Productivity of blast-fractured wells in liquid-rich shale gas formations

Productivity of blast-fractured wells in liquid-rich shale gas formations

Journal of Natural Gas Science and Engineering 18 (2014) 360e367 Contents lists available at ScienceDirect Journal of Natural Gas Science and Engine...

562KB Sizes 0 Downloads 60 Views

Journal of Natural Gas Science and Engineering 18 (2014) 360e367

Contents lists available at ScienceDirect

Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse

Productivity of blast-fractured wells in liquid-rich shale gas formations Boyun Guo*, Jia Shan, Yin Feng University of Louisiana at Lafayette, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 November 2013 Received in revised form 21 March 2014 Accepted 22 March 2014 Available online

The explosive stimulation by blast-fracturing was first used in well stimulation with pronounced results in the booming Pennsylvania oil field in 1865. The advent of hydraulic fracturing in the 1950’s caused explosive stimulation of oil wells to decline dramatically. Because the hydraulic fracturing process consumes a huge amount of water and imposes a threat to the ground water resources, it is highly desirable to revisit the feasibility of replacing hydraulic fracturing with blast-fracturing in the oil and gas well completion processes. This paper presents an analysis of well productivity of different types of well architectures to be completed with the blast-fracturing stimulation in liquid-rich shale gas formations. Compared to the hydraulic fracturing, blast-fracturing creates radial fractures with fractureorientations independent of formation stress anisotropy. This eliminates the requirement of in-situ stress orientation that is necessary for designing multi-hydraulic fracturing horizontal wells. An analytical model was developed in this investigation to predict the initial productivity of vertical wells, horizontal wells, and radial-lateral drain holes. Case studies indicate a good agreement between the result given by the models and field observation. Sensitivity analyses with the analytical model indicate that the initial productivity of blast-fractured wells increases non-linearly with the number of radial fractures and fracture penetration. The benefit of increasing fracture depth levels out as the amount of explosives increases. This paper presents well completion engineers a theoretical base and useful data for revolutionizing their well completion technology in developing liquid-rich shale gas basins. Ó 2014 Elsevier B.V. All rights reserved.

Keywords: Blast-fractured Well Productivity Liquid-rich Shale Gas

1. Introduction Unconventional shale gas throughout the country has revived the U.S. oil and natural gas industry, boosting regional economies, and providing an increasing share of domestically produced oil and gas. As the industry continues its march into the shale gas frontier, operators are discovering huge volumes of gas reserves. In some basins, they are also finding a steady stream of natural gas liquids (NGL), which can significantly enhance the value of the production. The question of whether gas shale contains more liquids than conventional reservoirs is still open for debate, but with U.S. gas production trending upward, NGL production is following in step. When gas volumes increase, NGL production can be expected to increase simply because more gas is being produced. As one of the world’s largest NGL producers, the United States accounts for roughly one-fifth of global supply. According to Radler

* Corresponding author. E-mail addresses: [email protected], [email protected] (B. Guo), [email protected] (J. Shan), [email protected] (Y. Feng). http://dx.doi.org/10.1016/j.jngse.2014.03.018 1875-5100/Ó 2014 Elsevier B.V. All rights reserved.

(2012), oil and liquids-rich shale dominated capital spending budgets for 2012. Capital expenditures for all oil and gas projects in North America will increase modestly in the few years to come. While upstream spending has climbed, the rate of growth has decelerated from 2 years ago. Oil and Gas Journal projects that total U.S. capital spending for upstream, midstream, downstream, and corporate activities would increase slightly this year to about $300 billion following a 12% surge in spending during 2011. The increase in upstream spending will be heavily weighted toward the development of oil and liquids-rich shales rather than dry gas production. In the Barnett Shale in North Texas that started shale gas production 10 years ago, the NGLs have been integral to development economics since the beginning. As Barnett production has grown from virtually nothing a decade ago to more than 4 billion cubic feet of gas a day, NGL output has increased correspondingly. To handle this growth, by some estimated the equivalent of a 100 millioncubic-foot-a-day, a cryogenic facility has been added in the Barnett every three months over the past 10 years to treat the liquidsrich gas. On average, the plants remove 3.5 gallons of NGLs for every

