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International Journal of Greenhouse Gas Control 19 (2013) 406–419

Contents lists available at ScienceDirect

International Journal of Greenhouse Gas Control

j ourna l ho me page: www.elsev ier .com/ locate / i jggc

umerical study of CO2 enhanced natural gas recovery andequestration in shale gas reservoirs

ai Sun, Jun Yao ∗, Sun-hua Gao, Dong-yan Fan, Chen-chen Wang, Zhi-xue Sunchool of Petroleum Engineering, China University of Petroleum (East China), Changjiang West Road 66, Economic Technical Development Zone,ingdao 266580, Shandong Province, PR China

r t i c l e i n f o

rticle history:eceived 13 April 2013eceived in revised form 4 September 2013ccepted 10 September 2013vailable online 1 November 2013

eywords:umerical simulationhale gas reservoir

a b s t r a c t

Due to the ultra-fine pore sizes, Darcy’s law is not applicable in shale gas reservoir. On the basis of themultiple binary gas transport mechanisms including viscous flow, Knudsen diffusion and ordinary diffu-sion, a new dual-porosity mathematical model is established to investigate the CO2 sequestration withenhanced natural gas recovery (CSEGR) in shale gas reservoirs. Meanwhile, the transfer function andwell flow model considering the multiple mechanisms are developed. The mathematical model is imple-mented by the finite element simulation software COMSOL, and the accuracy of the numerical solutionis verified by comparison with the classic analytical solution. Three sets of simulations were conductedto investigate the influence of CO2 injection on the development of shale gas reservoirs. The results show

SEGRual-porositynudsen diffusionrdinary diffusion

that CSEGR is feasible to achieve CO2 sequestration and enhance CH4 recovery in shale gas reservoirs,the CO2 storage and natural gas production rate can be improved highly with the increase of injectionpressure. The CSEGR process can be divided into three stages including the early depressurization pro-duction period, the intermediate CO2 adsorption and CH4 replacement period and the late period of CH4and CO2 produced simultaneously. In addition, the concentration variation and storage pattern of CH4and CO2 gas are analyzed during the CSEGR process in shale gas reservoirs.

. Introduction

The increasing carbon dioxide emission plays a prominent rolen global warming; therefore, carbon dioxide capture and storageCCS), as an important tool in reducing carbon dioxide emissionnto atmosphere, has become the hot spot in recent studies (Fusst al., 2008; Gunter et al., 1998; Luo et al., 2013). Harris et al. (2009)nd Luo et al. (2013) have shown that oil/gas reservoirs, not onlyonventional reservoirs but also coal bed methane (CBM) reservoirsnd shale gas reservoirs, are promising targets for CO2 sequestra-ion. Therefore, CO2 injection could enhance hydrocarbon recoveryf oil/gas reservoirs and sequestrate greenhouse gas. Methane isommonly stored in unconventional natural gas reservoirs as free

as in fractures and pores and as adsorbed gas on the surface ofrganic matter. The preferential adsorption of CO2 over CH4 inhese organic-rich rocks is the main reason that carbon dioxide

Abbreviations: CBM, coal bed methane; CCS, carbon dioxide capture and stor-ge; CSEGR, CO2 sequestration with enhanced natural gas recovery; DGM, dusty gasodel; ECBM, enhanced coal bed methane; PDE, partial differential equation.∗ Corresponding author. Tel.: +86 18605460123; fax: +86 53286981700.

E-mail addresses: sunhaiupc@sina.com (H. Sun), RCOGFR UPC@126.comJ. Yao), 123323234@qq.com (S.-h. Gao), fandongyan2010@126.com (D.-y. Fan),cc1220@163.com (C.-c. Wang), szx1979@126.com (Z.-x. Sun).

750-5836/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.ijggc.2013.09.011

© 2013 Elsevier Ltd. All rights reserved.

injection into CBM reservoirs and gas shales for CSEGR is feasible(Harris et al., 2009). Enhanced coal bed methane recovery (ECBM)with CO2 sequestration has been tested in several CBM fields. Asdemonstrated (Busch and Gensterblum, 2011; Chen et al., 2010;Ozdemir, 2009; Qin, 2008), several experimental and numericalinvestigations have been performed on ECBM.

Shale gas with huge reserve and extensive distribution repre-sents a significant portion of unconventional natural gas resourcesworldwide (EIA, 2011). There are two ways in which gas storedin shale: (1) adsorbed gas in organic matrix pores; (2) free gas inmatrix pores and natural fractures; a significant portion (20–85%) ofgas is stored as an adsorbed phase (Hill and Nelson, 2000; Vermylen,2011). The organic matter in gas shales which possesses large sur-face areas has significant gas adsorbed capacity. Nuttall et al. (2005)found that shale samples demonstrates Langmuir-type adsorptionof CO2 and CH4, there is very huge preferential adsorption of CO2over CH4 suggesting a significant potential for CO2 storage in gasshales. Kang et al. (2011) studied CO2 adsorption on two Bar-nett samples, the results show that the amount of CO2 adsorptionwas five to ten times greater than the amount of CH4 adsorption

and both adsorptions of CO2 and CH4 fit the Langmuir-adsorptionequation. Therefore, CO2 is preferentially adsorbed over CH4, shalegas reservoirs are promising CO2 storage targets. Dahaghi (2010)and Schepers et al. (2009) used COMET3 and Eclipse for reservoir

dx.doi.org/10.1016/j.ijggc.2013.09.011http://www.sciencedirect.com/science/journal/17505836http://www.elsevier.com/locate/ijggchttp://crossmark.crossref.org/dialog/?doi=10.1016/j.ijggc.2013.09.011&domain=pdfmailto:sunhaiupc@sina.commailto:RCOGFR_UPC@126.commailto:123323234@qq.commailto:fandongyan2010@126.commailto:wcc1220@163.commailto:szx1979@126.comdx.doi.org/10.1016/j.ijggc.2013.09.011

H. Sun et al. / International Journal of Greenhouse Gas Control 19 (2013) 406–419 407

Nomenclature

b Klinkenberg factor, PaBi the Langmuir constant of component i, 1/PaCinj,i injection molar concentration of component i,

mol/m3

Cprod,i production molar concentration of component i,mol/m3

Cs,i molarity of component i in the porous media s,mol/m3

D∗s,12 effective ordinary diffusion coefficient betweencomponent 1 and component 2 in the porous medias, m2/s

