The impact of diffusion type on multiscale discrete fracture model numerical simulation for shale gas

Journal of Natural Gas Science and Engineering 20 (2014) 74e81

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The impact of diffusion type on multiscale discrete fracture model numerical simulation for shale gas Lidong Mi*, Hanqiao Jiang, Junjian Li Key Laboratory of the Ministry of Education, China University of Petroleum, Beijing, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 May 2014 Received in revised form 14 June 2014 Accepted 16 June 2014 Available online

The development of unconventional gas reservoirs represents totally distinctive characteristics as compared with the conventional reservoirs. The complex pore structure in shale reservoir determines its special flow mechanism, which can be divided into several categories according to the size and type of pores- non Darcy flow, gas slippage, adsorption-desorption and gas diffusion effect. Based on the gas molecules diffusion form in porous media and combining with the multi-scale distribution structural characteristics of shale gas reservoirs, the shale gas diffusion mechanisms in the shale reservoir space including the diffusion of dissolved gases in the organic kerogen and the diffusion of free gas in the nanopores are analyzed in this paper. Meanwhile, the diffusion in the nanopores consists of Knudsen diffusion (KN
10), Fick diffusion (KN  0.1) and transition diffusion (0.1 < KN < 10) according to the Knudsen (KN) number. In this work, we set up new mathematical models for shale gas flow in matrix and fracture networks, and also for their mass transfer in between without neglecting its varying-scale nature following the concept of discrete fracture network (DFN). In addition, we also investigate the different diffusion mechanisms' influences on the production and pressure in the tight shale gas reservoir. Ultimately, concluding that the gas diffusion mechanisms in micro-and nano-scale matrix block have a greater impact on the distribution of shale gas production (especially the production at early time) and reservoir pressure. © 2014 Elsevier B.V. All rights reserved.

Keyword: Shale gas numerical simulation Discrete fracture network Diffusion type Knudsen diffusion Fick diffusion Transition diffusion

1. Introduction Huge technological advancements in the past decade in drill completion, hydraulic stimulation, microseismic fracture mapping as well as in numerical reservoir stimulation have paved the way for the exploration and production of shale gas in the world (Arthur et al., 2008; Wang and Krupnick, 2013). However, in actuality the development of shale gas reservoir are quite distinctively complex due to its nanopores, diverse pore structures (Loucks et al., 2009) and intricate in-place gas flow mechanisms which involve an array of phenomena, like desorption, diffusion and gas slippage etc. (Freeman, 2010). Development mechanisms and numerical methods applicable to conventional gas reservoirs apparently cannot withstand in this regard. Most conventional simulators available are based on the continuum model either on dual or multiple media, assuming a

* Corresponding author. E-mail address: [email protected] (L. Mi). http://dx.doi.org/10.1016/j.jngse.2014.06.013 1875-5100/© 2014 Elsevier B.V. All rights reserved.

homogeneous fracture distribution underground (Azom and Javadpour, 2010). Actually, numerous investigations have proved that a rather heterogeneous distribution of natural mature fractures in shale reservoirs (Gale et al., 2007). Fractures are the main flow path for gas production in the ultralow permeability and porosity shale matrix (Curtis, 2002; Darabi et al., 2012). Therefore, it becomes crucial for engineers to predict gas flow in the factures accurately for shale gas exploitation and production. Some commercial simulators, with CMG included, have done some progress in introducing a DFN to simulate real subterranean fracture systems (Computer Modeling Group, 2010). But their essentially continuum-based governing equations cannot tell fully the characteristics of predominant gas flow in fractures. NETL proposed a fracture flow model in an attempt to uncover the shale gas flow mechanisms (McKoy and Sams, 2010; Boyle and Sams, 2012). However it is also too simplified to encompass the various percolation mechanisms in the shale reservoir. In this present work, on the basis of DFN, we will build up mathematical models to characterize gas flow both in factures and shale matrix, and also mass transfer in between with a creative