B. Guo et al. / Journal of Natural Gas Science and Engineering 18 (2014) 360e367

Mcf of Barnett gas production. Some of the newest and least developed shales also appear to be some of the most significant in terms of reserves and productivity (i.e., the Marcellus in the Northeast and the Haynesville in Louisiana and Texas, as well as the Horn River in Canada). When the shale gas is processed, the NGLs go straight to the bottom line. This is especially advantageous when oil prices are higher than natural gas (NGLs track oil prices). Over the past few years, NGLs have been very profitable, with gas prices weaker on a Btu-equivalent basis than crude oil. A few years ago, historically high natural gas prices left little room for profits in liquids recovery, but that situation has changed with oil prices hovering at $90e$100 a barrel and natural gas in the $3 an Mcf range. Liquid-rich shale formations are essentially lithified clays with organic matter present in varying amounts. Quantities of hydrocarbons can be stored as an adsorbed phase on other materials within the shale, i.e., certain forms of illite. The phenomena of storage and flow of hydrocarbons in shale sediments are believed to be a combination of different controlling processes including molecular diffusion. According to Katsube (2000), hydrocarbon flows through a network of pores with different diameters ranging from nanometers (nm ¼ 109 m) to micrometers (mm ¼ 106 m). The fine-grained rocks in the shale formations are micro-porous with extremely low permeabilities (Javadpour et al., 2007). Carlson and Mercer (1991) found that for Devonian shale, the matrix is so tight that it takes a long time (many years) for the effects of a pressure drawdown in the fracture network to be felt deep in the interiors of the matrix. The challenge in shale gas and liquid development is the lowproductivity of wells with conventional completion methods. Due to the low-permeability nature of shale, hydraulic fracturing is frequently needed to improve well productivity. King (2010) summarized the evolution of the fracturing technique for shale gas formations. The recent engineering achievements in multi-stage fracturing horizontal wells (MFHW) have inevitably increased the interest in exploitation of shale gas reservoirs. McDaniel (2010) discussed this technology applied to all types of low-permeability oil and gas reservoirs. Baihly et al. (2010) presented a thorough investigation of the impact of MFHW technology on productivity of shale gas wells in several U.S. shale basins. Although the MFHW technology is effective for producing oil and gas from shale formations, this technology has disadvantages due to the huge amount of consumed water and the controversial issue of ground water damage by the hydraulic-fracturing fluids. Our recent research investigated the feasibility of replacing the hydraulic-fracturing with blast-fracturing (explosive fracturing) in liquid-rich shale formations. 2. Previous investigations The first wells were shot with black powder, and, in 1865, nitroglycerin was introduced with pronounced results in the booming Pennsylvania oil field. The advent of hydraulic fracturing in the 1950’s caused explosive stimulation of petroleum wells to decline dramatically (Brewer, 1957). The many problems associated with nitro blasting, such as safety and the limitations to working in open holes, caused the decline in the use of nitro shooting (Brandon, 1963). Renewed interest in explosive fracturing came about in 1960’s because hydraulic fracturing was not always successful in fields with unconnected permeability streaks. Explosive fracturing can connect these streaks with flow channels, and then, if desired, these streaks may be hydraulically fractured through these flow channels (Eakin and Miller, 1967). Experience showed conventional high-explosive fracturing to be more convenient and less costly than hydraulic fracturing for certain applications and, in