Ds,ki Knudsen diffusion coefficient of component i in theporous media s, m2/s

Fs,i mass flux of component i in the porous media s,kg/(m2 s)

ks,a the apparent permeability of the porous media s, m2

ks,∞ the intrinsic permeability of the porous media s, m2

k∞ Intrisic permeability, m2

Lx the fracture spacing in x direction, mLy the fracture spacing in y direction, mMi molar mass of component i, kg/molpi the initial pressure of shale gas reservoirs, Papinj pressure of production well, PapL,i the Langmuir pressure of component i, PapM,i the partial pressure of component i, Papprod pressure of injection well, Paps pressure of the porous media s, Pap(L) outlet boundary pressure, Paqads,i amount of component i absorbed per unit volume,

kg/m3

qstd,i the volume of component i absorbed per solid massat standard condition, std m3/kg

Qg,i the source/sink term of component i, kg/sQg,i,inj the source the source/sink term of component i at

the injection well, kg/sQg,i,prod the source the source/sink term of component i at

the production well, kg/sQm mass injection rate, kg/sQm,f,i the transfer volume of component i between frac-

ture and matrix, kg/srs the mean pore radii of the porous media s, mR gas constant, 8.314 J/(mol K)T absolute temperature, KVstd the molar volume at standard condition, std m3/molVL,i the Langmuir volume of component i, std m3/kgxs,i molar fraction of component i in the porous media

s, mol/molZS gas deviation factor of mixture gas in the porous

media sZt the initial gas deviation factor of mixture gas in shale

gas reservoirs

Greek symbols˛* shape factor of double porosity, 1/m2

̌ compressibility factor, 1/(Pa m)�g gas dynamic viscosity, Pa s�s gas viscosity of the porous media s, Pa s�s tortuosity of the porous media s� Lennard–Jones potential collision diameter of gas

Subscriptss porous media, s = m or s = fm Matrixf Fracturei gas component i

molecules, (Å)�s density of the shale core, kg/m3

� porosity�s porosity of the porous media s

simulation of CO2 injection in the Devonian gas shale of EasternKentucky based on the assumption that the gas transport in matrixstill is viscous flow.

Javadpour et al. (2007) reported that the main pore diameters inshale gas reservoirs range from 4 to 200 nm through core analysison 152 samples selected from nine reservoirs. Loucks et al. (2009)focused on the pore structure distribution of shale gas reservoirsin North America. They observed that most of pores are nanoporeswith pore diameters in the range of 5–800 nm and throat diame-ters between 10 nm and 20 nm. Bird et al. (2002) pointed out that,instead of conventional viscous flow, the gas transport in nanoporesis the combination of viscous flow, Knudsen diffusion and ordinarydiffusion, which cannot be described by Darcy equation. Javadpour(2009), Civan (2010) and Civan et al. (2011) presented a single com-ponent gas flow model in gas shales considering viscous flow andKnudsen diffusion, however, ordinary diffusion in multicomponentgas transport is not considered.

Due to the ultra-low matrix permeability, shale gas reser-voirs cannot be commercially developed without natural fractures,which are the main permeable channels with much higher per-meability than matrix. Therefore, in this paper we suppose thatthe shale gas reservoir is a dual-porosity system consisting of (1)a matrix and (2) a fracture system. We derive a multicomponentgas transport equation in the double porosity system using theDusty Gas Model (DGM) to incorporate the coupling mechanismsof viscous flow, Knudsen diffusion and ordinary diffusion. So, theordinary diffusion between binary components is considered in thisstudy. In the matrix system, we also consider the mechanism ofgas adsorption and desorption using extended Langmuir isother-mal equation. Subsequently we derive the matrix-fracture transferfunction and well flow model considering the coupled mechanismsof viscous flow, Knudsen diffusion and ordinary diffusion respec-tively. In addition, a mathematical model for CSEGR in shale gasreservoirs is established and numerically solved for case study.Finally, we discuss the feasibility of CSEGR in shale gas reservoirsand investigate the variation of CO2 sequestration and CH4 produc-tion, storage pattern of CH4 and CO2 as well as the variation of gasconcentration in shale gas reservoirs during the process of CSEGRin shale gas reservoirs.

2. Dual-porosity model of two-component gas for shale gasreservoirs

This model is based on the following assumptions: shale gasreservoirs consist of a large number of well-connected fracturesand lower-permeable matrix; gas is stored in natural fractures asa free phase, while in matrix as both a free phase and an adsorbedphase; the gas phase contains CH4 and CO2 only; the gas reservoirsare assumed to be isothermal with gas adsorption on the surface ofmatrix fitting Langmuir isotherm equation. Furthermore, 2D modelis used for simplicity to describe the real 3D reservoir due to the

thin reservoir thickness (Zou, 2012) and low vertical permeability(Zhang and Scherer, 2012).

408 H. Sun et al. / International Journal of Greenh

Knudsen diffusionViscous flow

2

cfafldpitvhgcfobwvac

gm

tfnpodt

F

wsanpdmgclm

mass at standard condition, V the Langmuir volume of compo-

Fig. 1. Schema of single component gas through porous media.

.1. Transport model of two-component gas in porous media

Under isothermal conditions, the gas transport in porous mediaonsists of three transport mechanisms: viscous flow, Knudsen dif-usion and ordinary diffusion (Ho and Webb, 2006). Viscous flownd Knudsen diffusion refer to individual gas species, with viscousow generated from collision between molecules while Knudseniffusion is generated from collision between molecules and theore walls (as shown in Fig. 1). Which mechanism that will dom-

nate depends on the pore scale of the porous media. Comparedo the gas molecular mean free path when the pore diameters areery large, the probability of collisions between molecules is muchigher than collisions between molecules and the pore walls, thusas transport is mainly governed by viscous flow resulting fromollisions between molecules, while less influenced by Knudsen dif-usion. As the pore diameters get smaller, reaching the same orderf magnitude of the gas molecular mean free path, the collisionsetween molecules and the pore walls become more importantith gas transport mainly governed by Knudsen diffusion than by

iscous flow. At the nanopore scale in gas shales, both viscous flownd Knudsen diffusion have impact on gas transport and should beonsidered for gas transport in shale gas reservoirs.