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consideration for a varying-scales gas flow. Furthermore, some important effects involved, which may contribute substantially to part of gas production, are also examined upon their influence to shale gas productivity, such as the different diffusion models (Knudsen diffusion, Fick diffusion and transition diffusion). 2. Flow in the fractures Shale gas formation is organic-rich and apparently the source rock as well as the reservoir (Kundert and Mullen, 2009). The matrix permeability of shale gas is generally not in the order of millidarcy or even microdarcy (103 md), but nanodarcy (106 md) typically 10 to 100 nanodarcy (Sakhaee-Pour and Bryant, 2012). Research has long revealed that shale gas flow primarily takes place in fractures due to the highly impermeable of the matrix (Barenblatt et al., 1960; Warren and Root, 1963). Actually both natural and induced fractures can be the paths for shale gas to flow through in production, which is commercially the main production strategies of shale gas by means of large-volume high-rate hydraulic fracture treatments that use water and small-mesh proppant to activate or stimulate the existing natural fractures or rock fabrics (Cipolla et al., 2009).Observations and research have further indicated that rather than falling in Darcy equation's regime, gas flow in fractures occurs quite differently. DFN deals with the gas movement in factures as flowing in a parallel-plate following the concept of Boussinesq equation (Yang and Wei, 2004). (1) Equation of motion (Yang and Wei, 2004)

w3 dp Qv ¼  12mg ds

(1)

(2) Auxiliary equation (Yang and Wei, 2004). The equation of state for real shale gas follows:

pV ¼ ZnRT

(2)

Shale gas density can be represented as:

rm ¼

m V

(3)

Combine Eq. (2) and Eq. (3) to yield further the equation of shale gas density:

rm ¼

M p RT Z

1 p RT Z


 r  ðQv ÞPw þ ðQv ÞPe  ðQv ÞPs þ ðQv ÞPn ¼ 0

(6)

3. Flow in shale matrix Microscopically shale matrix is quite limited in permeability for any fluid to flow through. However in a macroscopic perspective, it may be linked to a system of surrounding fractures, either natural or induced ones, making the fluid flow in the matrix a bit complex. Generally shale gas flow in the matrix can be governed by nonDarcy's flow principle. Mass transfer also occurs between matrix and fracture systems due to direct connections. To be more complex, several effects may also play in the nanopores, such as adsorption and desorption on pore surface, gas slippage and diffusion in nonopores, and diffusion in kerogen, etc. 3.1. Gas flow in nano-scale pore spaces

(4)

Now we can have the molar density of shale gas, which is defined as the molar mass for unit volume of shale gas at a specific pressure and temperature:

r¼

Fig. 1. Simplified fracture element schematic (McKoy and Sams, 2010).

(5)

(3) Continuity equation Assuming two arbitrary fractures intersect at point P as presented in Fig. 1, we try to set up the continuity equation for shale gas flow in the underground fracture networks (McKoy and Sams, 2010). Mass conservation principle applying to this case goes: Net mass inflow to the element per unit time ¼ mass variation in the element per unit time. Assume that the volumetric fluid flow rates into and out of point P in x-axis direction are ðQv ÞPe and ðQv ÞPw respectively. Likewise in y-axis direction, the rates are assumed to be ðQv ÞPn and ðQv ÞPs for coming into and out of the study point. If the mass variation at point is said to be nil at a certain time, then this following conservation applies (McKoy and Sams, 2010):

Gas flow in nanopores is typically subjected to two main mechanisms: pressure gradient and diffusion. Thus the mass flux for flow in nanopores mainly constitutes two parts (Javadpour, 2009):

J ¼ Ja þ JD

(7)

(1) Gas slippage Rushing et al. defined gas slippage as an effect giving birth to deviation of flow from viscous flow to a non-laminar flow. Usually gas slippage occurs when the dimensions of the subsurface pores are within the range of the mean free path of gas molecules which cause the gas molecules to slip when in contact with the rock surface and therefore accelerate (Rushing et al., 2007). The gas slippage phenomenon was first observed by Klinkenberg in 1941. So in the case of gas flow in the shale matrix, where pore spaces are predominantly in nanoscale, gas slippage is inevitably present. In the study of gas flow in a general capillary tube, the concept of slippage factor is introduced to account for gas slippage effect to the gross flow behavior. Neglecting the effect of entrance length, the Poiseuille principle gives a formula for the mass flux associated to single capillary tube (Bird et al., 2007):