361

some cases, the only way a reservoir can be stimulated (Levey, 1967; Anderson, 1968). Hydraulic fracturing, though an excellent method of stimulating wells, is limited to propagating a fracture only along the maximum stress orientation (fractures opens against the minimum stress). This leaves a large area which drains either inefficiently or not at all because it is untouched by stimulation attempts. Explosive fractures can propagate in all directions, resulting in more efficient drainage of oil and gas inside reservoirs. The fracture distance has been found to be proportional to the cubic root of the weight of the explosive. The coefficient of proportionality is rock-type-dependent, 5 for sandstone and 7 for limestone. A 3-lb of nitroglycerin explosive can produce a fracture of 10.8 ft in limestone (Dysart et al., 1969). Researchers outside the petroleum engineering have gained a great deal of knowledge on blast behavior of explosives in the last decade. Hao et al. (2002a, 2002b) analyzed blast-induced stress waves in a rock mass using the equivalent material property and stochastic approaches. They found that the peak particle velocity, peak particle acceleration, acceleration time history and Fourier spectra of acceleration from the numerical model all agreed favorably well with test results. It was also demonstrated that numerical results based on anisotropic damage model are more accurate than those based on isotropic damage model in predicting underground blasting-induced stress wave accelerations. The results of numerical analysis with the stochastic approach indicate that the method of combining the statistical initial damage and dynamic damage evolution can predict not only the stress wave intensities in a rock mass, but also give a range of lower and upper limits of peak values of stress wave. It also estimates the lower and upper limits of damage zones generated by the explosion in the rock mass. Wang et al. (2007) performed numerical simulation of tensile damage and blast crater in brittle rock due to underground explosion. They concluded that the damage model presented by Taylor et al. (1986) can well capture the brittle fracture and tensile damage of brittle rock mass due to underground explosions. Cho et al. (2008) carried out a numerical study of fracture plane control in laboratory-scale blasting. Their study showed that both notched guide hole and circular guide hole are effective in controlling crack propagation in blasting. Zhu et al. (2008) conducted a numerical investigation of blasting-induced damage in cylindrical rocks. They identified several factors affecting rock fracturing, including coupling medium, confinement, boundary condition, initiation location, and air ducking. Ma and An’s (2008) numerical simulation with commercial software LSDYNA indicates that the influences of key parameters on the rock fracture pattern can be quantified by the software package. Shao et al. (2009) investigated the critical characteristic of blastinduced rock fracture on the basis of percolation theory and renormalization group method. A damage evolution equation under statistical damage mechanics network was suggested by considering the effect of loading rate and initial damage in the rock. Balakrishnan et al. (2010) completed a numerical study of blast characteristics from detonation of homogeneous explosives. A generalized empirical scaling law based on detonation energy content was presented for explosive design. Bastante et al. (2012) presented a mathematical model for predicting the extent of blast-induced damage (BID) in rock masses. Zhou et al. (2012) carried out a study on fracture behavior of polymer-bonded explosive simulant subjected to uniaxial compression using digital image correlation method. The results showed that the samples with different aspect ratios were fractured in different ways. Onederra et al. (2013) conducted modeling of blast induced damage from a fully coupled explosive charge. They claimed that the coded model is capable of adequately predicting both the

362

B. Guo et al. / Journal of Natural Gas Science and Engineering 18 (2014) 360e367

extent and shape of the damage zone including the influence of point of initiation and free face boundary conditions. Kong et al. (2013) performed a numerical investigation on explosive fragmentation of metal casing using smoothed particle hydrodynamic method. Their simulation results show that the propagation and reflection of the detonation wave has direct relationship to the casing expansion and rupture. The size of fragments is related to the circumferential and axial spacing of the fractures. The blast-fracturing technique has been employed in China to increase productivity of vertical oil wells. Promising results have been reported (Bo et al., 2003; Wu and Sun, 2005; Sun et al., 2011). Production enhancement results from a few oil fields in China (Li, 2000; Li et al., 1988; Zhao and Ke, 1992; Zhang et al., 1994; Bo et al., 2003) are summarized in Table 1. People’s current knowledge to the productivity of wells in shale gas/oil formations is based on the horizontal wells completed with hydraulic fractures. This paper presents our newly gained knowledge of productivity of oil and gas wells completed by explosive fractures. This work will help develop and promote new technologies necessary for replacing the hydraulic fracturing with blastfracturing in the future. If successful, the new technologies will eliminate the environmentally sensitive issues associated with the hydraulic fracturing. 3. Well productivity models When an array of explosive-packs in a wellbore is fired, it is expected that multiple fractures of different lengths and penetrations in different directions are created. It would be very difficult, if not impossible, to predict the number and geometries of these fractures. This analysis assumes that these fractures can be represented by a few equivalent fractures with an average length being equal to the wellbore length packed by the array of explosive-packs. Penetrations of these equivalent fractures are assumed to be symmetrical around the wellbore. Fracture geometries and properties may be predicted using commercial software such as LS-DYNA. Analytical models were derived in this study to investigate the productivity of oil and gas wells in liquid-rich shale formations. Well architectures considered include vertical wells, horizontal wells, and radial-lateral drain hole. Since the growth of blast-fractures is radial and independent of formation stress, all productivity models for these three types of well architectures are based on the same analytical solution for a blast-fractured borehole section. The analytical solution is detailed in Appendix A. Productivity models for these wells are summarized as follows.