Molecular diffusion refers to the relative motion of differentas species and occurs in multicomponent gas transport in porousedia.The transport of multicomponent gas in shale gas reservoirs is

he combination of the mechanisms of viscous flow, Knudsen dif-usion and ordinary diffusion, and these mechanisms cannot beeglected in the transport model. The DGM is capable of incor-orating mixed mechanism of viscous flow, Knudsen diffusion andrdinary diffusion (Ho and Webb, 2006), therefore we use DGM toevelop the binary gas transport model in porous media (assuminghe two components are component 1 and component 2):

s,1 = −M1[Ds,k1D∗s,12(ps/(ZsRT))∇xs,1 + Ds,k1(D∗s,12 + Ds,k2)xs,1(∇ps/(ZsRT))]

(D∗s,12 + xs,1Ds,k2 + (1 − xs,1)Ds,k1)

− xs,1M1 ·ks,∞ps

�s

( ∇psZsRT

)(1)

here Fs,1 is the mass flux of component 1 in the porous media, s denotes porous media with s = m representing matrix systemnd s = f representing fracture system, Mi molar mass of compo-ent i, Ds,ki the Knudsen diffusion coefficient of component i in theorous media s, D∗s,12 presented in Eq. (4) is the effective ordinaryiffusion between component 1 and component 2 in the porousedia s (Cussler, 2009), ps the pressure of the porous media s, Zs

as deviation factor of mixture gas in the porous media s and can be

alculated as Dranchuk and Kassem (1975); R gas constant, T abso-ute temperature, xs,i molar fraction of component i in the porous

edia s, ks,∞ intrinsic permeability of the porous media s, �s gas

ouse Gas Control 19 (2013) 406–419

viscosity of the porous media s. The Knudsen diffusion coefficientis described in Eqs. (2) and (3) (Florence et al., 2007):

Ds,ki =�s�s

2rs3

√8RT�Mi

(2)

where �s is the tortuosity of the porous media s, �s porosity of theporous media s, rs the mean pore radii of the porous media s. Ds,kican also be represented in terms of the porosity �s and the intrinsicpermeability ks,∞:

Ds,ki =�s�s

4ks,∞

2.81708√

ks,∞�s

√�RT

2Mi(3)

D∗s,12 =�s�s

D12�s�s

× 1.882922475 · 10−2T3/2

×√

0.001 ×(

1M1

+ 1M1

)1

Ps�212˝12(4)

where � is the Lennard–Jones potential collision diameter of gasmolecules, �12 = 0.5(�1 + �2), ̋ is the dimensionless molecule con-stant with its detailed calculation presented in paper (Cussler,2009).

In terms of gas molarity Cs,i, xs,i and ps can be represented by:

xs,i =Cs,i∑j

Cs,j

(5)

ps =∑

j

Cs,jZsRT (6)

By substituting Eq. (5) and Eq. (6) into Eq. (1), we get:

Fs,1 = −M1[(Cs,1 + Cs,2)Ds,k1D∗s,12∇Cs,1 + Cs,1Ds,k1(D∗s,12 + Ds,k2)∇(Cs,1 + Cs,2)][

(Cs,1 + Cs,2)D∗s,12 + Cs,1Ds,k2 + Cs,2Ds,k1]

− M1 ·ks,∞Cs,1ZsRT

�s∇(Cs,1 + Cs,2) (7)

Similarly, we can obtain Fs,2, the mass flux of component 2.

2.2. Gas adsorption and desorption equation

Supposing gas is only adsorbed on the surface of matrix inshale gas reservoirs. Under a constant temperature, the amountof component i remaining absorbed on the surface of matrix can beestimated by the Langmuir isotherm equation (Civan et al., 2011):

qads,i =�sMiVstd

qstd,i =�sMiVstd

· VL,iBipm,i

1 +2∑

j=1Bjpm,j

= �sMiVstd

· VL,iBiCm,i · ZmRT

1 +2∑

j=1BjCm,j · ZmRT

(8)

where qads,i is the amount of component i absorbed per unit vol-ume, Vstd the molar volume at standard condition (273.15 K and101.325 kPa), qstd,i the volume of component i absorbed per solid

L,inent i, pL,i the Langmuir pressure of component i, Bi = 1/pL,i, pm,i thepartial pressure of component i, pm,i = xm,ipm, �S the density of theshale core.

Greenh

2r

2

atmsgfwnu

tc

1 ·km,∞Cm,1ZmRT

�m

)

(10)

f,2Df,

f,2Df,

Df,k1)

,k2 + [(C

Cf,2

]

Cf,2D

[(C

Cf,2

]

Cf,2D

]

Cf,2Df,k1

]⎠(Cf,2 − C) (15)

H. Sun et al. / International Journal of

.3. Dual-porosity model of two-component gas for shale gaseservoirs

.3.1. Continuity equation of matrix systemIn the dual-porosity model of shale gas reservoirs, CO2 and CH4

re stored in shale matrix both as a free phase and adsorbed phase,herefore free gas transport mechanisms as well as the desorption

echanism of adsorbed gas should be taken into consideration inhale matrix. Due to the ultra-low permeability of shale matrix, theas in matrix is assumed to transport into fractures only, there-ore we consider gas transfer between matrix and fracture onlyith ignoring gas transport from matrix to wellbore. The conti-uity equation of component i in matrix system can be presentedsing the principle of mass conservation as:

∂

∂t(MiCm,i�m + (1 − �m)qads,i) + ∇ · Fm,i + Qm,f,i = 0 (9)

By combining Eq. (7) and the principle of mass conservation, theransfer volume of component i between fracture and matrix Qm,f,ian be derived as:

Qm,f,1 = ˛∗(

M1[Cm,1Dm,k1(D∗m,12 + Dm,k2)][

(Cm,1 + Cm,2)D∗m,12 + Cm,1Dm,k2 + Cm,2Dm,k1] + M

[(Cm,1 + Cm,2) − (Cf,1 + Cf,2)

])

Qg,1,inj = −2�

ln(re/rw)

⎛⎝M1 [Cf,1Df,k1(D∗f,12 + Df,k2)][

(Cf,1 + Cf,2)D∗f,12 + Cf,1Df,k2 + C

− 2�ln(re/rw)

⎛⎝M1 [(Cf,1 + Cf,2)Df,k1D∗f,12][

(Cf,1 + Cf,2)D∗f,12 + Cf,1Df,k2 + C

Qg,2,inj = −2�

ln(re/rw)