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rr 2 Ja ¼  n Vp 8m

(8)

To include the effect of gas slippage, the corresponding equation for nanopores emerges:

rr 2 Ja ¼  n FVp 8m

(9)

(2) Gas Diffusion Currently, most researchers believe that there is only Knudsen diffusion in the shale matrix block, actually the diffusion forms which is directly determined by the pores radius can be divided into: Knudsen diffusion, Fick diffusion and transition diffusion (Fig. 2) according to the Knudsen number (Guo et al., 2007; Wang et al., 2013; Nie et al., 2000). Also the mathematical characterization of diffusion coefficient is different corresponding to the three diffusion forms. Knudsen number is given by:

Kn ¼

l dn

k T l ¼ pffiffiffi b 2 2pd p

! (10)

a. Knudsen diffusion model When Knudsen number Kn
10, The gas diffusion is mainly caused by the collision between gas molecules and the pore wall in nanopores (Fig. 2a), which is Knudsen diffusion. According to the molecular kinetic theory, the shale gas effective diffusion coefficient in which the surface porosity and tortuosity are considered is given by:



DK ¼

f dn 8RT t 3 pM

(11)

f kb T t 3pmg dg

(12)

c. Transition diffusion model when the Knudsen number 0.1 < Kn < 10, gas mean free path and nanopores diameter is similar so that the diffusion is caused by the collision among freedom molecules and the collision between molecules and pore walls (Fig. 2c), which is the Transition diffusion, as a result of the interaction between Knudsen diffusion and Fick diffusion, can be written as (Guo et al., 2012):



calculated by the Eq. (11) and the Knudsen diffusion coefficient, Fick diffusion coefficient and the Transition diffusion coefficient respectively calculated by Eq. (11), Eq. (12) and Eq. (13). As shown in Fig. 3, the three diffusion coefficients corresponding to the same pore radius vary greatly, and if the Knudsen diffusion is the only model used to characterize the diffusion phenomena in matrix pore, the influence of diffusion will expand artificially. If we dismiss the impact of gas viscosity, then the mass flux for diffusion can be expressed in terms of pressure gradient as (Roy et al., 2003):

1 DT ¼ D1 F þ DK

1

(13)

When temperature T ¼ 313, pore pressure p ¼ 1.72 MPa and methane gas deviation factor Z ¼ 0.97, the Knudsen number can be

8   q dn 8RT 0:5 > > > B > > t 3 pM B > > > B > > B < B q kb T BD ¼ B > t 3pmg dg > B > > B > > B > > @   > > : D1 þ D1 1 0

0:5

b. Fick diffusion model When the Knudsen number Kn  0.1, the gas diffusion is mainly caused by the collision among the gas molecules (Fig. 2b), which is Fick diffusion. The pores are interconnected and bending, the diffusion path is growth because of the pore channel tortuous, diffusion resistance increases due to the pore interface shrink. The shale gas effective diffusion coefficient in which the surface porosity and tortuosity are considered can be written as:

DF ¼

Fig. 3. The shale gas diffusion coefficient in different pore radius (1.72 MPa、313 K).

JD ¼ 

MD Vp RT

F

K

1 10 < Kn Kn < 0:1 0:1 < Kn < 10

C C C C C C C C C C A

(14) Let kapp denotes apparent permeability in mD, then mass flux in terms of apparent permeability is:

J¼

kapp rVp m

(15)

Thus we can derive the expression of apparent permeability for real gas from the above equations in line with the similar form of Eq. (16), which reveals in an integrated way the effects of gas slippage and diffusion in the gas flow (Javadpour, 2009):

kapp ¼ cg Dmg þ Fk∞

(16)

As shown in Fig. 4, when the Knudsen diffusion is the only diffusion model which is used in the matrix pores, the effects of diffusion on seepage will be magnified and result in apparent permeability increasing.

