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Triple-continuum modeling of shale gas reservoirs considering the effect of kerogen
Journal of Natural Gas Science and Engineering 24 (2015) 252e263
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Triple-continuum modeling of shale gas reservoirs considering the effect of kerogen Min Zhang a, Jun Yao a, *, Hai Sun a, b, Jian-lin Zhao a, Dong-yan Fan a, b, Zhao-qin Huang a, Yue-ying Wang a a b
School of Petroleum Engineering, China University of Petroleum, No. 66, Changjiang West Road, Huangdao District, Qingdao 266580, China School of Geosciences, China University of Petroleum, No. 66, Changjiang West Road, Huangdao District, Qingdao 266580, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 January 2015 Received in revised form 22 March 2015 Accepted 23 March 2015 Available online
Shale gas storage and transport mechanisms are notably different in kerogen systems and inorganic matrix systems. Based on these complex shale gas transport mechanisms, including viscous flow, Knudsen diffusion, surface diffusion and gas adsorption/desorption on the internal kerogen grain surfaces, a kerogen e inorganic matrix e fracture triple-continuum model is established. There are two transfer terms in the triple-continuum model in this paper. The kerogen e inorganic matrix transfer flow is simulated by the Warren-Root pseudo-steady state (PSS) transfer model, while the inorganic matrix e fracture transfer flow is simulated by the Vermeulen transient transfer model. To investigate the impact of the kerogen continuum on shale gas reservoir performance, a comparison between the matrix e fracture dual-continuum model and kerogen e inorganic matrix e fracture triplecontinuum model is conducted. The matrix e fracture transfer flow in the dual-continuum model is also simulated by the Vermeulen transient transfer model. In addition, two triple-continuum models with different inorganic matrix e fracture transfer models are compared to investigate the impact of the transfer model. One triple-continuum model uses the Warren-Root PSS transfer model. The other uses the Vermeulen transient transfer model. The mathematical model is solved by the PDE module of COMSOL Multiphysics. A sensitivity analysis of parameters affecting shale gas production, including kerogen pore volume, kerogen permeability, inorganic matrix permeability, fracture permeability and Langmuir parameters, is conducted. The results indicate that dividing the matrix system into a kerogen continuum and inorganic matrix continuum significantly influences shale gas reservoir performance. Not considering the kerogen continuum could lead to an overestimate in cumulative gas production of approximately 8%. The triplecontinuum model that uses the Vermeulen transient transfer model yields a higher recovery than that which uses the Warren-Root PSS transfer model. Moreover, natural fractures are the main permeable channels in shale gas reservoirs and play a more important role than the kerogen and inorganic matrix in shale gas recovery. Langmuir pressure and Langmuir volume also have significant effects on the cumulative production of desorbed gas, free gas and total gas, with the effect of Langmuir volume being relatively larger. In conclusion, a triple-continuum model with a transient transfer term should be incorporated into the numerical simulators of shale gas reservoirs to predict shale gas production more accurately. © 2015 Elsevier B.V. All rights reserved.
Keywords: Shale gas reservoir Triple-continuum model Numerical simulation Kerogen Transient transfer
1. Introduction
* Corresponding author. E-mail addresses: [email protected] (M. Zhang), [email protected] (J. Yao), [email protected] (H. Sun), [email protected] (J.-l. Zhao), [email protected] (D.-y. Fan), [email protected] (Z.-q. Huang), [email protected] (Y.-y. Wang). http://dx.doi.org/10.1016/j.jngse.2015.03.032 1875-5100/© 2015 Elsevier B.V. All rights reserved.
It is widely considered that conventional resource shortage has become a bottleneck of global economic development. As an important class of unconventional natural gas resources (EIA, 2013), shale gas reservoirs significantly contribute to the growing energy demand. In the past decade, shale gas resources
M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
have received additional attention and become a major focus of the petroleum industry, as well as energy resource industries worldwide (Wu et al., 2013). Compared to conventional reservoirs, shale has a relatively low porosity and ultra-low permeability at the nanoscale (Javadpour et al., 2007; Loucks et al., 2009). These characteristics mean that the traditional governing equation, Darcy's law, is no longer applicable. In addition, shale is a complex unconventional reservoir with coupled storage and transport mechanisms (Freeman et al., 2011; Hill and Nelson, 2000; Yao et al., 2013). Therefore, the simulation of shale gas reservoirs is far more difficult than that of conventional resources. Shale is a complex mixture of organic matter (i.e., porous kerogen pockets), inorganic matrix and natural fractures (Killough et al., 2013). The organic matter is a nanoscale porous medium that mainly consists of micropores (<2 nm) and mesopores (2e50 nm) (Kang et al., 2011), while inorganic pores with a much larger size (>100 nm) (Wasaki and Akkutlu, 2014). Because the organic material has a strong affinity (large molecular interaction) for the hydrocarbon fluids and large surface area associated with the pore walls, the organic pores are the ideal places for storage of shale gas in the adsorbed phase (Wasaki and Akkutlu, 2014). Due to the minor molecular interaction between the inorganic material and hydrocarbon fluids, and relatively larger pores in the inorganic material, the amount of gas adsorbed by the inorganic walls is considered negligible. In conclusion, free gas and adsorbed gas exist in the kerogen system, while only free gas exists in the inorganic matrix system. In addition, the transport mechanisms of shale gas are different in the kerogen and inorganic matrix. The transport mechanisms in the kerogen system include viscous flow, Knudsen diffusion, the surface diffusion and desorption mechanism of adsorbed gas. However, only viscous flow and Knudsen diffusion exist in the inorganic matrix system. Because the shale gas storage and transport mechanisms in the kerogen system and inorganic matrix system are notably different, it is necessary to divide the matrix system into a kerogen continuum and inorganic matrix continuum (Akkutlu and Fathi, 2012). The most critical issue faced when using a dual-continuum model is the handling of the matrixefracture transfer, which couples the matrix continuum with the fracture continuum. Generally, there are 3 ways to do so in a reservoir simulator framework (Azom and Javadpour, 2012). The first is the boundary condition approach, which is acceptable for near wellbore studies, such as well testing, but quite impractical for full field simulations. In addition, numerical instabilities will occur for large time steps when this method is used in a numerical simulator. The second approach is the Warren-Root method, which assumes that the transfer term between the matrix block and surrounding fractures is a pseudosteady state transfer and directly proportional to the difference between the matrix pressure and fracture pressure (Warren and Root, 1963). However, the Warren-Root PSS transfer model only holds for large times when the flow in matrix block can be represented by a pseudo-steady state flow regime. The errors will generally be quite large at small times (Zimmerman et al., 1993). The third method is the Vermeulen model, which provides a good approximation at all time scales. Additionally, the transient shape factor is not a constant, but a function of the reservoir pressure (Azom and Javadpour, 2012). In this study, the shale gas reservoir is simulated as a triplecontinuum (kerogen e inorganic matrix e natural fractures) system and the transport of shale gas takes place in the following sequence: kerogen / inorganic matrix / natural fractures (Akkutlu and Fathi, 2012; Kang et al., 2011). Therefore, there will be two transfer terms in the triple-continuum model. One term defines the transfer between the kerogen and inorganic matrix, and
253
the other term defines the transfer between the inorganic matrix and natural fractures. Because the Warren-Root PSS transfer model is relatively simple when compared to the Vermeulen model, which uses a nonlinear transfer formula, the present numerical simulations of the shale gas reservoirs typically use the Warren-Root PSS transfer model (Chawathe et al., 2014; Fathi and Akkutlu, 2014; Sun et al., 2013; Swami et al., 2013). However, due to the ultra-low matrix permeability and large fracture spacing, the transient shale gas flow from the inorganic matrix to natural fractures can take years in the triple-continuum model (Zimmerman et al., 1993). That is, it will take years for shale gas flow in the inorganic matrix block to reach a pseudosteady state flow regime. Hence, if we use the Warren-Root PSS transfer model for the entire inorganic matrix e natural fractures transfer period, the error will be quite large. Therefore, in this work we use the Vermeulen model for the inorganic matrix e fracture transfer. For the kerogen e inorganic matrix transfer, the permeability difference between the kerogen and inorganic matrix is very small (often one order of magnitude) when compared to the large permeability difference between the inorganic matrix and natural fractures. Moreover, the inorganic matrix spacing is quite small compared to fracture spacing. Therefore, the transient flow from the kerogen to inorganic matrix takes little time, and the shale gas flow in finely dispersed kerogen pockets will reach a pseudo-steady state flow regime relatively quickly. Hence, the Warren-Root PSS transfer model can be used to approximate the kerogen e inorganic matrix transfer period. This treatment of these two transfer terms in the triple-continuum model is justifiable and saves both computation time and memory. The mathematical model is solved by COMSOL Multiphysics. First, we compare a dual-continuum (matrix e fracture) model and triple-continuum (kerogen e inorganic matrix e fracture) model. Next, a comparison of two triple-continuum models with different inorganic matrix e fracture transfer models is conducted. Finally, a detailed sensitivity analysis of the impacts of parameters, including kerogen pore volume, kerogen permeability, inorganic matrix permeability, fracture permeability and Langmuir parameters, on the shale gas reservoir performance is conducted. 2. Triple-continuum model and numerical solution The assumptions are as follows: (1) the organic matter (kerogen pockets), inorganic matrix and natural fractures can be considered as continua in space; (2) only free gas exists in the inorganic matrix and natural fractures, while both free gas and adsorbed gas exist in kerogen; (3) the transport mechanisms in kerogen include viscous flow, Knudsen diffusion, surface diffusion and gas adsorption/desorption on the internal kerogen grain surfaces, while those in the inorganic matrix are viscous flow and Knudsen diffusion, and only viscous flow is considered in natural fractures; (4) the gas contains CH4 only; (5) shale gas flow in the reservoir is an isothermal process, with shale gas adsorption on the internal kerogen solid surfaces obeying the Langmuir isotherm equation. 2.1. Transport mechanisms and continuity equation of kerogen system 2.1.1. Viscous flow Viscous flow is the dominant transport mechanism in shale gas reservoirs, and the net flux generated by very small pressure gradients will exceed that caused by very large gradients in concentration (Freeman et al., 2011). The gas flux caused by viscous flow can be modeled by Darcy's law:
