104 m3/d, when VL is 40 m3/m3, the production duration will last approximately 2000 days. However, when VL is 0 m3/m3, the duration is reduced by half. 3.3.2. SRV region radius (rm) The variable rm is defined as the inner region radius, which is used as the SRV region radius. Figs. 10 and 11 show the influence of SRV radius (rm) on the well pressure and production decline performance. The SRV, which always improves the well performance dramatically, can be defined as a reservoir region containing the effectively stimulated fracture network and the main hydraulic fractures. This concept was first proposed for an unconventional reservoir and has an important impact on pressure and production performance. For the well testing type curves, rm mainly affects the pseudo-radial flow period of the SRV region. As rm increases from 1000 m to 2000 m, the duration of the pseudo-radial flow will be longer, which also means less pressure consumed when produced at a constant flow rate. When the well produces at a constant bottom-hole pressure, a larger rm results in a better stimulation and larger flow surface which definitely leads to greater production. 3.3.3. Permeability in SRV region (kf1) The variable kf1 is the permeability in the SRV region. Figs. 12 and 13 show the pressure and production performance for different permeability in the SRV region (kf1). The larger kf1 is, the smaller the pressure drop and lower the derivative curves are on the logelog plot. In other words, better fracture effects will result in a lower energy loss from seepage. When produced at a constant bottom-hole pressure, the large value of kf1 will lead to a large production rate. The effect of kf1 is more evident at early times. Notably, we can gain more insight from the detail in Fig. 15. As kf1 increases from 0.5 mD to 2 mD, the cumulative stimulated production improves only by approximately 7% (the yellow (in the web version) region in Fig. 13), which is much less than the stimulated production when kf1 increases from 0.1 mD to 0.5 mD (the green region in Fig. 13). This interesting phenomenon implies that there is
1.0E+0
ψWD , ψ’WD tD/CD
Dimensionless pseudo-pressure and derivative
1.0E+1
1.0E-1
1.0E-2 VL=0 VL=20
1.0E-3 1.0E-3
VL=40
1.0E-1
1.0E+1 1.0E+3 Dimensionless time, tD/CD
1.0E+5
Fig. 8. Effect of Langmuir volume (VL) on well testing type curves.
1.0E+7
D. Zhang et al. / Journal of Natural Gas Science and Engineering 26 (2015) 999e1010
100
1007
VL=0
Production, qsc (104m3/d)
VL=20 VL=40
10
1
0 1.0E+0
1.0E+1
1.0E+2
1.0E+3
1.0E+4
Time, t (Day) Fig. 9. Effect of Langmuir volume (VL) on well production performance curves.
rm=1000m rm=1500m rm=2000m
1.0E+0
ψWD , ψ’WD tD/CD
Dimensionless pseudo-pressure and derivative
1.0E+1
1.0E-1
1.0E-2
1.0E-3 1.0E-3
1.0E-1
1.0E+1 1.0E+3 Dimensionless time, tD/CD
1.0E+5
1.0E+7
Fig. 10. Effect of SRV radius (rm) on the pseudo-pressure and its derivative curves.
100
rm=1000m
Production, qsc (104m3/d)
rm=1500m rm=2000m
10
1
0 1.0E+0
1.0E+1
1.0E+2
1.0E+3
Time, t (Day) Fig. 11. Effect of SRV radius (rm) on the well production performance.
1.0E+4
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D. Zhang et al. / Journal of Natural Gas Science and Engineering 26 (2015) 999e1010
kf1=0.1mD kf1=0.5mD
1.0E+0
ψWD , ψ’WD tD/CD
Dimensionless pseudo-pressure and derivative
1.0E+1
kf1=2mD
1.0E-1
1.0E-2
1.0E-3
1.0E-4 1.0E-3
1.0E-1
1.0E+1 1.0E+3 Dimensionless time, tD/CD
1.0E+5
1.0E+7
Fig. 12. Effect of induced fracture network permeability in the SRV on pseudo-pressure and its derivative curves.
Production, qsc (104m3/d)
100
10
1 kf1=0.1mD kf1=0.5mD kf1=2mD
0 1.0E-1
1.0E+0
1.0E+1
1.0E+2
1.0E+3
Time, t (Day) Fig. 13. Effect of kf1 on the well production performance.