⎛⎝M2 [Cf,2Df,k2(D∗f,12 + [

(Cf,1 + Cf,2)D∗f,12 + Cf,1Df[(Cf,1 + Cf,2) − Cinj,1 + Cinj,2

]) − 2�

ln(re/rw)(M2 [

(Cf,1 +

Qg,1,prod = −2�

ln(re/rw)

⎛⎝M1 [Cf,1Df,k1(D∗f,12 + Df,k2)[

(Cf,1 + Cf,2)D∗f,12 + Cf,1Df,k2 +

[(Cf,1 + Cf,2) − (Cprod,1 + Cprod,2)

]) − 2�

ln(re/rw)

⎛⎝M1 [

(Cf,1 +

Qg,2,prod = −2�

ln(re/rw)

⎛⎝M2 [Cf,2Df,k2(D∗f,12 + Df,k1)[

(Cf,1 + Cf,2)D∗f,12 + Cf,1Df,k2 +

− 2�ln(re/rw)

⎛⎝M2 [(Cf,1 + Cf,2)Df,k2D∗f,12[

(Cf,1 + Cf,2)D∗f,12 + Cf,1Df,k2 +

+˛∗(M1[(Cm,1 + Cm,2)Dm,k1D∗m,12][

(Cm,1 + Cm,2)D∗m,12 + Cm,1Dm,k2 + Cm,2Dm,k1] (Cm,1 − Cf,1)

ouse Gas Control 19 (2013) 406–419 409

where ˛* is shape factor, ˛* = 4(1/Lx2) + (1/Ly2), Lx is the fracturespacing in x direction. The same procedure can be used to obtainQm,f,2.

2.3.2. Continuity equation of fracture systemAs only free gas is stored in fracture system, without consider-

ing the desorption mechanisms of adsorbed gas only the free gastransport mechanisms should be taken into account in fracture sys-tem. Meanwhile, the production well and the injection well areconnected to fracture system, according to the principle of massconservation, continuity equation of component i in fracture sys-tem is presented as:

∂

∂t(MiCf,i�f ) + ∇ · Ff,i − Qm,f,i = Qg,i (11)

where Qg,i is the source/sink term of component i. By combiningEq. (7) and well modeling method, the source/sink function of eachcomponent at the injection well can be derived as:

k1

] + M1 · kf,∞Cf,1Zf RT�f⎞⎠[(Cf,1 + Cf,2) − (Cinj,1 + Cinj,2)])

k1

]⎞⎠(Cf,1 − Cinj,1) (12)

]

Cf,2Df,k1

] + M2 · kf,∞Cf,2Zf RT�f⎞⎠

f,1 + Cf,2)Df,k2D∗f,12]

)D∗f,12 + Cf,1Df,k2 + Cf,2Df,k1

] (Cf,2 − Cinj,2)(13)

where Qg,i,inj is the source/sink term of component i at the injectionwell, Cinj,i the injection molar concentration of component i. Thesame procedure can be easily adapted to obtain the source/sinkfunction of each component at production well:

f,k1

] + M1 · kf,∞Cf,1Zf RT�f⎞⎠

f,1 + Cf,2)Df,k1D∗f,12]

)D∗f,12 + Cf,1Df,k2 + Cf,2Df,k1

]⎞⎠ (Cf,1 − Cprod,1)

(14)

f,k1

] + M2 · kf,∞Cf,2Zf RT�f⎞⎠[(Cf,1 + Cf,2) − (Cprod,1 + Cprod,2)])

⎞

4 Greenhouse Gas Control 19 (2013) 406–419

wti

2

cir

p

C

wg

swsu

F

p

(

3

3

mpimF(iCstmsa

3

cKttd

3

dbo

v

10 H. Sun et al. / International Journal of

here Qg,i,prod is the source/sink term of component i at the produc-ion well, Cprod,i is production molar concentration of component.

.3.3. Initial and boundary conditionIt is assumed that both production wells and injection wells have

onstant pressure, with production well pressure set to pprod andnjection well pressure set to pinj. The initial pressure of shale gaseservoirs is pi. The initial condition of dual-porosity model is:

m(x, y, z, t)|t=0 = pf (x, y, z, t)|t=0 = pi (16)Substituting Eq. (6) into Eq. (16) we get:

m,1|t=0 = Cf,1|t=0 =pi

ZiRT(17)

here Zi is the initial gas deviation factor of mixture gas in shaleas reservoirs.

The boundary for solution is = 1 + 2 + 3, where 1 repre-ents outer boundary, 2 represents inner boundary of productionell, and 3 represents inner boundary of injection well. In this

tudy, we suppose outer boundary is sealed and inner boundary isnder a constant pressure. The outer boundary condition is:

m,i · n|1 = 0, Ff,i · n|1 = 0, i = 1, 2 (18)Inner boundary condition is:

f (x, y, t)|2 = pprodpf (x, y, t)|3 = pinj (19)Substituting Eq. (6) into Eq. (19) we get:

Cf,1 + Cf,2)|2 =pprodZf RT

Cf,2|3 =pinj

Zf RT(20)

. Model verification and numerical solution

.1. Numerical solution

By substituting Eqs. (7), (8), (10) into continuity equation ofatrix system (9), we obtain the mathematical model for each com-

onent of matrix system. By substituting Eqs. (7), (10), (12)–(15)nto continuity equation of fracture system (11), we obtain the

athematical model for each component of fracture system.inally, by combining the initial and boundary condition Eqs. (17),18), (20), the mathematical model of dual-porosity model of CSEGRn shale gas reservoirs is established. This model is a function of theO2 and CH4 concentrations in the matrix system and the fractureystem Cm,i and Cf,i, and the mathematical model can be solved byhe finite element method software, COMSOL. We adopt the PDE

odule of COMSOL software and use Direct UMFPACK as the linearystem solver. The Tolerance factor of nonlinear solver is set as 0.1nd maximum iterative step as 10.

.2. Verification of numerical solution

The existing numerical simulation software modules could notonsider the coupling gas transport mechanisms of viscous flow,nudsen diffusion and ordinary diffusion in CSEGR, furthermore,

here is a lack of field data of CSEGR in shale gas reservoirs, thereforehe numerical solution in this paper is verified by validating theeveloped mathematical model and numerical method.