3.1. Vertical wells Fig. 1 illustrates the structure of a vertical well completed by blast-fracturing. Based on the general analytical solution (A.12) in Appendix A, the productivity of natural gas liquid (NGL) wells can be described by Eq. (1):

QNGL

    2:255  103 nkH Lf p  pf rw þ hf p ln ¼ rw mo Bo tan n

where QNGL is natural gas liquid production rate in stb/d, kH is horizontal permeability in mdp is reservoir pressure in psia, pf is the pressure in the fracture in psia, rw is the wellbore radius in ft, hf is fracture penetration in ft, mo is liquid viscosity in cp, Bo is formation volume factor in rb/stb, and n is the number of identical fractures created in a single borehole. The value of the fracture length is assumed to be equal to the length of wellbore packed with explosives. The values of the number of fractures and fracture penetration may be predicted using commercial software such as LS-DYNA. Based on the general analytical solution (A.15) in Appendix A, the productivity of gas wells can be expressed as

    2:24  104 nkH Lf p2  p2f rw þ hf p ln Qg ¼ rw mg zT tan n

Liaohe

3.2. Horizontal wells Fig. 2 shows a horizontal well completed by blast-fracturing. Based on the general analytical solution (A.12) in Appendix A, the productivity of natural gas liquid (NGL) wells can be described by Eq. (3):

Well number Pre-operation Post-operation Increase (%) production (t/d) production (t/d)

Jing 64-126 Jing 29-165 An 15-17 Qilicun 458 797 675 769 Ansai Sai 1-2 Sai 1-5 Sai 29-1 Wang 19-11 Zhongyuan Wen 13-267 Wei 22-19 Shengli Ying 11-60

6.4 5 5 N/A

19.8 15 40 N/A

309 300 800 850 (Average)

1.5 2.3 1.2 0.5 2.3 7.4 3

5.2 4.5 4 2.5 12 15.1 8

347 196 333 500 522 204 267

(2)

where Qg is natural gas production rate in Mscf/d and mg is gas viscosity in cp.

Table 1 Field applications of blast-fracturing in oil fields in China. Oil field

(1)

Fig. 1. Sketch of a vertical well completed by blast-fracturing.

B. Guo et al. / Journal of Natural Gas Science and Engineering 18 (2014) 360e367

363

where m is the number of drain holes and the subscript i is an index of drain hole. Based on the general analytical solution (A.15) in Appendix A, the productivity of gas wells can be expressed as

    m 2:24  104 ni kLfi p2  p2 X rwi þ hfi f   ln Qg ¼ rwi mg zT tan npi i¼1

(6)

4. Sensitivity analysis Equations (1) through (6) are from the same fracture model derived in Appendix A. The only difference between the productivity models for vertical wells and horizontal wells is the reservoir permeability. The productivity model for vertical wells uses horizontal permeability, while the productivity model for horizontal wells uses the geometric mean of horizontal and vertical permeabilities. The productivity model for radial-lateral drain holes is identical to that for horizontal wells except that the number of radial-laterals is considered. This section presents a sensitivity analysis only for horizontal NGL wells. The productivity index is thus defined as

Fig. 2. Sketch of a horizontal well completed by blast-fracturing.

QNGL

    2:255  103 nkLf p  pf rw þ hf p ln ¼ rw mo Bo tan n

(3)

pffiffiffiffiffiffiffiffiffiffiffi where k ¼ kH kV is the geometrical mean of horizontal and vertical permeabilities in md. Based on the general analytical solution (A.15) in Appendix A, the productivity of gas wells can be expressed as

    2:24  104 nkLf p2  p2f rw þ hf  p ln Qg ¼ rw mg zT tan n

(4)

3.3. Radial-lateral drain holes Fig. 3 shows radial-lateral drain holes completed by blastfracturing. Based on the general analytical solution (A.12) in Appendix A, the productivity of natural gas liquid (NGL) wells can be described by Eq. (5):

QNGL

    m 2:255  103 ni kLfi p  pf X rwi þ hfi   ln ¼ rwi mo Bo tan npi i¼1

PI ¼ 

QNGL p  pf

 ¼

  2:255  103 nkLf rw þ hf p ln rw mo Bo tan n

(7)

The following result of sensitivity analysis is valid for vertical wells where the permeability anisotropy ratio is 1. The result is valid for horizontal wells where the vertical and horizontal permeabilities are not equal, or the permeability anisotropy ratio is less than 1. The result is valid for radial-lateral drain holes after the number of radial-laterals is multiplied to the values predicted for horizontal wells, if the numbers of fractures in all the radial-laterals are the same. The sensitivity analysis was performed using the base data in Table 2. Eq. (7) indicates that the well productivity index is directly proportional to rock permeability and fracture length, and inversely proportional to fluid viscosity and formation volume factor. The effects of number of fractures, fracture penetration, and permeability anisotropy ratio are not obvious. These parameters were studied in the sensitivity analysis.