Fig. 2. Diffusion modes of shale gas in porous medium.

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Fig. 4. The apparent permeability dynamic curve.

3.2. Adsorption and desorption at the pore interface Research shows that natural gas exists in shale reservoir in diverse forms (Aguilera, 2010). Some are stored in the limited pore spaces of these rocks as free, compressed gas whilst a sizable fraction of the gas in place may present on the surface of organic materials and clays contained in the rocks as adsorbed gas. The classical Langmuir isothermal adsorption theory can be applied to the latter for interpreting the adsorption-desorption behavior of inplace shale gas as follows (Langmuir, 1918). The desorption flux of gas per unit time on unit area is proportional to the gas covered area ratio:

Jdes ¼ Kdes q

(17)

Likewise the gas adsorption flux per unit time on unit area is:

Jads ¼ Kads ð1  qÞpn

(18)

When adsorption and desorption reaches equilibrium, we have:

Jads ¼ Jdes

(19)

Further yield the equation for the adsorbed gas covered area ratio:

q¼

Kads pn Kdes þ Kads pn

(20)

Fig. 5. Representative physical diffusive model of kerogen.

3.3. Gas diffusion in kerogen Aside from being present in the nanoscale pore spaces, either as free gas or adsorbed gas, gas can also exist in a large amount as dissolved gas in organic materials, namely kerogen. During the process of gas production for shale, a substantial amount of free gas is released with declining pressure. At the same time, a fraction of gas will also be desorbed to be produced from the pore surfaces. With gas continuously being produced out of pores, difference in gas concentration between pore space and inside organic kerogen occurs. Then gas diffusion emerges from high-gas-concentration kerogen toward pore canals, offsetting its shrinking gas concentration in the production course. To account for the gas production from diffusion in kerogen, we intend to build up a physical model to represent kerogen in Fig. 5 for mathematical analysis. Without loss of generality, we postulate that kerogen circles pores with a 10-time-pore diameter. The mathematical model for that follows (Vivek et al., 2012):

  1 v vC vC Dkerogen r ¼ r vr vr vt C ¼ Ci ¼ kh pi ;

t ¼ 0;

During reservoir depletion, the decline in pressure will break the thermodynamic balance state. This non-equilibrium process causes gas to desorb vastly from the surface of the kerogen/clays, which contributes to part gas production. Apply mass conservation at the surface of pores to gain gas production for a time interval △t due to the off-balance of the adsorption-desorption system:

8 < C ¼ kh pn ; : vC ¼ 0; vt



Jdiff ¼ Dkerogen




  S M
Kdes qx;t  Kads 1  qx;t px;tþ1 Dt ¼ 0 qx;t  qx;tþ1 N

(21)

So the gas production due to the off-balance of adsorptiondesorption at unit pore surface can be computed as (Shabro et al., 2011):



  S M
q  q
 So M Dq x;t x;tþ1 Kdes qx;t  Kads 1  qx;t px;tþ1 ¼ 0 ¼ N N Dt Dt (22)

Therefore for area A, the corresponding mole mass flux should be ððS0 M=NÞðDq=DtÞÞA.

(23) 0  r  rk

r ¼ rn ;

t>0

r ¼ rk ;

t>0

(24)

(25)

Mass flux due to diffusion in kerogen can be approximated as:

vC vr ðr¼rn Þ

(26)

3.4. Non-Darcy’s flow in matrix Fluid flows in shale gas reservoirs accompanying other various phenomena, such as gas slippage, diffusion (Knudsen diffusion, Fick diffusion and Transition diffusion), gas desorption and gas diffusion in kerogen, etc. A more comprehensive mathematical model taking consideration of all those effects and mechanisms will be presented in this part. First we add the gas desorption term into the continuity equation, and integrate diffusion with gas slippage effects into the concept of apparent permeability.