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Nk ¼ ck
kk0 Vpk m
(1)
Where Nk is the gas flux in kerogen pores caused by viscous flow in mol/(m2,s), ck is the moles of free gas per kerogen pore volume in mol/m3, kk0 is the kerogen intrinsic permeability in m2, m is the gas viscosity in Pa,s and pk is the kerogen pressure in Pa. 2.1.2. Knudsen diffusion Knudsen diffusion occurs when the mean free path of gas molecules is at the same scale as the pore size of the porous medium. In shale gas reservoirs, the pore diameters mainly range from 4 nm to 200 nm (Javadpour et al., 2007), which are of similar magnitude as the mean free path of CH4 molecules. Therefore, Knudsen diffusion is significant in shale gas reservoirs and cannot be ignored. The gas flux generated by Knudsen diffusion is (Akkutlu and Fathi, 2012):
Nkk ¼ εkp fDkk Vck
(2)
Where Nkk is the gas flux in kerogen pores caused by Knudsen diffusion in mol/(m2,s), ɸ ¼ Vp/Vb is the total matrix porosity, Vp is the total pore volume for the matrix including the kerogen organic pore volume and inorganic pore volume, Vb is the bulk volume of shale core, εkp ¼ Vkp/Vp is the portion of the kerogen pore volume in total interconnected matrix pore volume, Vkp is the kerogen pore volume, ɸk ¼ εkpɸ is the kerogen porosity, ɸm ¼ (1 εkp)ɸ is the inorganic matrix porosity, Dkk is the tortuosity-corrected Knudsen diffusivity of the kerogen system in m2/s, which can be represented by Civan (2010); Javadpour et al. (2007); Millington and Quirk (1961):
Dkk
16 1 ¼ fk 3 3
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kk0 Rg T pM
(3)
Where Rg is the universal gas constant (8.314 J/(K,mol)), T is the reservoir temperature in K and M is the molecular weight in kg/mol. 2.1.3. Surface diffusion Surface diffusion occurs when adsorbed gas exists on the solid wall. Due to the considerable amount of gas adsorbed onto the internal kerogen solid surfaces, the surface diffusion is also a significant transport mechanism in the kerogen continuum (Fathi and Akkutlu, 2009). Surface diffusion is described by Akkutlu and Fathi (2012)
Nks
¼ εks 1 f ff Ds Vcm
(4)
Where Nks is the gas flux in kerogen caused by surface diffusion in mol/(m2,s), εks is the portion of the kerogen grain volume in total shale core grain volume, ɸf is the fracture porosity, εks(1ɸɸf) ¼ (Vks/Vms)(Vms/Vb) is the kerogen solid volume per unit bulk volume of shale core, Vks and Vms are the kerogen grain volume and matrix grain volume, respectively, Ds is the surface diffusion coefficient in m2/s and cm is the moles of adsorbed gas per solid volume of kerogen in mol/m3. 2.1.4. Gas adsorption and desorption equation The Langmuir isotherm equation is applied to calculate the moles of adsorbed gas on the kerogen solid surfaces (Langmuir, 1916):
cm ¼ cms
Kck 1 þ Kck
(5)
Where cms is the maximum monolayer adsorbed gas on the kerogen
surfaces in mol/m3 and K is the equilibrium distribution coefficient in m3/mol. In this paper, we adopt the following expressions for cms and K:
cms ¼
K¼
rgrain VsL εks Vstd
(6)
zRg T pL
(7)
Where rgrain is the grain density of shale core in kg/m3, VsL is the Langmuir volume in m3/kg, which shows the maximum adsorbed gas volume converted into the standard condition (273.15 K and 101.325 kPa) per unit total grain mass, Vstd is the molar volume of the gas at the standard condition in m3/mol, z is the gas compressibility factor and pL is the Langmuir pressure in Pa.
2.1.5. Continuity equation of kerogen system The transport mechanisms in the kerogen system include viscous flow, Knudsen diffusion and surface diffusion. As analyzed in part 1, the kerogen e inoganic matrix transfer term is modeled by the Warren-Root PSS transfer model. Therefore, according to the conservation of mass, the continuity equation of the kerogen system can be presented as
v εkp fck
i h v εks 1 f ff cm
þ vt h i ¼ V$ εkp fDkk Vck þ V$ εks 1 f ff Ds Vcm k þ V$ ck k0 Vpk Wkm m vt
(8)
Where Wkm is the transfer term between the kerogen and inorganic matrix in mol/(m3,s). It can be approximated by the Warren-Root PSS transfer model, and defined as (Kazemi et al., 1976):
Wkm ¼
zRg Tck kk sPSS ðck cm Þ m
(9)
Where cm is the moles of free gas per inorganic pore volume in mol/ m3 and kk is the apparent kerogen permeability in m2. For further details of the derivation, please refer to Appendix A.
kk ¼ kk0
εkp fDkk m 1þ ck zRg Tkk0
þ
εks 1 f ff Ds cms m
K
ck zRg T
ð1 þ Kck Þ2 (10)
Where sPSS is the pseudo-steady state shape factor in 1/m2, which will be a history match parameter due to the uncertainties in the matrixefracture geometry. For the kerogen e inorganic matrix transfer term, sPSS reflects the geometry of the kerogen pockets (Warren and Root, 1963). In this study, we adopt the following method (Kazemi et al., 1976):
sPSS ¼ 4
1 L2mx
þ
1 L2my
! (11)
Where Lmx,Lmy are the inorganic matrix spacing of the x-axis and y-axis, respectively in m. In our work, we assume that the organic material (kerogen pockets) forms 100 microns clusters, which are embedded in inorganic matrix (Chawathe et al., 2014). Hence, we adopt 100 mm as the inorganic matrix spacing in our following numerical simulation.
M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
We adopt the Z-factor equation of state to describe the behavior of gas:
p ¼ zcRg T
εkp f þ εks 1 f ff cms (" V$
εkp fDkk þ εks
#
K 2
vck vt
ð1 þ Kck Þ 1 f ff Ds cms
# ) k þ ck zRg T k0 Vck ¼ Wkm m
K ð1 þ Kck Þ2
(13)
i h 1 εkp fcm
(19)
h i k ¼ V$ 1 εkp fDkm Vcm þ V$ cm m0 Vpm m (14)
Where km0 is the inorganic matrix intrinsic permeability in m2, pm is the inorganic matrix pressure in Pa and Dkm is the tortuositycorrected Knudsen diffusivity of the inorganic matrix system in m2/s, which can be represented by:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 13 km0 Rg T ¼ fm 3 pM
Wmf ¼
m
km
vt
kf 0 ¼ V$ cf Vpf þ Wmf Qg m
(20)
Where kf0 is the fracture intrinsic permeability in m2, pf is the fracture pressure in Pa and Qg is the production rate in mol/(m3,s), which is defined as (Peaceman, 1983):
Qg ¼
kf 0 cf zRg T q cf cw lnðre =rw Þ m
(21)
In the above equation, when the production well is located in a corner, q ¼ p/2. When the production well is in the center, q ¼ 2p. In addition, cw ¼ pw/zRgT, where pw is the bottomhole flowing pressure in Pa, rw is the wellbore radius in m and re is the drainage radius in m, which can be expressed as (Peaceman, 1983):
8 > > > < > > > : 0:28
0:14
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i h ðDxÞ2 þ ðDyÞ2
kx ¼ ky
h i1=2 1=2 1=2 ðDxÞ2 þ kx ky ðDyÞ2 ky kx 1=4 1=4 þ kx ky ky kx
kx sky (22)
(16)
Where Dx, Dy are the grid sizes in the x and y directions, respectively and kx, ky are the permeabilities in the x and y directions, respectively. Substituting Eq. (12) into Eq. (20) yields:
ff
vcf kf 0 V$ cf zRg T Vcf ¼ Wmf Qg vt m
(23)
2.4. Initial and boundary conditions Assuming that the initial pressure of the shale gas reservoir is pi, then the initial condition used in this work is:
(17)
sTS is the transient shape factor, which can be represented as (Azom and Javadpour, 2012; Zimmerman et al., 1993):
v ff cf
(15)
Where cf is the moles of free gas per fracture pore volume in mol/m3 and km is the apparent inorganic matrix permeability in m2. For further details of the derivation, please refer to Appendix A.
f D m ¼ km0 1 þ m km cm zRg Tkm0
Due to the millimeter scale of the average pore diameters in the fracture system, the effect of Knudsen diffusion is notably small, and only viscous flow is considered in the fracture system. In addition, the inorganic matrix e fracture transfer term and the production well should be considered in fracture system. Then, the continuity equation of the fracture system is given by:
r¼
Where Wmf is the transfer term between the inorganic matrix and fractures in mol/(m3,s). It can be approximated by the Vermeulen transient transfer model and defined as (Kazemi et al., 1976):
zRg Tcm km sTS cm cf
vc k m 1 εkp f V$ 1 εkp fDkm þ cm zRg T m0 Vcm vt m
2.3. Continuity equation of fracture system
þ Wkm Wmf
Dkm
(18)
2ðci cm Þ
¼ Wkm Wmf
Because only free gas exists in inorganic matrix, viscous flow and Knudsen diffusion are considered. In addition, the inorganic matrix system involves two transfer terms. One occurs between the kerogen and inorganic matrix, while the other is between the inorganic matrix and natural fractures, which are modeled by the Vermeulen transient transfer model. Therefore, according to the principle of mass balance, the continuity equation of the inorganic matrix system can be presented as:
vt
2ci cm þ cf
Where Lfx,Lfy are the fracture spacing of the x-axis and y-axis, respectively in m, ci ¼ pi/zRgT and pi is the initial reservoir pressure in Pa. Substituting Eq. (12) into Eq. (14), the following equation can be obtained:
2.2. Continuity equation of inorganic matrix system
v
sTS
!
255
(12)
Substituting Eq. (5) and Eq. (12) into Eq. (8), a simpler form of the kerogen continuum continuity equation can be obtained:
"
1 1 ¼4 2 þ 2 Lfx Lfy
pk ðx; y; tÞjt¼0 ¼ pm ðx; y; tÞjt¼0 ¼ pf ðx; y; tÞ
t¼0
Substituting Eq. (12) into Eq. (24) yields:
¼ pi
(24)
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M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
ck ðx; y; tÞjt¼0 ¼ cm ðx; y; tÞjt¼0 ¼ cf ðx; y; tÞ
t¼0
¼
pi zRg T
(25)
The boundary conditions are G ¼ GI þ GO, where GI represents the inner boundary of the production well and GO represents the outer boundary. We applied the constant pressure inner boundary condition and the no-flow outer boundary condition. The inner boundary condition is:
pf ðx; y; tÞ
GI
¼ pw
(26)
Substituting Eq. (12) into Eq. (26) yields:
cf ðx; y; tÞ
GI
¼
pw zRg T
(27)
The outer boundary condition is:
vpk ¼ 0; vn GO
vpm ¼ 0; vn GO
vpf ¼0 vn GO
(28)
Substituting Eq. (12) into Eq. (28) yields:
vck ¼ 0; vn GO
vcm ¼ 0; vn GO
vcf ¼0 vn GO
Fig. 1. 2D shale gas reservoir production area and the simulation domain (gray area).