Fig. 14. Comparison of well testing type curves in homogeneous and composite reservoirs.
1.0E+4
D. Zhang et al. / Journal of Natural Gas Science and Engineering 26 (2015) 999e1010
1009
Fig. 15. Comparison of production performances in homogeneous and composite reservoirs.
an optimal SRV permeability in shale gas extraction, and the blind pursuit of higher fracture conductivity is not affordable or reasonable. We should therefore try our best to find a balance between obtaining better benefits and pursuing higher conductivity (McGuire and Sikora, 1960).
3.4. Comparison of homogeneous and composite reservoirs Figs. 14 and 15 compare the type curves and production performance of an MFH well in homogeneous and composite reservoirs. In general, the existence of the SRV can obviously improve the flow capacity, which leads to a less pressure drop and pressure derivative on the logelog scale for a constant rate and higher production under the constant pressure well condition. For example, with the consideration of SRV effects (kf1 ¼ 5 kf2 ¼ 0.1 mD), after 100 days, the production rate is nearly 3.2 104 m3/d, however, for a well without SRV (kf1 ¼ kf2 ¼ 0.02 mD), the rate is only 0.8 104 m3/d.
(3) The Langmuir volume (VL) mainly influences the well productivity in the later flow period. In the inter-porosity flow period especially, a large VL means that more gas will be desorbed at the same reservoir temperature and pressure, which will result in a higher production rate with a constant pressure drop. Acknowledgments This work is supported by the National Science Fund for Distinguished Young Scholars of China (Grant No. 51125019), the National Natural Science Foundation of China (Grant No. 51404206), the National Basic Research Program of China (Grant No. 2014CB239205) and the Scientific Research Fund of Sichuan Provincial Education Department (Grant No. 14ZA0038). Nomenclature cg ctf1
4. Conclusions ctf2 This paper investigates the transient pressure response and performance of a multistage fractured horizontal (MFH) well in a composite shale gas reservoir and analyzes the effects of the stimulated reservoir volume (SRV) and its characteristic properties. The main conclusions of this paper are as follows: (1) By dividing the flow region into two radial composite zones to reflect the effect of the SRV region and using the continuous linear source function to consider desorption and diffusion, a mathematical model was established to describe the flow behavior of MFH wells in shale, which is observed as an extension of the conventional MFH well model. (2) The fracture network may contribute significantly to the well production performance in shale gas reservoirs. A higher SRV region permeability (kf1) represents not only a higher flow rate but also an earlier decline. In a shale gas reservoir, there is an optimal SRV permeability. The blind pursuit of higher fracture conductivity is not affordable or reasonable. We should try to find a balance between obtaining better benefits and pursuing higher conductivity.
C h kf1 kf2 DLfi L М M12 pf1 pf2 psc ~fi q qfi qsc rm r rm R s
gas compressibility, MPa1 total compressibility in the natural fracture system of the inner region, MPa1 total compressibility in the natural fracture system of the outer region (for composite dual porosity model), MPa1 wellbore storage coefficient, m3/MPa formation thickness, m fracture network permeability in SRV region, D micro-fracture permeability in outer region, D length of the i-th discrete fracture, m half horizontal well length, m molar mass of natural gas, mol/kg mobility ratio, dimensionless fracture network system pressure in SRV, MPa micro-fracture system pressure in outer region, MPa pressure at standard condition, psc ¼ 0.10325 MPa continuous line source flow rate, m3/h production rate of the i-th discrete fracture, m3/d well production rate, m3/d radial distance of the SRV, m radius, m SRV region radius, m universal gas constant, R ¼ 0.008314 (MPa m3)/(K∙kmol) Laplace variable
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Skin t T Tsc VL zg
skin factor, dimensionless time, h temperature, K temperature at standard condition, Tsc ¼ 293 K Langmuir volume, dimensionless gas deviation factor, sm3/m3 j pseudo-pressure, MPa2/cp f formation porosity, fraction ∅f1 fracture network porosity in SRV, fraction ∅f2 micro-fracture porosity, fraction m gas viscosity, cp mgi gas viscosity at the initial reservoir condition, cp u1 storability coefficient in SRV, dimensionless u2 storability coefficient in outer region, dimensionless l dimensionless transfer constants, dimensionless s ad- and desorption coefficient, dimensionless g1, g 2, f1(s), f2(s) variables groups
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