.2.1. Verification of the developed mathematical modelThe accuracy and validity of our developed mathematical model

epend on the validation of the mass flux Eq. (1), transfer function

etween matrix and fracture system (10) and source/sink functionsf injection well and production well (12)–(15).

Firstly, we verify the mass flux Eq. (1). Based on authors’ obser-ation, there is no mass flux equation for two-component gas with

Fig. 2. The comparison of the variation of ks,a/ks,∞ with pressure between this studyand other references.

the combination of viscous flow, Knudsen diffusion and ordinarydiffusion, so we simplify our model by considering only Knudsendiffusion and viscous flow (assuming xs,1 = 1, xs,2 = 0) for a singlecomponent gas, and compare with other single component gasmodels presented by Javadpour (2009) and Civan (2010). The sim-plified model is described as:

Fs,1 = −M1ZRT

· ks,∞ps�s

(∇ps) − M1Ds,k1ZRT

∇ps = −�1ks,∞�s

(∇ps)

− �1Ds,k1ps

(∇ps) = −�1ks,a�s

(∇ps) (21)

where the apparent permeability ks,a in the porous media s is:

ks,a = ks,∞(

1 + bsps

), bs =

Ds,k1�sks,∞

(22)

The derived apparent permeability equation is compared withthe apparent permeability established by Javadpour (2009) andCivan (2010). Shale matrix porosity �s of 0.05, single componentof methane and temperature of 323.14 K are used for compari-son. The comparison of computed ks,a/ks,∞ versus pressure for themean pore diameter of 1 nm, 10 nm, 100 nm and 1 �m in this paperand other models of Javadpour (2009) as well as Civan (2010) areshown in Fig. 2. ks,a/ks,∞ indicates the proportion of viscous flowand Knudsen diffusion in the mass flux of a single component gasin porous media. When ks,a/ks,∞ approximates to 1, viscous flowhas more impact on gas transport in porous media than Knudsendiffusion; when ks,a/ks,∞ = 1, Knudsen diffusion can be neglected;when ks,a/ks,∞ becomes higher, the impact of Knudsen diffusionincreases:

ks,∞ = �sr2s

8�s(23)

A good agreement is shown in Fig. 2 between the mathematicalmodel for ks,a/ks,∞ in our work and other references. The value ofks,a/ks,∞ calculated by our model is a little higher than the resultsof Civan (2010) and a little lower than the results of Javadpour(2009); however, the difference is slight. Therefore, the mathemat-ical model developed in this study is correct.

Similarly, the matrix-fracture transfer function and source/sinkfunctions of injection well and production well are simplified into

that of single component gas condition for verification respectively.For a single component gas, the transfer function (10) is simplifiedinto Eq. (24), which is the transfer function of Warren-Root (Kazemiet al., 1976), while source/sink functions (12) and (14) are simplified

H. Sun et al. / International Journal of Greenhouse Gas Control 19 (2013) 406–419 411

Fg

i(

Q

Q

Q

3

upp

3itKa

s(

p

iT

itc

gtp1mu

Table 1Parameters for one-dimensional steady gas flow considering Klinkenberg effect.

Parameter Value

Klinkenberg factor, b 7.6 × 105 PaIntrisic permeability,k∞ 5.0 × 10−19 m2Length of the column, L 10.0 mOutlet boundary pressure, p(L) 1.0 × 105 PaAir mass injection rate,Qm 1.0 × 10−6 kg s−1Compressibility factor, ̌ 1.8 × 10−5 Pa−1 m−1Porosity,� 0.3Gas dynamic viscosity, �g 1.84 × 10−5 Pa s

ig. 3. Comparison of numerical and analytical solution of one-dimensional steadyas flow considering Klinkenberg effect.

nto Eq. (24) and Eq. (26), which are Peaceman well flow modelPeaceman, 1983):

m,f,1 = ˛∗km,a�m

�m(pm − pf ) (24)

g,1,inj = −2��f

ln(re/rw)kf,a�f

(pf − pinj) (25)

g,1,prod =2��f

ln(re/rw)kf,a�f

(pf − pprod) (26)

.2.2. Verification of numerical method with analytical solutionSince there is no analytical solution for CSEGR, this paper sim-

lates the gas flow considering Klingkenberg effects in singleorosity model and Darcy flow of single component gas in dual-orosity model to verify the numerical method.

.2.2.1. Verification of the gas flow of single porosity model consider-ng Klingkenberg effect. Under isothermal condition and assuminghat porosity � is constant, the continuity equation consideringlinkenberg effect for one-dimensional steady state is presenteds Eq. (27) (Wu et al., 1998):

∂

∂x(k∞ˇ(p + b)

�g

∂p

∂x) = 0 (27)

Fixing Qm the injection rate of entry end (x = 0) and p(L) the pres-ure of exit end (x = L), the analytical solution is obtained as Wu et al.1998):

(x) = −b +√

b2 + (p(L)2 + 2b[(p(L)]) + 2Qm�g(L − x)k∞ˇ

(28)

Fig. 3 compares the analytical solution with this paper’s numer-cal result by COMSOL. Parameters used in this paper are shown inable 1, which comes from the experiment of Yucca Mountain.

A good agreement of numerical solution and analytical solutions shown in Fig. 3, indicating that the numerical method used inhis paper is accurate in solving one-dimensional steady gas flowonsidering Klinkenberg effect.

Then we verify the accuracy of COMSOL in simulating unsteadyas flow with Klinkenberg effect. The model and parameters refero Wu et al. (1998). Fig. 4 compares the numerical solution of this

aper with analytical solution provided by Wu et al. (1998) after0 days of injection. A good agreement indicates that the numericalethod used in this paper is reliable in simulating two-dimensional

nsteady gas flow considering Klinkenberg effect.

Fig. 4. Comparison of numerical and analytical solutions of two-dimensional tran-sient gas flow considering Klinkenberg effect (This is the gas pressure distributionafter ten days of injection.).