(5) 4.1. Effect of number of fractures The effect of number of fractures evenly distributed around the wellbore on well productivity index was calculated with Eq. (7) and plotted in Fig. 4. It is indicated that well productivity increases with the number of fractures around the wellbore. This is because the drainage distances between fractures get shorter as the number of fractures increases. It is interesting that the rate of increase gets higher as the number of fractures grows. This implies that optimal designs of blast-fracturing should consider increasing the wellbore radius and the amount of explosives per unit length of wellbore to create more fractures around the wellbore in order to maximize well productivity. Table 2 Base data used in sensitivity analysis of well productivity.

Fig. 3. Sketch of a radial-lateral drain holes completed by blast-fracturing.

Horizontal permeability (kH) Permeability anisotropy ratio (kV/kH) Liquid viscosity (mo) Liquid formation volume factor (Bo) Wellbore radius (rw) Fracture length (Lf) Fracture penetration (hf) Number of identical fractures (n)

0.01 1 0.5 1.5 7.875 200 20 6

md cp rb/stb in. ft ft

364

B. Guo et al. / Journal of Natural Gas Science and Engineering 18 (2014) 360e367

Fig. 6. Effect of rock permeability anisotropy on well productivity. Fig. 4. Effect of number of fractures on well productivity.

4.2. Effect of fracture penetration The effect of fracture penetration into the rock on well productivity index was calculated with Eq. (7) and presented in Fig. 5. It shows that well productivity increases with fracture penetration. This is because deep fractures create more contact area to the reservoir rock and thus collect more fluid from the reservoir matrix. It is noticed that the rate of increase gets lower as the fracture penetration increases. This is because the average drainage distance between fractures increases as the fracture penetration grows. Therefore, the benefit of increasing the initial well productivity is not proportional to the effort of creating deep fractures. However, deep fractures will contribute to sustaining long-term well productivity due to the large coverage of reservoir body by the fractures. 4.3. Effect of rock permeability anisotropy The effect of rock permeability anisotropy on productivity of horizontal wells was calculated with Eq. (7) and presented in Fig. 6. It demonstrates that well productivity decreases as the permeability anisotropy ratio decreases. This is because the low permeability in the vertical direction reduces the draining efficiency in the reservoir body. 5. Case study Cunderman and Northrop (1986) conducted a field application of blast fracturing in the Meigs County, Ohio. The well was about 3400 ft deep with 50 ft of gas producing zone at the bottom.

Wellbore diameter is 7.875 in. and reservoir pressure is 600 psi. The well was producing at 6.7 Mscf/d before blast stimulation. After the stimulation, production was increased to 22 Mscf/d. Table 3 shows the estimated parameters for the reservoir and fractures. The fracture penetration was predicted to be 3 ft using the formula provided by Warpinski et al. (1979). Reservoir properties were given by Cunderman and Northrop (1986) and Lee et al. (1982) for the Devonian shale in the Meigs County. Because the tubing head pressure was not given in Cunderman and Northrop’s (1986) paper, several different values were assumed in this study to estimate the bottom hole pressure. Assuming a reasonable bottom hole pressure of 450 psia at the depth 3400 ft, the calculated production rate using the model presented in this paper is 20.7 Mscf/d, which is 5.89% lower than the observed value of 22 Mscf/d. 6. Conclusions Assuming uniformly distributed fractures around wellbore, analytical well productivity models were derived in this study to predict the initial productivity of vertical wells, horizontal wells, and radial-lateral drain holes completed with blast-fracturing in liquid-rich shale formations. Case studies indicate a good agreement between the result given by the models and field observation. The following conclusions are drawn based on the sensitivity studies with the models: 1. The initial productivity of blast-fractured wells increases with the number of fractures around the wellbore because the drainage distances between fractures get shorter as the number of fractures increases. The rate of increase gets higher as the number of fractures grows. Thus the optimal design of blastfracturing should consider increasing the wellbore radius and the amount of explosives per unit length of wellbore to create more fractures around the wellbore in order to maximize well productivity. 2. The initial productivity of blast-fractured wells increases with fracture penetration because deep fractures create more contact area to the reservoir rock and thus collect more fluid from the Table 3 Data used in case study.

Fig. 5. Effect of fracture penetration on well productivity.