(29)
2.5. Numerical solution The above equations are solved using the PDE module of COMSOL Multiphysics (a commercial software, which can directly implement partial differential equations and initial-boundary conditions). The time-dependent solver e backward differentiation formula (BDF) is used to solve the PDEs. Direct (UMFPACK) is used as the linear system solver in our work. 3. Results and discussion The impacts of the kerogen system, transfer model, kerogen permeability, inorganic matrix permeability, fracture permeability and Langmuir parameters on the shale gas reservoir performance were analyzed. We built two simulation scenarios to investigate the effect of the transfer model. In case 1, the intrinsic permeability difference between the inorganic matrix and fracture is small. In case 2, the intrinsic permeability difference between the inorganic matrix and fracture is large. As is shown in Fig. 1, because of the symmetry of the production area, only the gray portion of the production area is simulated and analyzed, for simplicity. The simulation parameters for case 1 and case 2 of the triple-continuum model are given in Table 1. These parameters were selected based on Akkutlu and Fathi (2012), Moridis et al. (2010) and Wasaki and Akkutlu (2014). 3.1. Comparison between dual-continuum model and triplecontinuum model To investigate the impact of the kerogen system on shale gas reservoir performance, a comparison was conducted between the dual-continuum model and triple-continuum model (for case 2). The simulation parameters of the dual-continuum model are given in Table 2. The matrix e fracture transfer term in the dualcontinuum model is described by the Vermeulen transient transfer model. Fig. 2 and Fig. 3 show the variations in cumulative gas production and reservoir pressure over time of the dual-continuum model and triple-continuum model. As illustrated in Figs. 2 and 3,
the cumulative gas production of the dual-continuum model is higher than that of triple-continuum model, and the matrix pressure of the dual-continuum model is lower than the kerogen pressure of the triple-continuum model. Moreover, cumulative gas production was overestimated by approximately 8% when the kerogen continuum was not considered. There are two main reasons for the above results. The first reason is that the mass exchange term between the kerogen and inorganic matrix can generate additional resistance, which leads to less cumulative gas production in the triple-continuum model. The second reason is that the intrinsic permeability of the kerogen system is smaller than that of the inorganic matrix system. It is also smaller than that of the matrix system in the dual-continuum model. Therefore, the reservoir pressure of the dual-continuum model decreases more rapidly than that of the triple-continuum model, which causes the matrix pressure of the dual-continuum model to be lower than the kerogen pressure of the triple-continuum model. Thus, it can be concluded that the matrix system should be divided into a kerogen continuum and inorganic matrix continuum. The triple-continuum model should be incorporated into the shale gas reservoir simulator.
3.2. Effect of the transfer model A comparison of two triple-continuum models with different inorganic matrix e fracture transfer models was conducted to investigate the impact of the transfer model. One triple-continuum model uses the Warren-Root PSS transfer model. The other uses the Vermeulen transient transfer model. The total cumulative gas production and reservoir pressure of two triple-continuum models for case 1 and case 2 are shown as Figs. 4 to 7. For both case 1 and case 2, the results show that the Vermeulen transient transfer model has a higher total cumulative gas production and lower reservoir pressure than those of the Warren-Root PSS transfer model. Apparently, our model gives a higher recovery for two cases, which implies that the present models might underestimate the recovery of shale gas reservoirs with natural fractures or fissures. In addition, the cumulative gas production and reservoir pressure differences between the Vermeulen transient transfer model and Warren-Root PSS transfer
M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
257
Table 1 Simulation parameters of the triple-continuum model for case 1 and case 2. Parameters
Value
Unit
Definition
εkp εks ɸ ɸf Rg T Ds kk0 km0 kf0 in case 1 kf0 in case 2 M Lmx ¼ Lmy Lfx ¼ Lfy pi
0.5 0.01 0.05 0.001 8.314 323.14 1 107 1 1020 1 1019 1 1014 1 1011 0.016 1 104 10 10.4 106 p/2 3.45 106 0.1 1.018 105 2600 3.121398 10-3 10.4 106 0.0224 400
e e e e J/(K,mol) K m2/s m2 m2 m2 m2 kg/mol m m Pa e Pa m Pa,s kg/m3 m3/kg Pa m3/mol m
portion of kerogen pore volume in total interconnected matrix pore volume portion of kerogen grain volume in total shale core grain volume total matrix porosity fracture porosity universal gas constant reservoir temperature surface diffusion coefficient kerogen intrinsic permeability inorganic matrix intrinsic permeability fracture intrinsic permeability in case 1 fracture intrinsic permeability in case 2 molecular weight inorganic matrix spacing fracture spacing initial reservoir pressure e bottomhole flowing pressure wellbore radius gas viscosity grain density of shale core Langmuir volume Langmuir pressure molar volume at standard condition well spacing
q
pw rw
m rgrain
VsL pL Vstd Lw
Table 2 Simulation parameters of the dual-continuum model. Parameters Value
Unit
Definition
ɸ ɸf Rg T Ds km0 kf0 M Lfx ¼ Lfy pi
e e J/(K,mol) K m2/s m2 m2 kg/mol m Pa e Pa m Pa,s kg/m3 m3/kg Pa m3/mol m
matrix porosity fracture porosity universal gas constant reservoir temperature surface diffusion coefficient matrix intrinsic permeability fracture intrinsic permeability molecular weight fracture spacing initial reservoir pressure e bottomhole flowing pressure wellbore radius gas viscosity grain density of shale core Langmuir volume Langmuir pressure molar volume at standard condition well spacing
q
pw rw
m rgrain VsL pL Vstd Lw
0.05 0.001 8.314 323.14 1 107 1 1019 1 1011 0.016 10 10.4 106 p/2 3.45 106 0.1 1.018 105 2600 3.121398 10-3 10.4 106 0.0224 400
model are larger for case 2 than for case 1. The results indicate that as the permeability difference between the matrix system and fracture system becomes larger, the impact of the transfer model on shale gas recovery gets larger. 3.3. Sensitivity analysis A set of data is prepared and given in Table 3 to investigate the impacts of kerogen pore volume, kerogen permeability, inorganic matrix permeability, fracture permeability and Langmuir parameters on shale gas reservoir performance. 3.3.1. Kerogen pore volume Organic nanopores constitute the most widespread and numerous pore type in shale gas reservoirs. Hence, the effect of kerogen pore volume on shale gas production (for case 1) was examined. The parameter εkp represents the portion of the kerogen
Fig. 2. Comparison of cumulative gas production between the dual-continuum model and triple-continuum model.
pore volume in the total interconnected matrix pore volume. Figs. 8 to 10 show the variation of cumulative production of desorbed gas, free gas and total gas over time for different kerogen pore volumes. As illustrated in Figs. 8 to 10, the cumulative production of desorbed gas, free gas and total gas decreases as kerogen pore volume increases. The reasons are as follows. As the kerogen pore volume increases, the amount of free gas increases in the kerogen nanopores and decreases in the inorganic pores. However, compared to the free gas in the inorganic pores, the free gas in the kerogen nanopores needs to overcome an additional resistance, which is caused by the mass exchange term between the kerogen and inorganic matrix to flow into the fractures. Therefore, the cumulative free gas production decreases. The adsorbed gas desorbs from the kerogen solid surfaces into the free gas phase and contributes to the total production only as the reservoir pressure decreases with production. When the production rate of free gas in the kerogen nanopores decreases, the desorption rate of adsorbed
258
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Fig. 3. Comparison of reservoir pressure between the dual-continuum model and triple-continuum model.
Fig. 6. Effect of the transfer model on total cumulative gas production for case 2.
Fig. 7. Effect of the transfer model on kerogen pressure for case 2. Fig. 4. Effect of the transfer model on total cumulative gas production for case 1.
gas also decreases. Hence, the cumulative production of desorbed gas and total gas decreases in turn. 3.3.2. Intrinsic permeability Figs. 11 to 13 show the variation of total cumulative gas production over time for different kerogen intrinsic permeabilities, inorganic matrix intrinsic permeabilities and fracture intrinsic permeabilities (for case 1). As illustrated in Figs. 11 to 13, the kerogen intrinsic permeability has almost no effect on total cumulative gas production. However, the total cumulative gas production increases as the intrinsic permeability of the inorganic matrix and fractures increases. In addition, the impact of fracture permeability is the most obvious, which reveals that natural fractures are the main permeable channels in shale gas reservoirs and significantly contribute to shale gas production.
Fig. 5. Effect of the transfer model on kerogen pressure for case 1.
3.3.3. Langmuir parameters Figs. 14 to 19 show the variation in the cumulative production of desorbed gas, free gas and total gas over time for different Langmuir pressures and volumes. As illustrated in Figs. 14 to 16, as Langmuir pressure increases, the cumulative production of desorbed gas and total gas decreases, while the cumulative free gas production
M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
259
Table 3 List of parameters for the sensitivity analysis of triple-continuum model. Parameters
Unit
Range
εkp kk0 km0 kf0 pL VsL
e m2 m2 m2 Pa m3/kg
0.25 ~ 0.75 1 1019 ~ 1 1021 1 1018 ~ 1 1020 1 1014 ~ 1 1016 5.2 106 ~ 15.6 106 1.560699 103 ~ 4.682097 103
Fig. 10. Comparison of total cumulative gas production for different kerogen pore volumes.
Fig. 8. Comparison of cumulative desorbed gas production for different kerogen pore volumes.
Fig. 11. Effect of the kerogen intrinsic permeability on shale gas production.
Fig. 9. Comparison of cumulative free gas production for different kerogen pore volumes.
slightly increases. The reasons are as follows. As Langmuir pressure increases, the desorption rate of the adsorbed gas decreases and the cumulative desorbed gas production decreases accordingly. As the adsorbed gas desorption rate decreases, the rate of pore pressure decline increases. In turn, this causes the cumulative free gas production to increase. However, the effect of cumulative desorbed gas production on total cumulative gas production is larger. The total cumulative gas production decreases as Langmuir pressure increases. As illustrated in Figs. 17 to 19, as Langmuir volume
Fig. 12. Effect of the inorganic matrix intrinsic permeability on shale gas production.
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Fig. 13. Effect of the fracture intrinsic permeability on shale gas production.
Fig. 14. Cumulative desorbed gas production under different Langmuir pressures.
Fig. 15. Cumulative free gas production under different Langmuir pressures.
Fig. 16. Total cumulative gas production under different Langmuir pressures.
Fig. 17. Cumulative desorbed gas production under different Langmuir volumes.
Fig. 18. Cumulative free gas production under different Langmuir volumes.