3.2.2.2. Verification of Darcy flow of single component gas in dual-porosity model. We use the following case to verify the accuracyof COMSOL in simulating Darcy flow of single component gas indual-porosity model. A gas production well is in the center of thereservoirs with a constant pressure. The basic parameters are asfollows: the initial reservoir pressure is 1 × 107 Pa, bottom holepressure is 1 × 106 Pa, initial reservoir temperature is 323 K, gascomposition is 100% CH4, matrix permeability is 1 × 10−19 m2,matrix porosity is 0.05, fracture porosity is 0.001, fracture per-meability is 1 × 10−15 m2, CH4 molar mass is 0.016 kg/mol, gascompressibility factor is 1, molar volume under standard con-dition is 0.0224 m3/mol, wellbore radius is 0.1 m, reservoirthickness is 1 m, the viscosity of CH4 is 1.2 × 10−5 Pa s, ideal gasconstant R = 8.314 J/(K mol), compressibility coefficient of CH4 is1 × 10−7 Pa−1. Fig. 5 compares different variations of production ofvertical well producing under a constant pressure in dual-porositymodel. As shown in Fig. 5, the numerical solution of this paper is ingood consistent with analytical solution (Warren and Root, 1963),and it is reliable to simulate single component gas flow in dual-porosity model by COMSOL, therefore COMSOL can be used to solvethe model of CSEGR developed in this paper.

4. Results and discussion

The performance of a production well with well spacing of 200 min a five-spot well pattern (as shown in Fig. 6) is modeled. Withsymmetry, only a quarter of the domain needs to be modeled.Parameters used in this modeling are given as follows: the initialpressure of the gas reservoir is 5.38 MPa, the pressure of produc-

tion well is 0.1 MPa, the reservoir temperature is 303.15 K, thematrix porosity is 0.05, the intrinsic permeability of matrix sys-tem is 10−19 m2, the average fracture spacing is 0.05 m, the averagefracture width is 5 �m, the shale density is 2600 kg/m3, the molar

412 H. Sun et al. / International Journal of Greenhouse Gas Control 19 (2013) 406–419

Fig. 5. Comparison of numerical and analytical solutions of dual-porosity in gasreservoirs with Darcy flow (This is the well production.).

vrmfL3msa(

vivgtpi

of CO2, the production of CH4 declines.

Fig. 6. Top view of shale gas production area.

olume under standard condition is 0.0224 m3/mol, the wellboreadius is 0.1 m, CH4’s molar mass is 0.016 kg/mol, CO2’s molarass is 0.044 kg/mol. Typical CH4 and CO2 Langmuir Isotherms

or the Lower Huron shale are used (Schepers et al., 2009): CH4’sangmuir volume is 0.98 × 10−3 m3/kg, CH4’s Langmuir pressure is.05 MPa,CO2’s Langmuir volume is 1.91 × 10−3 m3/kg, CO2’s Lang-uir pressure is 1.68 MPa. The porosity and permeability of fracture

ystem is calculated by the match-stick model (Bustin et al., 2008),nd gas viscosity is calculated according to the method in Lee et al.1966).

In order to investigate the potential of CSEGR in shale gas reser-oir, three sets of simulations are conducted to investigate thenfluence of CO2 injection on the development of shale gas reser-oirs. The detailed simulation scenarios are as follows: Case 1: shaleas reservoirs are developed by production well without CO2 injec-

ion well; Case 2: a five-spot well pattern is used with CO2 injectionressure of 6 MPa; Case 3: a five-spot well pattern is used with CO2

njection pressure of 7 MPa. Under these three production Cases,

Fig. 7. Variation of CO2 storage rate and CH4 production rate with time.

the dynamic change of CO2 sequestration and CH4 recovery duringthe process of CSEGR is studied.

4.1. Variation of the amounts of CO2 sequestration and CH4production in shale gas reservoirs

In order to investigate the potential of CSEGR, we compare theamounts of CO2 sequestration which is the total of free phase andabsorbed phase of CO2 in formation, and CH4 production underthree different production cases. The calculated CO2 storage rateand CH4 production rate with time are shown in Fig. 7.

As illustrated in Fig. 7, under a constant injection pressure, CO2storage rates of Cases 2 and 3 decrease with time, and their curvestend to have two different inflection points (point a and b, point aand c) due to different decreasing rates; meanwhile, under a con-stant production pressure, the CH4 production rate curves for Cases2 and 3 also have two inflection points indicating that the daily pro-duction undergoes the variation of reduce-increase-reduce. In Case1 without CO2 injection, production rate gradually decreases withtime.

According to the relative positions of inflection points, theCSEGR process is divided into three stages: (1) an early depres-surization production period. In the first twenty days of initialproduction period, the pressure wave caused by CO2 injection hasnot reached production well because of the relative small amountof CO2 injection, thus Case 2 and Case 3 are similar to Case 1 inthat they all developed by depressurization with faster decline ofCH4 production rate. However, because of CO2 injection, the pres-sure drop rates of Cases 2 and 3 are slower than the rate of Case1, therefore CH4 production rate of Case 1 is higher than the othertwo Cases at this stage. (2) An intermediate period when CH4 isdisplaced by adsorbed CO2. The duration of this stage varies withthe injection pressure, such as a–b section under CO2 injection pres-sure of 7 MPa and a–c section under CO2 injection pressure of 6 MPa.With increasing amount of CO2 injection, the very large preferentialadsorption of CO2 over CH4 results in an increasing amount of CO2absorbed on the surface of shale pore. Meanwhile, a large amountof CH4 desorbed from surface of the shale pore, therefore the CH4production rate gradually increases at this stage. (3) A late periodwhen CH4 and CO2 are produced simultaneously. As time goes on,CO2 steadily flows to production well with its breakthrough timeat the point b (approximately 224 days) and c (approximately 291days) in the Case 2 and Case 3, respectively. Due to the production

Fig. 8 shows the position of the displacement front of CO2 alongthe injection well-production well line (see Fig. 9) at different timesfor Case 2 and Case 3. As illustrated in Fig. 8, point b and point

H. Sun et al. / International Journal of Greenhouse Gas Control 19 (2013) 406–419 413

Fig. 8. Dynamic variation of displacement front during production.

Fig. 9. Diagonal line of production well and injection well.

Fig. 10. CO2 mass fraction in the production area af

Fig. 11. Cumulative CH4 production and CO2 storage vs. time.

c are the times when displacement front reaches the productionwell under injection pressure of 7 MPa and 6 MPa, respectively. Inaddition, the CO2 displacement front has an earlier breakthroughtime under injection pressure of 7 MPa.