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L. Mi et al. / Journal of Natural Gas Science and Engineering 20 (2014) 74e81

(1) Continuity equation

  v So Dq v ðAc ðxÞruÞ þ A ¼ Ac ðxÞ ð4rÞ vx vt N Dt

(27)

(2) Dynamic equation

kapp vp u¼ mg vx

(28)

4. Mass transfer between matrix and fractures In shale gas production process, gas will flow from the matrix system to fracture networks, natural and stimulated, for much larger permeability and porosity. For simplicity, in this work we assume gas flows toward the midpoint of a fracture as demonstrated in Fig. 6. Then we can establish a mass conservative form for point e in Fig. 6 (McKoy and Sams, 2010):


 1  RC Qr ðFe Þ þ QlRC ðFe Þ r  ðQv ÞPe þ ðQv ÞEe þ RT Z d rdV ¼ dt

Fig. 6. The element of fracture-matrix node (McKoy and Sams, 2010).

(29)

Vfe

5. Mathematical model for fluid flow in horizontal wells It has been a progressive innovation to use the concept of discrete fracture network (DFN) to characterize the fluid flow behavior in shale sequences. Actually DFN deals with wells as large fractures. For this reason, the treating of fluid flow from fractures to the producers in DFN doesn't have much difference from that for fractures. A horizontal well, which penetrates many fractures, is shown in Fig. 7. Apply the mass conservation principle to production well to get (McKoy and Sams, 2010):

X Q RC Q RC r þ l RT RT ks

! þ ks

Z X

 d r ðQv Þl  ðQv Þr j ¼ rdV þ rsc QVsc f dt j f

Vh

(30) 6. Solutions to the mathematical model Finite difference has been a powerful analytical tool in solving partial differential equations in engineering, and also far beyond. In this part, it is also utilized to discretize the differential equations to yield their difference forms for analytical analysis.

with various shapes, are transformed to corresponding the-samevolume rectangles. For this we can have the length of pathway for each rectangular block in line with each fracture length. The gas flow inside each block is seen as one-dimensional. Gas flows out of each block from the midpoint of fracture toward fracture. The difference form of Eq. (27) can be written as Eq. (32) by discretizing the effective matrix volume into nb blocks centered at xi with length △xi where i ¼ 1 … nb.

 ðnþ1Þ  n  Vi p p RTASo  nþ1 qi  qni   Z i Dtnþ1 Z i NDtnþ1     ðnþ1Þ ðnþ1Þ ðnþ1Þ ðnþ1Þ þ TXi1=2 fi ¼0  TXiþ1=2 fiþ1  fi  fi1

6.3. Difference form for fluid transfer between the matrix and fractures Directly take the difference form of Eq. (29):



    Vfe pnþ1 pn þ QlRC fnþ1  ¼ QrRC fnþ1 e e Z e Dtnþ1 Z e   f þ fnþ1  2fnþ1 þ 2TX e fnþ1 e P E

6.4. Difference form for fluid flow in horizontal wells

6.1. Difference form for fluid flow in fractures Eq. (6) describes the shale gas flow in the fracture networks. Gas transfer between midpoint and intersection of fractures proceeds in cubic law. Along the fracture do the finite difference upon Eq. (6) to get:

The difference form for Eq. (30) is:





 2TX fw fw  fp þ 2TX fe fe  fp þ 2TX fs fs  fp þ 2TX fn fn  fp ¼0 (31) 6.2. Difference form for fluid flow in the matrix The analytical treatment of matrix blocks is based on the concept of equivalent volume. According to this, matrix blocks,

(32)

Fig. 7. Schematic of horizontal wells.

(33)

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79

Table 1 Geological properties and parameters for a Surrogate model.

2

Volume/feet3

Depth/feet

Porosity

Matrix Permeability/mD

Temperature/F

Initial pressure/PSI

Reserve/MSCF

No. of fractures

Fracture notes

30  30  20

10,000

0.05

0.008/0.006

285

10,000

0.31

4

6

i Xh f f TXr ðfr  fw Þ þ TXl ðfl  fw Þ  Qcomp jf

jf

¼

VH Dtnþ1

 ðnþ1Þ p Z

w



pðnÞ Z

w



X

QrRC þ QlRC



ks

(34)

ks

Combining Eq. (31), Eq. (32) and Eq. (33) gives birth to a large group of sparse linear equations. Assume an initial gas reservoir pressure; Newton iteration method can be quite useful in solving the equations for reservoir pressure distribution at varying times. With Eq. (34) we can further gain the pressure distribution in wellbores or the production rate.