M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
261
3. The natural fractures are the main permeable channels in shale gas reservoirs. They play a more important role than the kerogen and inorganic matrix during shale gas production. 4. Langmuir pressure and Langmuir volume also have significant effects on the cumulative production of desorbed gas, free gas and total gas, with the effect of Langmuir volume being relatively larger. Therefore, accurate lab measurements of Langmuir parameters are necessary for the numerical simulation of shale gas reservoirs. Currently, hydraulic fracturing technology is being widely applied for shale gas development. In future work, it will be both interesting and challenging to incorporate hydraulic fractures into the triple-continuum model to investigate their effects on shale gas recovery. Acknowledgments
Fig. 19. Total cumulative gas production under different Langmuir volumes.
increases, the cumulative production of desorbed gas and total gas increases, while the cumulative free gas production slightly decreases. The reasons are as follows. As Langmuir volume increases, the desorption rate of the adsorbed gas increases and the cumulative desorbed gas production increases accordingly. As adsorbed gas desorption rate increases, the rate of pore pressure decline decreases and the cumulative free gas production decreases. However, the effect of cumulative desorbed gas production on total cumulative gas production is larger, so the total cumulative gas production increases as Langmuir volume increases.
The authors thank the National Natural Science Foundation of China (No.51490654, No.51234007, No.51404291), the Natural Science Foundation of Shandong Province (ZR2014EL016, ZR2014 EEP018), and China Postdoctoral Science Foundation (2014M55 1989) for supporting this work. Nomenclature ck cm cf ci cw cm
4. Conclusions cms In this paper, based on the complex gas transport mechanisms, including viscous flow, Knudsen diffusion and surface diffusion, the kerogen e inorganic matrix e fracture triple-continuum model for shale gas reservoirs has been presented. In addition, the gas adsorption/desorption on the internal kerogen grain surfaces was also considered. The Vermeulen transient transfer model was incorporated into our triple-continuum model. Next, a finite element simulation software, COMSOL Multiphysics, was used to solve the mathematical model. Subsequently, the impacts of the kerogen system, transfer model, kerogen permeability, inorganic matrix permeability, fracture permeability and Langmuir parameters on the shale gas reservoir performance were examined. The specific conclusions of this study are summarized below: 1. Dividing the matrix system into a kerogen continuum and inorganic matrix continuum has a significant effect on shale gas reservoir performance. Not considering the kerogen continuum could lead to an overestimate in cumulative gas production of approximately 8%. Hence, the triple-continuum model should be incorporated into the shale gas reservoir simulator. 2. The triple-continuum model using the Vermeulen transient transfer model for inorganic matrix e fracture transfer flow yields a higher recovery than using the Warren-Root PSS transfer model, which may underestimate shale gas recovery. A larger difference between the results of the Vermeulen transient transfer model and the current model will occur if properties of the matrix system and fracture system become increasingly different.
Dkk Dkm Ds kk0 km0 kf0 kk km K Lmx,Lmy Lfx,Lfy Lw M Nk Nkk Nks pk pm pf pL pi
the moles of free gas per kerogen organic pore volume [mol/m3] the moles of free gas per inorganic pore volume [mol/m3] the moles of free gas per fracture pore volume [mol/m3] the initial moles of free gas per reservoir pore volume [mol/m3] the moles of free gas per reservoir pore volume in bottomhole [mol/m3] moles of adsorbed gas per solid volume of kerogen [mol/ m3 ] maximum monolayer adsorbed gas on the kerogen surfaces [mol/m3] tortuosity-corrected Knudsen diffusivity of kerogen system [m2/s] tortuosity-corrected Knudsen diffusivity of inorganic matrix system [m2/s] surface diffusion coefficient [m2/s] kerogen intrinsic permeability [m2] inorganic matrix intrinsic permeability [m2] fracture intrinsic permeability [m2] apparent kerogen permeability [m2] apparent inorganic matrix permeability [m2] equilibrium distribution coefficient [m3/mol] inorganic matrix spacing of x axis and y axis, respectively [m] fracture spacing of x axis and y axis, respectively [m] well spacing [m] molecular weight [kg/mol] gas flux in kerogen pores caused by viscous flow [mol/ (m2,s)] gas flux in kerogen pores caused by Knudsen diffusion [mol/(m2,s)] gas flux in kerogen caused by surface diffusion [mol/ (m2,s)] kerogen pressure [Pa] inorganic matrix pressure [Pa] fracture pressure [Pa] Langmuir pressure [Pa] initial reservoir pressure [Pa]
262
pw Qg Rg rw re T VsL Vstd
M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
bottomhole flowing pressure [Pa] production rate [mol/(m3,s)] universal gas constant [J/(K,mol)] wellbore radius [m] drainage radius [m] reservoir temperature [K] Langmuir volume [m3/kg] gas molar volume at the standard condition (273.15 K and
Appendix A In this appendix, we derive the apparent permeabilities of the kerogen system and inorganic matrix system, which are given in Eqs. (10) and (17). Starting from the continuity equation of the kerogen system (Eq. (8)), we manipulate the mass flux terms of the right-hand-side (RHS) to obtain the apparent kerogen permeability:
h i k ðRHSÞ ¼ V$ εkp fDkk Vck þ V$ εks 1 f ff Ds Vcm þ V$ ck k0 Vpk Wkm m pk Kck k þ V$ εks 1 f ff Ds V cms þ V$ ck k0 Vpk Wkm ¼ V$ εkp fDkk V zRg T 1 þ Kck m " # pk K kk0 þ V$ εks 1 f ff Ds cms Vp ¼ V$ εkp fDkk V Vc þ V$ c k k k Wkm zRg T m ð1 þ Kck Þ2 " # pk K pk kk0 þ V$ εks 1 f ff Ds cms þ V$ c Vp ¼ V$ εkp fDkk V V k k Wkm zRg T zRg T m ð1 þ Kck Þ2 3 2 εks 1 f ff Ds cms εkp fDkk K 5 þ V$ ck kk0 Vpk Wkm Vpk þ V$4 ¼ V$ Vp k zRg T zRg T m ð1 þ Kck Þ2
(A-1)
82 9 3 < ε fD = 1 f f Ds cms ε ks f K kk0 5 kp kk 4 þ Vpk Wkm ¼ V$ þ ck 2 : zRg T ; zRg T m ð1 þ Kck Þ 8 9 2 3 < 1 ε fD m εks 1 f ff Ds cms m = K kp kk þ ¼ V$ ck 4 þ kk0 5Vpk Wkm 2 : m ck zRg T ; ck zRg T ð1 þ Kck Þ 8 9 2 3 < 1 = εkp fDkk m εks 1 f ff Ds cms m K 5Vpk Wkm þ ¼ V$ ck 4kk0 þ 2 : m ; ck zRg T ck zRg T ð1 þ Kck Þ
Wkm Wmf z
101.325 kPa) [m3/mol] transfer term between kerogen and inorganic matrix [mol/(m3,s)] transfer term between inorganic matrix and fractures [mol/(m3,s)] gas compressibility factor
Greek symbols m gas viscosity [Pa,s] εkp portion of kerogen pore volume in total interconnected matrix pore volume εks portion of kerogen grain volume in total shale core grain volume ɸ total matrix porosity ɸk kerogen porosity ɸm inorganic matrix porosity ɸf fracture porosity rgrain grain density of shale core [kg/m3] sPSS pseudo-steady state shape factor [1/m2] sTS transient shape factor [1/m2] Subscripts k related to kerogen m related to inorganic matrix f related to fracture
Hence, the total mass flux of the kerogen system, J, can be expressed as follows:
2 εkp fDkk m 14 k þ J ¼ ck m k0 ck zRg T 3 εks 1 f ff Ds cms m K 5Vpk þ ck zRg T ð1 þ Kck Þ2
(A-2)
Then, by binding terms which have the same dimension as kk0, we can obtain a new expression of apparent kerogen permeability:
kk ¼ kk0
εkp fDkk m 1þ ck zRg Tkk0
þ
εks 1 f ff Ds cms m
K
ck zRg T
ð1 þ Kck Þ2 (A-3)
In the same way, starting from the continuity equation of the inorganic matrix system (Eq. (14)), we can obtain the expression for apparent inorganic matrix permeability:
f D m km ¼ km0 1 þ m km cm zRg Tkm0
(A-4)
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References Akkutlu, I.Y., Fathi, E., 2012. Multiscale gas transport in shales with local Kerogen heterogeneities. SPE J. 17, 1002e1011. Azom, P.N., Javadpour, F., 2012. Dual-continuum modeling of shale and tight gas reservoirs. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, San Antonio, Texas, USA. Chawathe, A., Sun, H., Shi, X., Li, L., Hoteit, H., 2014. Understanding shale Gas production mechanisms through reservoir simulation. In: SPE/EAGE European Unconventional Resources Conference and Exhibition. Society of Petroleum Engineers. Civan, F., 2010. Effective correlation of apparent gas permeability in tight porous media. Transp. Porous Media 82, 375e384. EIA, U., 2013. Technically Recoverable Shale Oil and Shale Gas Resources: an Assessment of 137 Shale Formations in 41 Countries outside the United States. US Department of Energy/EIA, Washington, DC. Fathi, E., Akkutlu, I.Y., 2009. Nonlinear sorption kinetics and surface diffusion effects on gas transport in low-permeability formations. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, New Orleans, Louisiana. Fathi, E., Akkutlu, I.Y., 2014. Multi-component gas transport and adsorption effects during CO2 injection and enhanced shale gas recovery. Int. J. Coal Geol. 123, 52e61. Freeman, C.M., Moridis, G.J., Blasingame, T.A., 2011. A numerical study of microscale flow behavior in tight gas and shale gas reservoir systems. Transp. Porous Media 90, 253e268. Hill, D.G., Nelson, C.R., 2000. Gas productive fractured shales: an overview and update. Gas. Tips 6, 4e13. Javadpour, F., Fisher, D., Unsworth, M., 2007. Nanoscale gas flow in shale gas sediments. J. Can. Petrol. Technol. 46, 55e61. Kang, S.M., Fathi, E., Ambrose, R.J., Akkutlu, I.Y., Sigal, R.F., 2011. Carbon dioxide storage capacity of organic-rich shales. SPE J. 16, 842e855. Kazemi, H., Merrill, L.S., Porterfield, K.L., Zeman, P.R., 1976. Numerical simulation of water-oil flow in naturally fractured reservoirs. SPE J. 16, 317e326. Killough, J.E., Wang, Y.-h., Yan, B.-c, 2013. Beyond dual-porosity modeling for the simulation of complex flow mechanisms in shale reservoirs. In: SPE Reservoir
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Simulation Symposium. Society of Petroleum Engineers, The Woodlands, Texas, USA. Langmuir, I., 1916. The constitution and fundamental properties of solids and liquids. part i. solids. J. Am. Chem. Soc. 38, 2221e2295. Loucks, R.G., Reed, R.M., Ruppel, S.C., Jarvie, D.M., 2009. Morphology, genesis, and distribution of nanometer-scale pores in siliceous mudstones of the Mississippian Barnett Shale. J. Sediment. Res. 79, 848e861. Millington, R.J., Quirk, J.P., 1961. Permeability of porous solids. Trans. Faraday Soc. 57, 1200e1207. Moridis, G.J., Blasingame, T.A., Freeman, C.M., 2010. Analysis of mechanisms of flow in fractured tight-gas and shale-gas reservoirs. In: SPE Latin American and Caribbean Petroleum Engineering Conference. Society of Petroleum Engineers. Peaceman, D.W., 1983. Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability. SPE J. 23, 531e543. Sun, H., Yao, J., Gao, S.-h., Fan, D.-y., Wang, C.-c., Sun, Z.-x, 2013. Numerical study of CO2 enhanced natural gas recovery and sequestration in shale gas reservoirs. Int. J. Greenh. Gas Control 19, 406e419. Swami, V., Javadpour, F., Settari, A., 2013. A numerical model for multi-mechanism flow in shale Gas reservoirs with application to laboratory scale testing. In: EAGE Annual Conference & Exhibition Incorporating SPE. Europec. Society of Petroleum Engineers, London, UK. Warren, J., Root, P.J., 1963. The behavior of naturally fractured reservoirs. SPE J. 3, 245e255. Wasaki, A., Akkutlu, I.Y., 2014. Permeability of organic-rich shale. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. Wu, Y.-S., Li, N., Wang, C., Ran, Q.-q., Li, J.-f., Yuan, J.-r, 2013. A multiple-continuum model for simulation of Gas production from shale Gas reservoirs. In: SPE Reservoir Characterization and Simulation Conference and Exhibition. Society of Petroleum Engineers, Abu Dhabi, UAE. Yao, J., Sun, H., Fan, D.-y., Wang, C.-c., Sun, Z.-x, 2013. Numerical simulation of gas transport mechanisms in tight shale gas reservoirs. Petrol. Sci. 10, 528e537. Zimmerman, R.W., Chen, G., Hadgu, T., Bodvarsson, G.S., 1993. A numerical dualporosity model with semianalytical treatment of fracture/matrix flow. Water Resour. Res. 29, 2127e2137.