The variations of CO2 mass fractions in the production area areshown in Fig. 10 for Cases 2 and 3. As illustrated in Fig. 10, theamount of CO2 storage increases with production time, and theproportion of CO2 in total storage increase when getting closer tothe injection well. The displacements rate of Case 3 is faster thanthat of Case 2; the displacement front of Case 3 reaches production

well at the time of 224 days, while the displacement front of Case2 reaches at the time of 291 days. It indicates that the increase ininjection pressure accelerates the CO2 transport and shortens theCO2 breakthrough time.

ter 20, 50, 224 and 291 days of CO2 injection.

414 H. Sun et al. / International Journal of Greenhouse Gas Control 19 (2013) 406–419

Fig. 12. The ratio of CO2 storage rate to CH4 production rate vs. time.

Fig. 13. CH4 recovery vs. time (three cases).

Fig. 14. Average storages of adsorbed gas, free gas and total gas of CH4 and CO2 perunit volume vs. time.

Fig. 15. The proportions of CO2 and CH4 in each storage pattern in production areavs. time.

Fig. 16. The mole fractions of free gas and adsorbed gas in total CH4 production andCO2 storage during CO2 sequestration vs. time.

This paper compares CO2 cumulative storage and CH4 cumula-tive production at different times in Fig. 11. As illustrated in Fig. 11,CO2 storage is far more than CH4 production, i.e., shale gas reser-voirs are able to store a large amount of CO2 as primary greenhousegas; meanwhile, the increase in CO2 injection pressure enlargesCO2 storage in formation. For instance, cumulative CO2 storage is5000 kg under the injection pressure of 6 MPa, while cumulativeCO2 storage is 7000 kg under the injection pressure of 7 MPa. Fig. 12shows the variation of the ratio of CO2 storage rate to CH4 produc-tion rate with time for Cases 2 and 3. The ratio increases in earlydepressurization production period, because the CH4 productionrate declines much faster than CO2 storage rate (shown in Fig. 7).In the intermediate period, when CH4 is displaced by adsorbed CO2,the ratio of CO2 storage rate to CH4 production rate decreases dueto the growing CH4 production rate. In Case 2, the ratio of CO2 stor-age rate to CH4 production rate reaches its maximum of 6.54 after13 days of CO2 injection into the reservoir, i.e. 6.54 kg CO2 is storedin the formation while 1 kg CH4 is produced, the ratio graduallyreduces to 4.14 after 350 days with cumulative ratio of CO2 storage

to CH4 production of 5.11; In Case 3, the ratio of CO2 storage rate toCH4 production rate reaches its maximum of 10.28 after 9 days ofCO2 injection into the reservoir and reduces to 3.96 after 350 dayswith cumulative ratio of storage to production of 5.73. Therefore,

H. Sun et al. / International Journal of Greenhouse Gas Control 19 (2013) 406–419 415

Ftl

atis

odaCia

4s

ftftttc

gai

Fig. 18. CO2 mole storage of free gas and adsorbed gas per unit volume at varioustimes and distances to the injection well along the production well-injection wellline.

ig. 17. CH4 mole storage of free gas and adsorbed gas per unit volume at variousimes and distances to the injection well along the production well-injection welline.

s the injection pressure increases, the ratio of CO2 storage rateo CH4 production rate becomes greater, i.e., if injection pressurencreases, more CO2 can be stored in the formation with increasingtorage rate for the same CH4 production.

Fig. 13 shows the variation of enhanced CH4 recovery with timef the three production cases. As illustrated in Fig. 13, after 350ays of production, the enhanced CH4 recovery of Cases 1, 2 and 3re 27.3%, 36.7% and 41.2%, respectively. CO2 injection can enhanceH4 recovery in shale gas reservoirs, with 9.3% in Case 2 and 13.9%

n Case 3. As CO2 injection pressure increases, a higher recovery ischieved as more methane is replaced.

.2. Gas storage pattern in shale gas reservoirs during CO2equestration

Fig. 14 shows the variation of average storages of adsorbed gas,ree gas and total gas of CH4 and CO2 per unit volume with time inhe production area of Case 2. As illustrated in Fig. 14, the storages ofree gas, adsorbed gas and total gas of CO2 increase with productionime respectively, on the contrary, the free gas, adsorbed gas andotal gas storage of CH4 decrease with time respectively. It indicateshat in the process of CSEGR, CH4 is displaced, and CO2 is storedontinually.

Fig. 15 shows the variation of proportions of CH4 and CO2 in freeas, adsorbed gas and total gas storage with time in the productionrea of Case 2. As illustrated in Fig. 15, the average storage of CH4n free gas and adsorbed gas storages per unit volume decrease

Fig. 19. CH4 free gas storage to all gas storage ratio distributions at various timesand distances to the injection well along the production well-injection well line.

with time respectively, on the contrary, the average free gas andadsorbed gas storage per unit volume for CO2 increase with timerespectively. Because the adsorption capacity of CO2 is higher thatof CH4 in shale gas reservoirs, the mass fraction of CO2 in adsorbed

gas storage increases faster than that in free gas storage. After 350days of production, the mass of CO2 occupies 84.752% in absorbedgas storage, 61.467% in free gas storage and 74.599% in total gas

416 H. Sun et al. / International Journal of Greenhouse Gas Control 19 (2013) 406–419

Fa

st

iFaii(tepC

adis

ig. 20. CO2 free gas storage to all gas storage ratio distributions at various timesnd distances to the injection well along the production well-injection well line.

torage. For CH4, its mass fraction in adsorbed gas, free gas andotal gas storage are 15.248%, 38.533% and 25.401%, respectively.

The variation of the mole fractions of free gas and adsorbed gasn total CH4 production and CO2 storage with time are shown inig. 16. The mole fractions of free gas and adsorbed gas in CO2 stor-ge show little change, with adsorbed gas of 56% and free gas of 35%n shale. However, the mole fractions of free gas and adsorbed gasn CH4 production vary with time. In the initial production periodduring the first 130 days), CH4 production is mainly from desorp-ion, and then the mole fraction of free gas gradually increases andxceeds 50% after 130 days of production. In the late productioneriod of CSEGR, free gas gradually becomes the main source ofH4 production, from 42% at 20 days to 60% at 350 days.