7. Simulation analysis A surrogate model is specifically selected for shale gas reservoir numerical case study. Some geological properties and parameters associated with this reservoir, fractures and well are respectively tabulated in Table 1, Table 2 and Table 3. The surrogate model is 30  30  20 feet3, which contains four fractures (two parallel vertical and two vertical fractures) and a horizontal wells that penetrating the two horizontal fractures and the 2-D chart is shown in Fig. 8. The well is pressure controlled (Bottom hole flowing pressure Pwf ¼ 5000PSI), and the simulation period is carried out for half a day, then investigating the production rate, cumulative production and the average matrix pressure variation. As shown in Fig. 9, the gas production rate decrease rapidly when T is less than 0.1day. And the gas velocity difference demonstrates that: when T is less than 0.03day, the higher matrix permeability the faster production rate; when T is between 0.03day and 0.18day, the higher matrix permeability the slower production rate. This is because the mass transfer speed is larger when the matrix permeability higher and the pore pressure drop faster than the lower-permeability matrix (Fig. 10). So, at the late stage of production the lower permeability matrix production rate will larger than the higher permeability matrix's. The average matrix pressure drop curve corresponding to different permeability is shown in Fig. 10. The higher matrix permeability, the faster pressure drop at the early production phase, and meantime the conduction velocity decreased faster because of the higher mass transfer rate, so it take long time to

achieve the abandonment pressure for the lower permeability reservoirs. The cumulative production curve corresponding to different permeability is shown in Fig. 11. The higher permeability matrix's (Km ¼ 0.008mD) cumulative production increase faster than the lower permeability matrix's (Km ¼ 0.006mD), without considering the effects of adsorption and retention, etc., the cumulative production equal to the geological reserves (0.1075MMCF). Based on the above numerical model, we find that: when the matrix permeability difference is 0.002mD, the impact on the calculation results cannot be ignored, and the result of the section 3.1 shows us that the diffusion and gas slippage are the main factors that affect the matrix apparent permeability (greater impact) in shale matrix micro-and nano-scale pores. Therefore, if the numerical simulation is carried in the tight shale gas reservoir, the influence of matrix pore size and the corresponding diffusion model should be considered. 8. Conclusion (1) Gas flow in shale reservoirs is distinctively complex as compare to conventional gas reservoirs. The ultralow permeability and porosity nature of shale matrix rendering gas flow at varying scales. (2) DFN is superior to continuity model in describing shale gas flow. Numerical analysis on shale gas production using DFN should also take into account the varying-scale gas flow. (3) The diffusion forms which are directly determined by the pores radius, in the shale matrix Nano-scale pores, can be divided into: Knudsen diffusion, Fick diffusion and transition diffusion according to the Knudsen number. (4) Various effects or mechanisms are present in shale gas reservoir depletion, such as adsorption and desorption on rock surfaces, gas slippage and gas diffusion (Knudsen

Table 2 Fractures parameters for the surrogate model. Number

x-left

y-left

x-right

y-right

Aperture(feet)

1 2 3 4

0 0 10 20

10 20 0 0

30 30 10 20

10 20 30 30

0.1 0.1 0.5 0.5

   

103 103 103 103

Table 3 Well parameters for the surrogate model. Number

x-left

y-left

x-right

y-right

Well bore radius(feet)

1

11

7

18

25

0.29

Fig. 8. 2-D representatives DFN for a shale gas reservoir.

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L. Mi et al. / Journal of Natural Gas Science and Engineering 20 (2014) 74e81

Acknowledgments This study was supported by the National Key Technology Research and Development Program (973 Program) “The Basic Research of South China Marine Shale Gas Efficient Development ”（No. 2013CB228000） Nomenclature and units h w

Fig. 9. The production rate corresponding to different permeability.