Horizontal permeability (k) Wellbore radius (rw) Fracture length (Lf) Fracture penetration (hf) Number of identical fractures (n) Gas viscosity (mg) Temperature (T) Gas deviation factor (z)

0.02 7.875 48 3 4 0.02 583 0.95

md in. ft ft cp  R ft

B. Guo et al. / Journal of Natural Gas Science and Engineering 18 (2014) 360e367

reservoir matrix. The rate of increase gets lower as the fracture penetration increases because the average drainage distance between fractures increases as the fracture penetration grows. Thus the benefit of increasing the initial well productivity is not proportional to the effort of creating deep fractures. However, deep fractures are still desirable for they contribute to sustaining long-term well productivity due to the large coverage of reservoir body by the fractures. 3. The initial productivity of blast-fractured well horizontal and radial-lateral wells decreases as the permeability anisotropy ratio decreases because the low permeability in the vertical direction reduces the draining efficiency in the reservoir body. More fractures should be created in shale formations that have low permeability anisotropy ratios to improve well productivity.

365

5. Fluid flow from the rock directly to the wellbore is negligible. 6. Pseudo-steady state flow in the pay zone.

Governing equation Fig. 7 shows a sketch of fracture configuration around a wellbore. Consider a fluid particle at point P which is on the line of stagnation where PA ¼ PB. The flow distance to fracture A is expressed as

PA ¼ x tanðaÞ

(A.1)

where the angle a relates to the number of fractures n (n > 2) around the wellbore by

Acknowledgments This research was supported by the China National Natural Science Foundation Founding No. 51274220, No. 51134004, No. 51221003, 51274045, 51274221, and No. 51334003. Partial support from the Major State Basic Research Development Program of China (973 Program Grant No. 2010CB226704) is acknowledged. The authors are grateful to Chevron USA for providing the LA Board of Regents Chevron I and II Professorships in Petroleum Engineering throughout this study. Nomenclature Bo hf k kH kV Lf n p pf Qg QNGL rw T z

Formation volume factor of oil (rb/STB) Fracture penetration (ft) Geometric mean of permeabilities (md) Horizontal permeability (md) Vertical permeability (md) Fracture length (ft) Number of fractures created in a single borehole Reservoir pressure (psia) Pressure in the fracture (psia) Natural gas production rate (Mscf/d) Natural gas liquid production rate (stb/d) Wellbore radius (ft) Temperature ( R) Compressibility factor

Fig. 7. A sketch of fracture configuration around a wellbore.

Greeks

mg mo

Gas viscosity (cp) Liquid viscosity (cp)

Subscripts f Fracture i Index of drain hole m Number of drain holes sc Standard condition Appendix A. Derivation of inflow equation for blast-fractured wells Assumptions

a ¼

p 2p 2 ¼ : n n

(A.2)

where n is the number of fractures around the wellbore. Consider the fluid flowing from point P to a fracture segment of width dx, according to Darcy’s law, the fluid influx rate to the fracture segment can be expressed as:

  kLf p  pf dx dq ¼ mðPAÞ

(A.3)

The following assumptions are made in deriving a well inflow model:

where k is mean permeability of rock, Lf is fracture length along the wellbore, p reservoir pressure, pf is pressure in the fracture, and m is fluid viscosity. Substituting Eqs. (A.1) and (A.2) into Eq. (A.3) gives:

1. 2. 3. 4.

  kLf p  pf   dx: dq ¼ mx tan pn

The rock in the pay zone is homogeneous and anisotropic. Linear flow of fluid governed by Darcy’s law prevails in the rock. All fractures are identical in geometry. Pressure drop in the fractures is negligible.

(A.4)

366

B. Guo et al. / Journal of Natural Gas Science and Engineering 18 (2014) 360e367

2

Boundary condition

4

Assuming linear flow of fluid to the fracture surface with a length of Lf and a total width between x ¼ rw and x ¼ hf, the fluid influx rate to the fracture is null at x ¼ rw, i.e.,

q ¼ 0

at

x ¼ rw

p þ pf

(A.5)

Solution

rZ w þhf

q ¼ rw

  kLf p  pf   dx mx tan pn

Qsc



  kLf p  pf rw þ hf p ln rw m tan n

(A.7)

Since each fracture has 2 faces, the total fluid influx rate into n fractures is expressed as

    2nkLf p  pf rw þ hf p ln : Q ¼ 2nq ¼ rw m tan n

(A.8)

Unit conversion Eq. (A.8) is valid in Darcy’s units. In U.S. field units, it becomes