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Triple-continuum modeling of shale gas reservoirs considering the effect of kerogen Min Zhang a, Jun Yao a, *, Hai Sun a, b, Jian-lin Zhao a, Dong-yan Fan a, b, Zhao-qin Huang a, Yue-ying Wang a a b
School of Petroleum Engineering, China University of Petroleum, No. 66, Changjiang West Road, Huangdao District, Qingdao 266580, China School of Geosciences, China University of Petroleum, No. 66, Changjiang West Road, Huangdao District, Qingdao 266580, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 January 2015 Received in revised form 22 March 2015 Accepted 23 March 2015 Available online
Shale gas storage and transport mechanisms are notably different in kerogen systems and inorganic matrix systems. Based on these complex shale gas transport mechanisms, including viscous flow, Knudsen diffusion, surface diffusion and gas adsorption/desorption on the internal kerogen grain surfaces, a kerogen e inorganic matrix e fracture triple-continuum model is established. There are two transfer terms in the triple-continuum model in this paper. The kerogen e inorganic matrix transfer flow is simulated by the Warren-Root pseudo-steady state (PSS) transfer model, while the inorganic matrix e fracture transfer flow is simulated by the Vermeulen transient transfer model. To investigate the impact of the kerogen continuum on shale gas reservoir performance, a comparison between the matrix e fracture dual-continuum model and kerogen e inorganic matrix e fracture triplecontinuum model is conducted. The matrix e fracture transfer flow in the dual-continuum model is also simulated by the Vermeulen transient transfer model. In addition, two triple-continuum models with different inorganic matrix e fracture transfer models are compared to investigate the impact of the transfer model. One triple-continuum model uses the Warren-Root PSS transfer model. The other uses the Vermeulen transient transfer model. The mathematical model is solved by the PDE module of COMSOL Multiphysics. A sensitivity analysis of parameters affecting shale gas production, including kerogen pore volume, kerogen permeability, inorganic matrix permeability, fracture permeability and Langmuir parameters, is conducted. The results indicate that dividing the matrix system into a kerogen continuum and inorganic matrix continuum significantly influences shale gas reservoir performance. Not considering the kerogen continuum could lead to an overestimate in cumulative gas production of approximately 8%. The triplecontinuum model that uses the Vermeulen transient transfer model yields a higher recovery than that which uses the Warren-Root PSS transfer model. Moreover, natural fractures are the main permeable channels in shale gas reservoirs and play a more important role than the kerogen and inorganic matrix in shale gas recovery. Langmuir pressure and Langmuir volume also have significant effects on the cumulative production of desorbed gas, free gas and total gas, with the effect of Langmuir volume being relatively larger. In conclusion, a triple-continuum model with a transient transfer term should be incorporated into the numerical simulators of shale gas reservoirs to predict shale gas production more accurately. © 2015 Elsevier B.V. All rights reserved.
Keywords: Shale gas reservoir Triple-continuum model Numerical simulation Kerogen Transient transfer
1. Introduction
* Corresponding author. E-mail addresses: [email protected] (M. Zhang), [email protected] (J. Yao), [email protected] (H. Sun), [email protected] (J.-l. Zhao), [email protected] (D.-y. Fan), [email protected] (Z.-q. Huang), [email protected] (Y.-y. Wang). http://dx.doi.org/10.1016/j.jngse.2015.03.032 1875-5100/© 2015 Elsevier B.V. All rights reserved.
It is widely considered that conventional resource shortage has become a bottleneck of global economic development. As an important class of unconventional natural gas resources (EIA, 2013), shale gas reservoirs significantly contribute to the growing energy demand. In the past decade, shale gas resources
M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
have received additional attention and become a major focus of the petroleum industry, as well as energy resource industries worldwide (Wu et al., 2013). Compared to conventional reservoirs, shale has a relatively low porosity and ultra-low permeability at the nanoscale (Javadpour et al., 2007; Loucks et al., 2009). These characteristics mean that the traditional governing equation, Darcy's law, is no longer applicable. In addition, shale is a complex unconventional reservoir with coupled storage and transport mechanisms (Freeman et al., 2011; Hill and Nelson, 2000; Yao et al., 2013). Therefore, the simulation of shale gas reservoirs is far more difficult than that of conventional resources. Shale is a complex mixture of organic matter (i.e., porous kerogen pockets), inorganic matrix and natural fractures (Killough et al., 2013). The organic matter is a nanoscale porous medium that mainly consists of micropores (<2 nm) and mesopores (2e50 nm) (Kang et al., 2011), while inorganic pores with a much larger size (>100 nm) (Wasaki and Akkutlu, 2014). Because the organic material has a strong affinity (large molecular interaction) for the hydrocarbon fluids and large surface area associated with the pore walls, the organic pores are the ideal places for storage of shale gas in the adsorbed phase (Wasaki and Akkutlu, 2014). Due to the minor molecular interaction between the inorganic material and hydrocarbon fluids, and relatively larger pores in the inorganic material, the amount of gas adsorbed by the inorganic walls is considered negligible. In conclusion, free gas and adsorbed gas exist in the kerogen system, while only free gas exists in the inorganic matrix system. In addition, the transport mechanisms of shale gas are different in the kerogen and inorganic matrix. The transport mechanisms in the kerogen system include viscous flow, Knudsen diffusion, the surface diffusion and desorption mechanism of adsorbed gas. However, only viscous flow and Knudsen diffusion exist in the inorganic matrix system. Because the shale gas storage and transport mechanisms in the kerogen system and inorganic matrix system are notably different, it is necessary to divide the matrix system into a kerogen continuum and inorganic matrix continuum (Akkutlu and Fathi, 2012). The most critical issue faced when using a dual-continuum model is the handling of the matrixefracture transfer, which couples the matrix continuum with the fracture continuum. Generally, there are 3 ways to do so in a reservoir simulator framework (Azom and Javadpour, 2012). The first is the boundary condition approach, which is acceptable for near wellbore studies, such as well testing, but quite impractical for full field simulations. In addition, numerical instabilities will occur for large time steps when this method is used in a numerical simulator. The second approach is the Warren-Root method, which assumes that the transfer term between the matrix block and surrounding fractures is a pseudosteady state transfer and directly proportional to the difference between the matrix pressure and fracture pressure (Warren and Root, 1963). However, the Warren-Root PSS transfer model only holds for large times when the flow in matrix block can be represented by a pseudo-steady state flow regime. The errors will generally be quite large at small times (Zimmerman et al., 1993). The third method is the Vermeulen model, which provides a good approximation at all time scales. Additionally, the transient shape factor is not a constant, but a function of the reservoir pressure (Azom and Javadpour, 2012). In this study, the shale gas reservoir is simulated as a triplecontinuum (kerogen e inorganic matrix e natural fractures) system and the transport of shale gas takes place in the following sequence: kerogen / inorganic matrix / natural fractures (Akkutlu and Fathi, 2012; Kang et al., 2011). Therefore, there will be two transfer terms in the triple-continuum model. One term defines the transfer between the kerogen and inorganic matrix, and
253
the other term defines the transfer between the inorganic matrix and natural fractures. Because the Warren-Root PSS transfer model is relatively simple when compared to the Vermeulen model, which uses a nonlinear transfer formula, the present numerical simulations of the shale gas reservoirs typically use the Warren-Root PSS transfer model (Chawathe et al., 2014; Fathi and Akkutlu, 2014; Sun et al., 2013; Swami et al., 2013). However, due to the ultra-low matrix permeability and large fracture spacing, the transient shale gas flow from the inorganic matrix to natural fractures can take years in the triple-continuum model (Zimmerman et al., 1993). That is, it will take years for shale gas flow in the inorganic matrix block to reach a pseudosteady state flow regime. Hence, if we use the Warren-Root PSS transfer model for the entire inorganic matrix e natural fractures transfer period, the error will be quite large. Therefore, in this work we use the Vermeulen model for the inorganic matrix e fracture transfer. For the kerogen e inorganic matrix transfer, the permeability difference between the kerogen and inorganic matrix is very small (often one order of magnitude) when compared to the large permeability difference between the inorganic matrix and natural fractures. Moreover, the inorganic matrix spacing is quite small compared to fracture spacing. Therefore, the transient flow from the kerogen to inorganic matrix takes little time, and the shale gas flow in finely dispersed kerogen pockets will reach a pseudo-steady state flow regime relatively quickly. Hence, the Warren-Root PSS transfer model can be used to approximate the kerogen e inorganic matrix transfer period. This treatment of these two transfer terms in the triple-continuum model is justifiable and saves both computation time and memory. The mathematical model is solved by COMSOL Multiphysics. First, we compare a dual-continuum (matrix e fracture) model and triple-continuum (kerogen e inorganic matrix e fracture) model. Next, a comparison of two triple-continuum models with different inorganic matrix e fracture transfer models is conducted. Finally, a detailed sensitivity analysis of the impacts of parameters, including kerogen pore volume, kerogen permeability, inorganic matrix permeability, fracture permeability and Langmuir parameters, on the shale gas reservoir performance is conducted. 2. Triple-continuum model and numerical solution The assumptions are as follows: (1) the organic matter (kerogen pockets), inorganic matrix and natural fractures can be considered as continua in space; (2) only free gas exists in the inorganic matrix and natural fractures, while both free gas and adsorbed gas exist in kerogen; (3) the transport mechanisms in kerogen include viscous flow, Knudsen diffusion, surface diffusion and gas adsorption/desorption on the internal kerogen grain surfaces, while those in the inorganic matrix are viscous flow and Knudsen diffusion, and only viscous flow is considered in natural fractures; (4) the gas contains CH4 only; (5) shale gas flow in the reservoir is an isothermal process, with shale gas adsorption on the internal kerogen solid surfaces obeying the Langmuir isotherm equation. 2.1. Transport mechanisms and continuity equation of kerogen system 2.1.1. Viscous flow Viscous flow is the dominant transport mechanism in shale gas reservoirs, and the net flux generated by very small pressure gradients will exceed that caused by very large gradients in concentration (Freeman et al., 2011). The gas flux caused by viscous flow can be modeled by Darcy's law:
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M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
Nk ¼ ck
kk0 Vpk m
(1)
Where Nk is the gas flux in kerogen pores caused by viscous flow in mol/(m2,s), ck is the moles of free gas per kerogen pore volume in mol/m3, kk0 is the kerogen intrinsic permeability in m2, m is the gas viscosity in Pa,s and pk is the kerogen pressure in Pa. 2.1.2. Knudsen diffusion Knudsen diffusion occurs when the mean free path of gas molecules is at the same scale as the pore size of the porous medium. In shale gas reservoirs, the pore diameters mainly range from 4 nm to 200 nm (Javadpour et al., 2007), which are of similar magnitude as the mean free path of CH4 molecules. Therefore, Knudsen diffusion is significant in shale gas reservoirs and cannot be ignored. The gas flux generated by Knudsen diffusion is (Akkutlu and Fathi, 2012):
Nkk ¼ εkp fDkk Vck
(2)
Where Nkk is the gas flux in kerogen pores caused by Knudsen diffusion in mol/(m2,s), ɸ ¼ Vp/Vb is the total matrix porosity, Vp is the total pore volume for the matrix including the kerogen organic pore volume and inorganic pore volume, Vb is the bulk volume of shale core, εkp ¼ Vkp/Vp is the portion of the kerogen pore volume in total interconnected matrix pore volume, Vkp is the kerogen pore volume, ɸk ¼ εkpɸ is the kerogen porosity, ɸm ¼ (1 εkp)ɸ is the inorganic matrix porosity, Dkk is the tortuosity-corrected Knudsen diffusivity of the kerogen system in m2/s, which can be represented by Civan (2010); Javadpour et al. (2007); Millington and Quirk (1961):
Dkk
16 1 ¼ fk 3 3
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kk0 Rg T pM
(3)
Where Rg is the universal gas constant (8.314 J/(K,mol)), T is the reservoir temperature in K and M is the molecular weight in kg/mol. 2.1.3. Surface diffusion Surface diffusion occurs when adsorbed gas exists on the solid wall. Due to the considerable amount of gas adsorbed onto the internal kerogen solid surfaces, the surface diffusion is also a significant transport mechanism in the kerogen continuum (Fathi and Akkutlu, 2009). Surface diffusion is described by Akkutlu and Fathi (2012)
Nks
¼ εks 1 f ff Ds Vcm
(4)
Where Nks is the gas flux in kerogen caused by surface diffusion in mol/(m2,s), εks is the portion of the kerogen grain volume in total shale core grain volume, ɸf is the fracture porosity, εks(1ɸɸf) ¼ (Vks/Vms)(Vms/Vb) is the kerogen solid volume per unit bulk volume of shale core, Vks and Vms are the kerogen grain volume and matrix grain volume, respectively, Ds is the surface diffusion coefficient in m2/s and cm is the moles of adsorbed gas per solid volume of kerogen in mol/m3. 2.1.4. Gas adsorption and desorption equation The Langmuir isotherm equation is applied to calculate the moles of adsorbed gas on the kerogen solid surfaces (Langmuir, 1916):
cm ¼ cms
Kck 1 þ Kck
(5)
Where cms is the maximum monolayer adsorbed gas on the kerogen
surfaces in mol/m3 and K is the equilibrium distribution coefficient in m3/mol. In this paper, we adopt the following expressions for cms and K:
cms ¼
K¼
rgrain VsL εks Vstd
(6)
zRg T pL
(7)
Where rgrain is the grain density of shale core in kg/m3, VsL is the Langmuir volume in m3/kg, which shows the maximum adsorbed gas volume converted into the standard condition (273.15 K and 101.325 kPa) per unit total grain mass, Vstd is the molar volume of the gas at the standard condition in m3/mol, z is the gas compressibility factor and pL is the Langmuir pressure in Pa.
2.1.5. Continuity equation of kerogen system The transport mechanisms in the kerogen system include viscous flow, Knudsen diffusion and surface diffusion. As analyzed in part 1, the kerogen e inoganic matrix transfer term is modeled by the Warren-Root PSS transfer model. Therefore, according to the conservation of mass, the continuity equation of the kerogen system can be presented as
v εkp fck
i h v εks 1 f ff cm
þ vt h i ¼ V$ εkp fDkk Vck þ V$ εks 1 f ff Ds Vcm k þ V$ ck k0 Vpk Wkm m vt
(8)
Where Wkm is the transfer term between the kerogen and inorganic matrix in mol/(m3,s). It can be approximated by the Warren-Root PSS transfer model, and defined as (Kazemi et al., 1976):
Wkm ¼
zRg Tck kk sPSS ðck cm Þ m
(9)
Where cm is the moles of free gas per inorganic pore volume in mol/ m3 and kk is the apparent kerogen permeability in m2. For further details of the derivation, please refer to Appendix A.
kk ¼ kk0
εkp fDkk m 1þ ck zRg Tkk0
þ
εks 1 f ff Ds cms m
K
ck zRg T
ð1 þ Kck Þ2 (10)
Where sPSS is the pseudo-steady state shape factor in 1/m2, which will be a history match parameter due to the uncertainties in the matrixefracture geometry. For the kerogen e inorganic matrix transfer term, sPSS reflects the geometry of the kerogen pockets (Warren and Root, 1963). In this study, we adopt the following method (Kazemi et al., 1976):
sPSS ¼ 4
1 L2mx
þ
1 L2my
! (11)
Where Lmx,Lmy are the inorganic matrix spacing of the x-axis and y-axis, respectively in m. In our work, we assume that the organic material (kerogen pockets) forms 100 microns clusters, which are embedded in inorganic matrix (Chawathe et al., 2014). Hence, we adopt 100 mm as the inorganic matrix spacing in our following numerical simulation.
M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
We adopt the Z-factor equation of state to describe the behavior of gas:
p ¼ zcRg T
εkp f þ εks 1 f ff cms (" V$
εkp fDkk þ εks
#
K 2
vck vt
ð1 þ Kck Þ 1 f ff Ds cms
# ) k þ ck zRg T k0 Vck ¼ Wkm m
K ð1 þ Kck Þ2
(13)
i h 1 εkp fcm
(19)
h i k ¼ V$ 1 εkp fDkm Vcm þ V$ cm m0 Vpm m (14)
Where km0 is the inorganic matrix intrinsic permeability in m2, pm is the inorganic matrix pressure in Pa and Dkm is the tortuositycorrected Knudsen diffusivity of the inorganic matrix system in m2/s, which can be represented by:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 13 km0 Rg T ¼ fm 3 pM
Wmf ¼
m
km
vt
kf 0 ¼ V$ cf Vpf þ Wmf Qg m
(20)
Where kf0 is the fracture intrinsic permeability in m2, pf is the fracture pressure in Pa and Qg is the production rate in mol/(m3,s), which is defined as (Peaceman, 1983):
Qg ¼
kf 0 cf zRg T q cf cw lnðre =rw Þ m
(21)
In the above equation, when the production well is located in a corner, q ¼ p/2. When the production well is in the center, q ¼ 2p. In addition, cw ¼ pw/zRgT, where pw is the bottomhole flowing pressure in Pa, rw is the wellbore radius in m and re is the drainage radius in m, which can be expressed as (Peaceman, 1983):
8 > > > < > > > : 0:28
0:14
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i h ðDxÞ2 þ ðDyÞ2
kx ¼ ky
h i1=2 1=2 1=2 ðDxÞ2 þ kx ky ðDyÞ2 ky kx 1=4 1=4 þ kx ky ky kx
kx sky (22)
(16)
Where Dx, Dy are the grid sizes in the x and y directions, respectively and kx, ky are the permeabilities in the x and y directions, respectively. Substituting Eq. (12) into Eq. (20) yields:
ff
vcf kf 0 V$ cf zRg T Vcf ¼ Wmf Qg vt m
(23)
2.4. Initial and boundary conditions Assuming that the initial pressure of the shale gas reservoir is pi, then the initial condition used in this work is:
(17)
sTS is the transient shape factor, which can be represented as (Azom and Javadpour, 2012; Zimmerman et al., 1993):
v ff cf
(15)
Where cf is the moles of free gas per fracture pore volume in mol/m3 and km is the apparent inorganic matrix permeability in m2. For further details of the derivation, please refer to Appendix A.
f D m ¼ km0 1 þ m km cm zRg Tkm0
Due to the millimeter scale of the average pore diameters in the fracture system, the effect of Knudsen diffusion is notably small, and only viscous flow is considered in the fracture system. In addition, the inorganic matrix e fracture transfer term and the production well should be considered in fracture system. Then, the continuity equation of the fracture system is given by:
r¼
Where Wmf is the transfer term between the inorganic matrix and fractures in mol/(m3,s). It can be approximated by the Vermeulen transient transfer model and defined as (Kazemi et al., 1976):
zRg Tcm km sTS cm cf
vc k m 1 εkp f V$ 1 εkp fDkm þ cm zRg T m0 Vcm vt m
2.3. Continuity equation of fracture system
þ Wkm Wmf
Dkm
(18)
2ðci cm Þ
¼ Wkm Wmf
Because only free gas exists in inorganic matrix, viscous flow and Knudsen diffusion are considered. In addition, the inorganic matrix system involves two transfer terms. One occurs between the kerogen and inorganic matrix, while the other is between the inorganic matrix and natural fractures, which are modeled by the Vermeulen transient transfer model. Therefore, according to the principle of mass balance, the continuity equation of the inorganic matrix system can be presented as:
vt
2ci cm þ cf
Where Lfx,Lfy are the fracture spacing of the x-axis and y-axis, respectively in m, ci ¼ pi/zRgT and pi is the initial reservoir pressure in Pa. Substituting Eq. (12) into Eq. (14), the following equation can be obtained:
2.2. Continuity equation of inorganic matrix system
v
sTS
!