Fig. 10. The average matrix pressure corresponding to different permeability.

diffusion, Fick diffusion, Transition diffusion), and diffusion in bulk kerogen, etc. Apparent permeability can be used to account for the effects of gas slippage and diffusion on shale gas production, while adsorption-desorption phenomenon can be depicted by conventional Langmuir isothermal theory. (5) The simulation of the model shows that: the select of the diffusion model has a great effect on the matrix apparent permeability, and the small differences in permeability (0.002mD) will lead to large difference in results.

Fig. 11. The cumulative production corresponding to different permeability.

formation thickness, m fracture aperture, m mg gas viscosity, MPa$s dp/ds pressure gradient, MPa/m Qv volumetric gas flow rate, m3/s T shale gas reservoir temperature, K Tsc temperature at standard condition, K R universal gas constant, 8.134  106 MPa m3/mol$K; Z shale gas compressibility factor, fractional; V shale gas volume, m3; P shale gas reservoir pressure, MPa; psc pressure at standard condition, K n gas amount in moles, mol. m mass corresponding to n mol gas, kg M molecular mass for shale gas, M ¼ m/n, kg/mol. r shale gas molar density, mol/m3. rn mean pore space radius, m; l average free path of gas molecules, m dn pore diameter, m kb Boltzmann constant, 1.38  1023, J$K1 d molecular collision diameter, m 4 matrix porosity, fraction t tortuosity, correction the changes of diffusion path. dg gas molecule diameter, m. Kn Knudsen number, dimensionless DT transition effective diffusion coefficient, m2 s1 DK Knudsen effective diffusion coefficient, m2 s1 DF Fick effective diffusion coefficient, m2 s1; D diffusion coefficients, m2/s cg gas compressibility, MPa1; Ja mass influx caused by pressure gradient, kg/m2 s JD mass influx caused by diffusion, kg/m2 s Jdes desorption flux per unite time, kg/(m2 s) Kdes equilibrium desorption factor, kg/(m2 s) Jads adsorption flux per unit time, kg/(m2 s) Kads equilibrium adsorption factor, kg/(MPa m2 s) pn nanopores pressure, MPa. S0 total number of surface sites available for adsorption, m2 N Avogadro's constant, 6.02  1023/mol. Dkerogen diffusion coefficients for gas diffusion in kerogen, m2/s; rk mean kerogen radius, m; C gas concentration in kerogen, kg/m3 Ci gas concentration in kerogen at initial reservoir pressure and temperature, kg/m3 kh Henry constant Ac(x) Cross sectional area used in one-dimensional model, m2 A matrix pore wall surface area, m2 QRC l (Fe) recharge rate from the left, mol/s QRC r (Fe) recharge rate from the right, mol/s; Vfe fracture segment volume, m3. rsc gas molar density at standard condition, mol/m3 Qvsc volumetric flow rate in horizontal wells at standard condition, m3/s ks well segments index jf intersecting fracture index Vh volume of horizontal wellbore segment, m3

L. Mi et al. / Journal of Natural Gas Science and Engineering 20 (2014) 74e81

nb kapp k∞ F

q F Qcomp TXf TXi

number of blocks in a special one-dimensional model apparent permeability, mD absolute permeability, 10-3mD, k∞ ¼ rn2 =8 gas slippage factor, dimensionless,F ¼ 1 þ ð8pRT=MÞ0:5 ðmg =pn rn Þðð2=f Þ  1Þ fraction of total available surface sites occupied by molecules, qtþ1 ¼ q þ ðrN=S0 ÞJdiff Dt Z p real gas pseudo potential,f ¼ p=mZdp 0

scaled surface production rate, Qcomp ¼ ðpsc TQVsc =Tsc Þ fracture transmissibility, TX f ¼ w3 =12Ds matrix block transmissibility 8 > > 2ðkapp Ac Þ1þ1=2 > i¼1 > > > Dx1 þ Dx2 > < 0 i ¼ nb TXiþ1=2 ¼ > > > > 2ðkapp Ac Þiþ1=2 > > > i ¼ 2; /; nb  1 : Dxi þ Dxiþ1

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