ð60Þð60Þð24Þ cm3 sec

#

ð2:54Þ3 ð12Þ3 ft3 day i h i h i h Darcy   Atm L ð12Þð2:54Þ 2nk 1000 p  pf 14:696 rw þ hf psi md f ft p ln :  rw m tan n (A.9)

or

    nkLf p  pf rw þ hf p ln : Q ¼ 0:01266 rw m tan n

(A.10)

where the permeability is in md, viscosity is in cp, pressures are in psia, lengths are in ft, and fluid flow rate is in cubic feet per day in reservoir condition (rcfd). If the reservoir fluid is natural gas liquid (NGL), or oil, the flow rate should be converted to stb/d

    nkL p  p f f rw þ hf 0:01266 p ln ¼ 5:615 mo Bo tan n rw

(A.11)

or

QNGL

zT

:

(A.14)

    2:24  104 nkLf p2  p2f rw þ hf p ln : ¼ rw mg zT tan n

(A.15)

References



QNGL

  0:01769 p þ pf Q

(A.6)

which results in

Q ¼

(A.13)

Substituting Eq. (A.10) into Eq. (A.12) results in:

Integration of Eq. (A.4) using the boundary condition (A.5) gives

"

5 Q ¼ psc Qsc : zT zsc Tsc

Substituting psc ¼ 14.696 psia, zsc ¼ 1.0, Tsc ¼ 520  R, and Qcs in Mscf/d into this equation gives:

Qsc

q ¼

2

3

    2:255  103 nkLf p  pf rw þ hf p ln ¼ rw mo Bo tan n

(A.12)

If the reservoir fluid is gas, the flow rate should be converted to Mscf/d using real gas law:

Anderson, A.L., May 3, 1968. Blast-Frac Report and Recommendations. The Western Company Publication. Baihly, J., Alttman, R., Malpani, R., Luo, F., 2010. Shale Gas Production Decline Trend Comparison Over Time and Basins. Paper SPE 135555 presented at the SPE Annual Technical Conference and Exhibition held 19e22 September 2010 in Florence, Italy. Balakrishnan, K., Genin, F., Nance, D.V., Menon, S., 2010. Numerical study of blast characteristics from detonation of homogeneous explosives. Shock Waves 20, 147e162. Bastante, F.G., Alejano, L., Gonzalez-Cao, J., 2012. Predicting the extent of blastinduced damage in rock masses. Int. J. Rock Mech. Min. Sci. 56, 44e53. Brandon, C.W. Method of Explosively Fracturing a Productive Oil and Gas Formation. U.S. Patent 3,066,733, Dec 4, 1963. Bo, Q., Ge, G., Ma, G., 2003. High-energy gas fracturing technique and its applications. J. China Ocean Pet. 23 (3), 69e71. Brewer, B., February 1957. Stimulation of oil production by the use of explosives after hydraulic fracturing. Producers Monthly 21 (4), 22e23. Carlson, E.S., Mercer, J.C., April 1991. Devonian shale gas production: mechanisms and simple models. J. Pet. Technol., 476e482. Cho, H., Nakamura, Y., Mohanty, B., Yang, H., Kaneko, K., 2008. Numerical study of fracture plane control in laboratory-scale blasting. Eng. Fract. Mech. 75, 3966e 3984. Cunderman, J.F., Northrop, D.A., 1986. A propellant-based technology for multiplefracturing wellbores to enhance gas recovery: application and results in Devonian shale. SPE Prod. Eng. 1 (2), 97e103. Dysart, G.R., Spencer, A.M., Anderson, A.L., April 1969. Blast-Fracturing. paper presented at the spring meeting of the Mid-Continent District, API Division of Production. Eakin, J.L., Miller, J.S., Nov 1967. Explosives research to improve flow through low permeability rock. J. Pet. Technol. 1431. Hao, H., Wu, C., Zhou, Y., 2002a. Numerical analysis of blast-induced stress waves in a rock mass with anisotropic continuum damage models e part 1: equivalent material property approach. Rock Mech. Rock Eng. 35 (2), 79e94. Hao, H., Wu, C., Seah, C.C., 2002b. Numerical analysis of blast-induced stress waves in a rock mass with anisotropic continuum damage models e part 2: part 2: stochastic approach. Rock Mech. Rock Eng. 35 (2), 95e108. Javadpour, F., Fisher, D., Unsworth, M., Oct. 2007. Nanoscale gas flow in shale gas sediments. JCPT 46 (10), 55e61. Katsube, T.J., 2000. Shale Permeability and Pore-Structure Evolution Characteristics. Research 2000 E15, Geological Survey of Canada, Ottawa. King, G.E., Sept. 19e22, 2010. Thirty Years of Gas Shale Fracturing: What Have We Learned. paper SPE 133456 presented at the SPE ATCE held in Florence, Italy. Kong, X., Wu, W., Li, J., Liu, F., Chen, P., Li, Y., 2013. A numerical investigation on explosive fragmentation of metal casing using smoothed particle hydrodynamic method. Mater. Des. 51, 729e741. Lee, B., Alam, J., Sawyer, W.K., Horan, K., Frohne, K., 1982. Evaluation of Devonian Shale Reservoir Using Multi-well Pressure Transient Testing Data presented at the SPE Unconventional Gas Recovery Symposium, 16e18 May, Pittsburgh, Pennsylvania. Levey, D., Aug 30, 1967. Explosive Stimulation Report. The Western Company Publication. Li, D., Xue, Z., Liu, F., Sep. 1988. Preliminary analysis of high energy gas fracture effect at Qili village oil field. J. Xi’an Pet. Inst. 3 (2), 7e14. Li, W., Jun. 2000. Researching and application on high energy gas fracturing technique used in the development of oil/gas resources. J. Xi’an Eng. Univ. 22 (2), 60e62. Ma, G., An, X., 2008. Numerical simulation of blasting-induced rock fractures. Int. J. Rock Mech. Min. Sci. 45, 966e975. McDaniel, B.W., 18e20 October 2010. Horizontal Wells with Multi-stage Fracs Provide Better Economics for Many Lower permeability Reservoirs. paper SPE 133427 presented at the SPE Asia Pacific Oil & Gas Conference and Exhibition held in Brisbane, Queensland, Australia.