255
(12)
Substituting Eq. (5) and Eq. (12) into Eq. (8), a simpler form of the kerogen continuum continuity equation can be obtained:
"
1 1 ¼4 2 þ 2 Lfx Lfy
pk ðx; y; tÞjt¼0 ¼ pm ðx; y; tÞjt¼0 ¼ pf ðx; y; tÞ
t¼0
Substituting Eq. (12) into Eq. (24) yields:
¼ pi
(24)
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M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
ck ðx; y; tÞjt¼0 ¼ cm ðx; y; tÞjt¼0 ¼ cf ðx; y; tÞ
t¼0
¼
pi zRg T
(25)
The boundary conditions are G ¼ GI þ GO, where GI represents the inner boundary of the production well and GO represents the outer boundary. We applied the constant pressure inner boundary condition and the no-flow outer boundary condition. The inner boundary condition is:
pf ðx; y; tÞ
GI
¼ pw
(26)
Substituting Eq. (12) into Eq. (26) yields:
cf ðx; y; tÞ
GI
¼
pw zRg T
(27)
The outer boundary condition is:
vpk ¼ 0; vn GO
vpm ¼ 0; vn GO
vpf ¼0 vn GO
(28)
Substituting Eq. (12) into Eq. (28) yields:
vck ¼ 0; vn GO
vcm ¼ 0; vn GO
vcf ¼0 vn GO
Fig. 1. 2D shale gas reservoir production area and the simulation domain (gray area).
(29)
2.5. Numerical solution The above equations are solved using the PDE module of COMSOL Multiphysics (a commercial software, which can directly implement partial differential equations and initial-boundary conditions). The time-dependent solver e backward differentiation formula (BDF) is used to solve the PDEs. Direct (UMFPACK) is used as the linear system solver in our work. 3. Results and discussion The impacts of the kerogen system, transfer model, kerogen permeability, inorganic matrix permeability, fracture permeability and Langmuir parameters on the shale gas reservoir performance were analyzed. We built two simulation scenarios to investigate the effect of the transfer model. In case 1, the intrinsic permeability difference between the inorganic matrix and fracture is small. In case 2, the intrinsic permeability difference between the inorganic matrix and fracture is large. As is shown in Fig. 1, because of the symmetry of the production area, only the gray portion of the production area is simulated and analyzed, for simplicity. The simulation parameters for case 1 and case 2 of the triple-continuum model are given in Table 1. These parameters were selected based on Akkutlu and Fathi (2012), Moridis et al. (2010) and Wasaki and Akkutlu (2014). 3.1. Comparison between dual-continuum model and triplecontinuum model To investigate the impact of the kerogen system on shale gas reservoir performance, a comparison was conducted between the dual-continuum model and triple-continuum model (for case 2). The simulation parameters of the dual-continuum model are given in Table 2. The matrix e fracture transfer term in the dualcontinuum model is described by the Vermeulen transient transfer model. Fig. 2 and Fig. 3 show the variations in cumulative gas production and reservoir pressure over time of the dual-continuum model and triple-continuum model. As illustrated in Figs. 2 and 3,
the cumulative gas production of the dual-continuum model is higher than that of triple-continuum model, and the matrix pressure of the dual-continuum model is lower than the kerogen pressure of the triple-continuum model. Moreover, cumulative gas production was overestimated by approximately 8% when the kerogen continuum was not considered. There are two main reasons for the above results. The first reason is that the mass exchange term between the kerogen and inorganic matrix can generate additional resistance, which leads to less cumulative gas production in the triple-continuum model. The second reason is that the intrinsic permeability of the kerogen system is smaller than that of the inorganic matrix system. It is also smaller than that of the matrix system in the dual-continuum model. Therefore, the reservoir pressure of the dual-continuum model decreases more rapidly than that of the triple-continuum model, which causes the matrix pressure of the dual-continuum model to be lower than the kerogen pressure of the triple-continuum model. Thus, it can be concluded that the matrix system should be divided into a kerogen continuum and inorganic matrix continuum. The triple-continuum model should be incorporated into the shale gas reservoir simulator.
3.2. Effect of the transfer model A comparison of two triple-continuum models with different inorganic matrix e fracture transfer models was conducted to investigate the impact of the transfer model. One triple-continuum model uses the Warren-Root PSS transfer model. The other uses the Vermeulen transient transfer model. The total cumulative gas production and reservoir pressure of two triple-continuum models for case 1 and case 2 are shown as Figs. 4 to 7. For both case 1 and case 2, the results show that the Vermeulen transient transfer model has a higher total cumulative gas production and lower reservoir pressure than those of the Warren-Root PSS transfer model. Apparently, our model gives a higher recovery for two cases, which implies that the present models might underestimate the recovery of shale gas reservoirs with natural fractures or fissures. In addition, the cumulative gas production and reservoir pressure differences between the Vermeulen transient transfer model and Warren-Root PSS transfer
M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
257
Table 1 Simulation parameters of the triple-continuum model for case 1 and case 2. Parameters
Value
Unit
Definition
εkp εks ɸ ɸf Rg T Ds kk0 km0 kf0 in case 1 kf0 in case 2 M Lmx ¼ Lmy Lfx ¼ Lfy pi
0.5 0.01 0.05 0.001 8.314 323.14 1 107 1 1020 1 1019 1 1014 1 1011 0.016 1 104 10 10.4 106 p/2 3.45 106 0.1 1.018 105 2600 3.121398 10-3 10.4 106 0.0224 400
e e e e J/(K,mol) K m2/s m2 m2 m2 m2 kg/mol m m Pa e Pa m Pa,s kg/m3 m3/kg Pa m3/mol m
portion of kerogen pore volume in total interconnected matrix pore volume portion of kerogen grain volume in total shale core grain volume total matrix porosity fracture porosity universal gas constant reservoir temperature surface diffusion coefficient kerogen intrinsic permeability inorganic matrix intrinsic permeability fracture intrinsic permeability in case 1 fracture intrinsic permeability in case 2 molecular weight inorganic matrix spacing fracture spacing initial reservoir pressure e bottomhole flowing pressure wellbore radius gas viscosity grain density of shale core Langmuir volume Langmuir pressure molar volume at standard condition well spacing
q
pw rw
m rgrain
VsL pL Vstd Lw
Table 2 Simulation parameters of the dual-continuum model. Parameters Value
Unit
Definition
ɸ ɸf Rg T Ds km0 kf0 M Lfx ¼ Lfy pi
e e J/(K,mol) K m2/s m2 m2 kg/mol m Pa e Pa m Pa,s kg/m3 m3/kg Pa m3/mol m
matrix porosity fracture porosity universal gas constant reservoir temperature surface diffusion coefficient matrix intrinsic permeability fracture intrinsic permeability molecular weight fracture spacing initial reservoir pressure e bottomhole flowing pressure wellbore radius gas viscosity grain density of shale core Langmuir volume Langmuir pressure molar volume at standard condition well spacing
q
pw rw
m rgrain VsL pL Vstd Lw
0.05 0.001 8.314 323.14 1 107 1 1019 1 1011 0.016 10 10.4 106 p/2 3.45 106 0.1 1.018 105 2600 3.121398 10-3 10.4 106 0.0224 400
model are larger for case 2 than for case 1. The results indicate that as the permeability difference between the matrix system and fracture system becomes larger, the impact of the transfer model on shale gas recovery gets larger. 3.3. Sensitivity analysis A set of data is prepared and given in Table 3 to investigate the impacts of kerogen pore volume, kerogen permeability, inorganic matrix permeability, fracture permeability and Langmuir parameters on shale gas reservoir performance. 3.3.1. Kerogen pore volume Organic nanopores constitute the most widespread and numerous pore type in shale gas reservoirs. Hence, the effect of kerogen pore volume on shale gas production (for case 1) was examined. The parameter εkp represents the portion of the kerogen
Fig. 2. Comparison of cumulative gas production between the dual-continuum model and triple-continuum model.
pore volume in the total interconnected matrix pore volume. Figs. 8 to 10 show the variation of cumulative production of desorbed gas, free gas and total gas over time for different kerogen pore volumes. As illustrated in Figs. 8 to 10, the cumulative production of desorbed gas, free gas and total gas decreases as kerogen pore volume increases. The reasons are as follows. As the kerogen pore volume increases, the amount of free gas increases in the kerogen nanopores and decreases in the inorganic pores. However, compared to the free gas in the inorganic pores, the free gas in the kerogen nanopores needs to overcome an additional resistance, which is caused by the mass exchange term between the kerogen and inorganic matrix to flow into the fractures. Therefore, the cumulative free gas production decreases. The adsorbed gas desorbs from the kerogen solid surfaces into the free gas phase and contributes to the total production only as the reservoir pressure decreases with production. When the production rate of free gas in the kerogen nanopores decreases, the desorption rate of adsorbed
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Fig. 3. Comparison of reservoir pressure between the dual-continuum model and triple-continuum model.
Fig. 6. Effect of the transfer model on total cumulative gas production for case 2.
Fig. 7. Effect of the transfer model on kerogen pressure for case 2. Fig. 4. Effect of the transfer model on total cumulative gas production for case 1.
gas also decreases. Hence, the cumulative production of desorbed gas and total gas decreases in turn. 3.3.2. Intrinsic permeability Figs. 11 to 13 show the variation of total cumulative gas production over time for different kerogen intrinsic permeabilities, inorganic matrix intrinsic permeabilities and fracture intrinsic permeabilities (for case 1). As illustrated in Figs. 11 to 13, the kerogen intrinsic permeability has almost no effect on total cumulative gas production. However, the total cumulative gas production increases as the intrinsic permeability of the inorganic matrix and fractures increases. In addition, the impact of fracture permeability is the most obvious, which reveals that natural fractures are the main permeable channels in shale gas reservoirs and significantly contribute to shale gas production.