Fig. 17 shows the variation of CH4 mole storage of free gas and

dsorbed gas per unit volume with time at each point along the pro-uction well-injection well line (the diagonal line). As illustrated

n Fig. 17, from injection well to displacement front, CH4 free gastorage increases when getting closer to the displacement front

Fig. 22. Variation of CH4 con

Fig. 21. Average CO2 storage mole fraction in production area of two different pro-duction Cases vs. time.

and reaches its maximum at the displacement front. Same resultscan be obtained for the area from displacement front to produc-tion well. Same results can be obtained for CH4 adsorbed gas. Thisis because that displacement front is the place where the pressureis maximum in the production area without CO2, CH4 free gas andadsorbed gas storage reach their maximum at the displacementfront. For a certain point along the diagonal line, a longer displace-ment time results in less CH4 free gas and adsorbed gas storage inthe reservoir.

Fig. 18 shows the variation of CO2 mole storage of free gas andadsorbed gas per unit volume with time at each point along the pro-duction well-injection well line (the diagonal line). As illustrated inFig. 18, for each location from injection well to displacement front,the closer to injection well, the greater the CO2 free gas storage.

And there is no CO2 storage from displacement front to productionwell. Same result can be obtained for adsorbed gas. For a certainpoint along the diagonal line, a longer displacement time results inmore CO2 free gas and adsorbed gas storage in the reservoir.

centration (mol/m3).

H. Sun et al. / International Journal of Greenhouse Gas Control 19 (2013) 406–419 417

Fig. 23. Variation of CO2 con

Fig. 24. CH4 and CO2 mole storage distributions at various times and distances tothe injection well along the production well-injection well line for case 2.

centration (mol/m3).

Fig. 19 shows CH4 free gas storage to all gas storage ratio distri-butions at various times along the production well-injection wellline .At the same point in time, the more closer to injection well, thelarger proportion of free gas in CH4 total storage; the more closerto production well, the smaller proportion of free gas in CH4 totalstorage. The proportion of free gas in CH4 total storage decreasesrelatively fast from injection well to displacement front, while fromdisplacement front to production well, the proportion of free gas inCH4 total storage declines at different rates from slow to fast.

Fig. 20 shows CO2 free gas storage to all gas storage ratio distri-butions at various times along the production well-injection wellline .At the same point in time, the closer to injection well, the largerproportion of free gas in CO2 total storage; the closer to displace-ment front, the smaller proportion of free gas in CO2 total storage.In the process of CSEGR, a longer displacement time results in moreCO2 free gas storage in the reservoir.

4.3. The variation of gas concentration in shale gas reservoirsduring the process of CO2 sequestration

Fig. 21 shows the variation of average CO2 storage mole fractionwith time in production area of two cases, the fraction increasingwith production time. After 350 days of production, average CO2storage mole fraction and mass faction for Case 2 are 51.57% and74.54%, respectively. While for Case 3, the average CO2 storage molefraction and mass faction are 60.30% and 80.68%, respectively.

Figs. 22 and 23 show the distributions of CH4 and CO2 concen-tration after 20, 50, 224, 291 days of production for different cases,respectively. As illustrated in figures, with continuous injection ofCO2, CH4 is displaced out from the reservoir continually resultingin decreasing CH4 concentration and increasing CO2 concentra-tion, which can be accelerated by increased injection pressure. CO2breakthrough time for Case 3 is at the time of t = 224 days. Whilefor Case 2, the production well begins to produce CO2 after 291

days of CO2 injection into the reservoir. As CO2 injection pressureincreases, the CO2 injection rate becomes faster, therefore displace-ment front reaches production well earlier and production wellbegins to produce CO2 earlier.

418 H. Sun et al. / International Journal of Greenh

Fac

awpgttitmCodfiapts

5

Cnwm

ig. 25. CH4 (lower) and CO2 (upper) mole storage distributions at various timesnd distances to the injection well along the production well-injection well line forase 3.

Figs. 24 and 25 show the total mole storage of free gas anddsorbed gas of CH4 and CO2 at different time along the productionell-injection well line for Cases 2 and 3 respectively. At the sameoint of time, CO2 mole storage per unit volume increases whenetting close to injection well, while decreases when getting closeo the displacement front, and there is no CO2 stored at the loca-ion from displacement front to production well. At the same pointn time, CH4 storage per unit volume increases when getting closeo displacement front in the area from injection well to displace-

ent front, That is because the closer to injection well, the higherO2 concentration resulting in more CH4 displaced. We can alsobtain similar result for the area from displacement front to pro-uction well, that is because there is only CH4 stored at the locationrom displacement front to production well, therefore the pressurencreases when getting close to displacement front, resulting largermount of CH4 storage per unit volume. At each point along theroduction well-injection well line, with the increase of produc-ion time, CO2 storage per unit volume becomes larger while CH4torage per unit volume becomes smaller.

. Conclusions

This paper presents a new dual-porosity mathematical model of

SEGR for shale gas reservoirs incorporating the multiple mecha-isms of viscous flow, Knudsen diffusion and ordinary diffusion,here the desorption mechanism of adsorbed gas in the shaleatrix is also considered. The matrix-fracture transfer function

ouse Gas Control 19 (2013) 406–419

and source/sink models of production and injection wells underthe coupling mechanisms are developed.

Subsequently, numerical modeling is conducted by the finiteelement simulation software COMSOL and the accuracy of thenumerical solution is theoretically verified by the classic analyticalsolution. Three sets of simulations have been conducted to investi-gate the feasibility of CO2 sequestration and enhanced CH4 recoveryin shale gas reservoirs (CSEGR). The numerical result shows thatCSEGR may have significant impact that should be tested underreal conditions.

Acknowledgements

This project was supported by the National Natural ScienceFoundation of China (No. 51234007), the National Natural Sci-ence Foundation of China (No. 11072268), Program for ChangjiangScholars and Innovative Reserch Team in University (IRT1294),the Major Programs of Ministry of Education of China (No.311009), Specialized Research Fund for the Doctoral Program ofHigher Education (No. 20110133120012), the National NaturalScience Foundation of Shandong Province (No. ZR2011EEQ002),the Fundamental Research Funds for the Central Universities (No.11CX05007A), the Fundamental Research Funds for the CentralUniversities (No. 11CX04022A), Introducing Talents of Disciplineto Universities (B08028), and The Open Project of Key Laboratoryof Ministry of Education for EOR.

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