B. Guo et al. / Journal of Natural Gas Science and Engineering 18 (2014) 360e367 Onederra, I.A., Furtney, J.K., Sellers, E., Iverson, S., 2013. Modelling blast induced damage from a fully coupled explosive charge. Int. J. Rock Mech. Min. Sci. 58, 73e84. Radler, M., March 5, 2012. Oil, liquids-rich shales dominate capital spending budgets for 2012. OGJ Online. Shao, P., Xu, Z., Zhang, H., He, Y., 2009. Evolution of blast-induced rock damage and fragmentation prediction. Procedia Earth Planet. Sci. 1, 585e591. Sun, Z., Li, Z., Su, J., Liu, C., January 2011. An analysis of the effects of formation parameters on the fracture extension in explosive-gas fracturing. China Pet. Geol. Recovery Effic. 18 (1). Taylor, L.M., Chen, E.P., Kuszmaul, J.S., 1986. Micro-crack induced damage accumulation in brittle rock under dynamic loading. Comp. Methods Appl. Mech. Eng. 55 (3), 301e320. Wang, Z., Li, Y., Shen, R., 2007. Numerical simulation of tensile damage and blast crater in brittle rock due to underground explosion. Int. J. Rock Mech. Min. Sci. 44, 730e738.

367

Warpinski, N.R., Schmidt, R.A., Cooper, P.W., Walling, H.C., Northrop, D.A., 1979. High-energy Gas Frac: Multiple Fracturing in a Wellbore. Paper ARMA 79e0143 presented at the 20th U.S. Symposium on Rock Mechanics (USRMS), June 4e6, 1979, Austin, Texas. Wu, J., Sun, A., 2005. An investigation of multi-pulse gas fracturing and its applications. China J. China Pet. Mach. 34 (1), 77e80. Zhang, Q., Zhao, W., Wang, F., May 1994. The high-energy gas fracturing technology. Fault Block Oil Gas Fields 1 (3), 50e60. Zhao, X., Ke, Z., Apr. 1992. The application of high Energy Gas fracturing at Ansai oil field. Pet. Drill. Prod. Technol. 14 (6), 83e85. Zhou, Z., Chen, P., Duan, Z., Huang, F., 2012. Study on fracture behaviour of a polymer-bonded explosive simulant subjected to uniaxial compression using digital image correlation method. Strain 48, 326e332. Zhu, Z., Xie, H., Mohanty, B., 2008. Numerical investigation of blasting-induced damage in cylindrical rocks. Int. J. Rock Mech. Min. Sci. 45, 111e121.