Fig. 5. Effect of the transfer model on kerogen pressure for case 1.
3.3.3. Langmuir parameters Figs. 14 to 19 show the variation in the cumulative production of desorbed gas, free gas and total gas over time for different Langmuir pressures and volumes. As illustrated in Figs. 14 to 16, as Langmuir pressure increases, the cumulative production of desorbed gas and total gas decreases, while the cumulative free gas production
M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
259
Table 3 List of parameters for the sensitivity analysis of triple-continuum model. Parameters
Unit
Range
εkp kk0 km0 kf0 pL VsL
e m2 m2 m2 Pa m3/kg
0.25 ~ 0.75 1 1019 ~ 1 1021 1 1018 ~ 1 1020 1 1014 ~ 1 1016 5.2 106 ~ 15.6 106 1.560699 103 ~ 4.682097 103
Fig. 10. Comparison of total cumulative gas production for different kerogen pore volumes.
Fig. 8. Comparison of cumulative desorbed gas production for different kerogen pore volumes.
Fig. 11. Effect of the kerogen intrinsic permeability on shale gas production.
Fig. 9. Comparison of cumulative free gas production for different kerogen pore volumes.
slightly increases. The reasons are as follows. As Langmuir pressure increases, the desorption rate of the adsorbed gas decreases and the cumulative desorbed gas production decreases accordingly. As the adsorbed gas desorption rate decreases, the rate of pore pressure decline increases. In turn, this causes the cumulative free gas production to increase. However, the effect of cumulative desorbed gas production on total cumulative gas production is larger. The total cumulative gas production decreases as Langmuir pressure increases. As illustrated in Figs. 17 to 19, as Langmuir volume
Fig. 12. Effect of the inorganic matrix intrinsic permeability on shale gas production.
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Fig. 13. Effect of the fracture intrinsic permeability on shale gas production.
Fig. 14. Cumulative desorbed gas production under different Langmuir pressures.
Fig. 15. Cumulative free gas production under different Langmuir pressures.
Fig. 16. Total cumulative gas production under different Langmuir pressures.
Fig. 17. Cumulative desorbed gas production under different Langmuir volumes.
Fig. 18. Cumulative free gas production under different Langmuir volumes.
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261
3. The natural fractures are the main permeable channels in shale gas reservoirs. They play a more important role than the kerogen and inorganic matrix during shale gas production. 4. Langmuir pressure and Langmuir volume also have significant effects on the cumulative production of desorbed gas, free gas and total gas, with the effect of Langmuir volume being relatively larger. Therefore, accurate lab measurements of Langmuir parameters are necessary for the numerical simulation of shale gas reservoirs. Currently, hydraulic fracturing technology is being widely applied for shale gas development. In future work, it will be both interesting and challenging to incorporate hydraulic fractures into the triple-continuum model to investigate their effects on shale gas recovery. Acknowledgments
Fig. 19. Total cumulative gas production under different Langmuir volumes.
increases, the cumulative production of desorbed gas and total gas increases, while the cumulative free gas production slightly decreases. The reasons are as follows. As Langmuir volume increases, the desorption rate of the adsorbed gas increases and the cumulative desorbed gas production increases accordingly. As adsorbed gas desorption rate increases, the rate of pore pressure decline decreases and the cumulative free gas production decreases. However, the effect of cumulative desorbed gas production on total cumulative gas production is larger, so the total cumulative gas production increases as Langmuir volume increases.
The authors thank the National Natural Science Foundation of China (No.51490654, No.51234007, No.51404291), the Natural Science Foundation of Shandong Province (ZR2014EL016, ZR2014 EEP018), and China Postdoctoral Science Foundation (2014M55 1989) for supporting this work. Nomenclature ck cm cf ci cw cm
4. Conclusions cms In this paper, based on the complex gas transport mechanisms, including viscous flow, Knudsen diffusion and surface diffusion, the kerogen e inorganic matrix e fracture triple-continuum model for shale gas reservoirs has been presented. In addition, the gas adsorption/desorption on the internal kerogen grain surfaces was also considered. The Vermeulen transient transfer model was incorporated into our triple-continuum model. Next, a finite element simulation software, COMSOL Multiphysics, was used to solve the mathematical model. Subsequently, the impacts of the kerogen system, transfer model, kerogen permeability, inorganic matrix permeability, fracture permeability and Langmuir parameters on the shale gas reservoir performance were examined. The specific conclusions of this study are summarized below: 1. Dividing the matrix system into a kerogen continuum and inorganic matrix continuum has a significant effect on shale gas reservoir performance. Not considering the kerogen continuum could lead to an overestimate in cumulative gas production of approximately 8%. Hence, the triple-continuum model should be incorporated into the shale gas reservoir simulator. 2. The triple-continuum model using the Vermeulen transient transfer model for inorganic matrix e fracture transfer flow yields a higher recovery than using the Warren-Root PSS transfer model, which may underestimate shale gas recovery. A larger difference between the results of the Vermeulen transient transfer model and the current model will occur if properties of the matrix system and fracture system become increasingly different.
Dkk Dkm Ds kk0 km0 kf0 kk km K Lmx,Lmy Lfx,Lfy Lw M Nk Nkk Nks pk pm pf pL pi
the moles of free gas per kerogen organic pore volume [mol/m3] the moles of free gas per inorganic pore volume [mol/m3] the moles of free gas per fracture pore volume [mol/m3] the initial moles of free gas per reservoir pore volume [mol/m3] the moles of free gas per reservoir pore volume in bottomhole [mol/m3] moles of adsorbed gas per solid volume of kerogen [mol/ m3 ] maximum monolayer adsorbed gas on the kerogen surfaces [mol/m3] tortuosity-corrected Knudsen diffusivity of kerogen system [m2/s] tortuosity-corrected Knudsen diffusivity of inorganic matrix system [m2/s] surface diffusion coefficient [m2/s] kerogen intrinsic permeability [m2] inorganic matrix intrinsic permeability [m2] fracture intrinsic permeability [m2] apparent kerogen permeability [m2] apparent inorganic matrix permeability [m2] equilibrium distribution coefficient [m3/mol] inorganic matrix spacing of x axis and y axis, respectively [m] fracture spacing of x axis and y axis, respectively [m] well spacing [m] molecular weight [kg/mol] gas flux in kerogen pores caused by viscous flow [mol/ (m2,s)] gas flux in kerogen pores caused by Knudsen diffusion [mol/(m2,s)] gas flux in kerogen caused by surface diffusion [mol/ (m2,s)] kerogen pressure [Pa] inorganic matrix pressure [Pa] fracture pressure [Pa] Langmuir pressure [Pa] initial reservoir pressure [Pa]
262
pw Qg Rg rw re T VsL Vstd
M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
bottomhole flowing pressure [Pa] production rate [mol/(m3,s)] universal gas constant [J/(K,mol)] wellbore radius [m] drainage radius [m] reservoir temperature [K] Langmuir volume [m3/kg] gas molar volume at the standard condition (273.15 K and
Appendix A In this appendix, we derive the apparent permeabilities of the kerogen system and inorganic matrix system, which are given in Eqs. (10) and (17). Starting from the continuity equation of the kerogen system (Eq. (8)), we manipulate the mass flux terms of the right-hand-side (RHS) to obtain the apparent kerogen permeability:
h i k ðRHSÞ ¼ V$ εkp fDkk Vck þ V$ εks 1 f ff Ds Vcm þ V$ ck k0 Vpk Wkm m pk Kck k þ V$ εks 1 f ff Ds V cms þ V$ ck k0 Vpk Wkm ¼ V$ εkp fDkk V zRg T 1 þ Kck m " # pk K kk0 þ V$ εks 1 f ff Ds cms Vp ¼ V$ εkp fDkk V Vc þ V$ c k k k Wkm zRg T m ð1 þ Kck Þ2 " # pk K pk kk0 þ V$ εks 1 f ff Ds cms þ V$ c Vp ¼ V$ εkp fDkk V V k k Wkm zRg T zRg T m ð1 þ Kck Þ2 3 2 εks 1 f ff Ds cms εkp fDkk K 5 þ V$ ck kk0 Vpk Wkm Vpk þ V$4 ¼ V$ Vp k zRg T zRg T m ð1 þ Kck Þ2
(A-1)
82 9 3 < ε fD = 1 f f Ds cms ε ks f K kk0 5 kp kk 4 þ Vpk Wkm ¼ V$ þ ck 2 : zRg T ; zRg T m ð1 þ Kck Þ 8 9 2 3 < 1 ε fD m εks 1 f ff Ds cms m = K kp kk þ ¼ V$ ck 4 þ kk0 5Vpk Wkm 2 : m ck zRg T ; ck zRg T ð1 þ Kck Þ 8 9 2 3 < 1 = εkp fDkk m εks 1 f ff Ds cms m K 5Vpk Wkm þ ¼ V$ ck 4kk0 þ 2 : m ; ck zRg T ck zRg T ð1 þ Kck Þ
Wkm Wmf z
101.325 kPa) [m3/mol] transfer term between kerogen and inorganic matrix [mol/(m3,s)] transfer term between inorganic matrix and fractures [mol/(m3,s)] gas compressibility factor
Greek symbols m gas viscosity [Pa,s] εkp portion of kerogen pore volume in total interconnected matrix pore volume εks portion of kerogen grain volume in total shale core grain volume ɸ total matrix porosity ɸk kerogen porosity ɸm inorganic matrix porosity ɸf fracture porosity rgrain grain density of shale core [kg/m3] sPSS pseudo-steady state shape factor [1/m2] sTS transient shape factor [1/m2] Subscripts k related to kerogen m related to inorganic matrix f related to fracture
Hence, the total mass flux of the kerogen system, J, can be expressed as follows:
2 εkp fDkk m 14 k þ J ¼ ck m k0 ck zRg T 3 εks 1 f ff Ds cms m K 5Vpk þ ck zRg T ð1 þ Kck Þ2
(A-2)
Then, by binding terms which have the same dimension as kk0, we can obtain a new expression of apparent kerogen permeability:
kk ¼ kk0
εkp fDkk m 1þ ck zRg Tkk0
þ
εks 1 f ff Ds cms m
K
ck zRg T
ð1 þ Kck Þ2 (A-3)
In the same way, starting from the continuity equation of the inorganic matrix system (Eq. (14)), we can obtain the expression for apparent inorganic matrix permeability:
f D m km ¼ km0 1 þ m km cm zRg Tkm0
(A-4)
M. Zhang et al. / Journal of Natural Gas Science and Engineering 24 (2015) 252e263
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