Research ArticleNanopore Confinement Effect on the Phase Behavior ofCO2/Hydrocarbons in Tight Oil Reservoirs considering CapillaryPressure, Fluid-Wall Interaction, and Molecule Adsorption

Zhixue Zheng ,1,2 Yuan Di ,1,2 and Yu-Shu Wu3

1College of Engineering, Peking University, Beijing 100871, China2Institute of Energy, Peking University, Beijing 100871, China3Department of Petroleum Engineering, Colorado School of Mines, Golden CO 80401, USA

Correspondence should be addressed to Yuan Di; diyuan@mech.pku.edu.cn

Received 8 July 2021; Revised 25 July 2021; Accepted 9 August 2021; Published 21 August 2021

Academic Editor: Steffen Berg

Copyright © 2021 Zhixue Zheng et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The pore sizes in tight reservoirs are nanopores, where the phase behavior deviates significantly from that of bulk fluids inconventional reservoirs. The phase behavior for fluids in tight reservoirs is essential for a better understanding of themechanics of fluid flow. A novel methodology is proposed to investigate the phase behavior of carbon dioxide(CO2)/hydrocarbons systems considering nanopore confinement. The phase equilibrium calculation is modified by couplingthe Peng-Robinson equation of state (PR-EOS) with capillary pressure, fluid-wall interaction, and molecule adsorption. Theproposed model has been validated with CMG-Winprop and experimental results with bulk and confined fluids. Subsequently,one case study for the Bakken tight oil reservoir was performed, and the results show that the reduction in the nanopore sizecauses noticeable difference in the phase envelope and the bubble point pressure is depressed due to nanopore confinement,which is conductive to enhance oil recovery with a higher possibility of achieving miscibility in miscible gas injection. As thepore size decreases, the interfacial tension (IFT) decreases whereas the capillary pressure increases obviously. Finally, therecovery mechanisms for CO2 injection are investigated in terms of minimum miscibility pressure (MMP), solution gas-oilratio, oil volume expansion, viscosity reduction, extraction of lighter hydrocarbons, and molecular diffusion. Results indicatethat nanopore confinement effect contributes to decrease MMP, which suppresses to 650 psi (65.9% smaller) as the pore sizedecreases to 2 nm, resulting in the suppression of the resistance of fluid transport. With the nanopore confinement effect, theCO2 solution gas-oil ratio and the oil formation volume factor of the oil increase with the decrease of pore size. In turn, the oilviscosity reduces as the pore size decreases. It indicates that considering the nanopore confinement effect, the amount of gasdissolved into crude oil increases, which will lead to the increase of the oil volume expansion and the decrease of the viscosityof crude oil. Besides, considering nanopore confinement effect seems to have a slightly reduced effect on extraction of lighterhydrocarbons. On the contrary, it causes an increase in the CO2 diffusion coefficient for liquid phase. Generally, the nanoporeconfinement appears to have a positive effect on the recovery mechanisms for CO2 injection in tight oil reservoirs. Thedeveloped novel model could provide a better understanding of confinement effect on the phase behavior of nanoscale porousmedia in tight reservoirs. The findings of this study can also help for better understanding of a flow mechanism of tight oilreservoirs especially in the case of CO2 injection for enhancing oil recovery.

1. Introduction

According to the predictions, global energy demand is pro-jected to grow by about a third by 2040, particularly in India,China, and across Asia [1]. Tight oil reservoirs have beenincreasingly concerned due to its abundant reserves, huge

development, and utilization potential. As illustrated inEIA’s crude oil production forecast graph from 2020 annualreport, tight oil in the United States is predicted to increasethe total amount of crude oil production by 160% from2010 to 2050. Tight oil production will more than doublefrom 2015 to 2040 as shown in Figure 1 [2]. To meet oil

HindawiGeofluidsVolume 2021, Article ID 2435930, 18 pageshttps://doi.org/10.1155/2021/2435930

demand, substantial ongoing investment in tight oil reser-voir development will be required.

Despite horizontal drilling and multistage hydraulicfracturing technologies achieving tremendous success foreconomic development of tight oil reservoirs, the oil indus-try still faces challenges such as low oil recovery and rapidlydeclining production rate due to unknown fluid phasebehavior and recovery mechanisms in tight oil reservoirsaccurately [3–6]. Therefore, understanding the phase behav-ior for fluids in nanopores is essential for a better developingtight oil reservoirs and predicting well performance.

The pore sizes in tight reservoirs are nanopores, wherethe phase behavior deviates significantly from that of bulkfluids in conventional reservoirs [7–9]. Wang et al. [10]conducted nanofluidic device experiments with pure alkaneand showed that the vaporization of the liquid phase innanochannels is obviously suppressed compared to that inmicrochannels. Nojabaei et al. [11] found that the PVTproperties of crude oil were significantly different underthe two conditions comparing the PVT properties of crudeoil in a PVT cylinder with those in nanoscale porous media.Luo et al. [12, 13] studied nanopore confinement effect usingdifferential scanning calorimetry (DSC) and found that thebubble point alteration is significant when the pore sizeunder 4.1 nm. Pinho et al. [14] introduced a novel techniqueto conduct a microfluidic multicomponent phase behaviorand showed that multicomponent P-T diagrams are alteredunder nanopore confinement effect. Other experimentaltechniques including temperature-programmed desorption(TPD) [15], neutron diffraction [16], volumetric measure-ment [17], X-ray diffraction [18], scanning electron micro-scope (SEM) [19], and micro-CT scanning [20] also havefound similar phenomena in nanopores. All these experi-ments showed that nanoscale interaction had a remarkableeffect on the gas-liquid equilibrium of hydrocarbon compo-nents, but the results were influenced by experimental mate-rials, and there were few experimental studies on the phasebehavior in nanopores with a radius smaller than 50nmwas present in the literature due to the unconventional char-acteristics of tight oil reservoirs and the limitations of labo-ratory equipment. Molecular simulation is also widely used

to investigate the fluid phase behavior under nanoporeconfinement. Wang et al. [21] presented Monte Carlo simu-lation (MC) to investigate the adsorption behavior of pen-tane, heptane, and their mixtures in slit-nanopores andfound that multiple adsorption layers properties depend onpore size and fluid compositions. Jin and Firoozabadi [22]analyzed the effect of pore size distribution by gauge-GCMC simulation and revealed that fluids in tiny pores con-dense before that in large pores, and the shift of the phasediagram would increase with the proportion of small pores.Despite these simulation studies providing details of thebehavior of confined fluids, it is not applicable to employthe molecular simulation method to analyze the nanoporeconfinement effect on the phase behavior at an engineeringscale due to their high computation costs. Therefore,researchers have focused on developing thermodynamicsmodels to characterize the fluid phase behavior of confinedtight oil reservoirs.

The equation of state (EOS) is one of the most usedapproaches in thermodynamics models, and it has accom-plished a huge success in modeling bulk phase behavior.Recently, experimental and theoretical studies have shownthe existence of capillary pressure effect in nanopores. Inan effort to consider the nanopore confinement effect onthe phase behavior in tight oil reservoirs, Zhang et al. [23]modified flash calculation with capillary pressure and per-formed the studies of the phase behavior of CO2/hydrocar-bons systems in Bakken formations, and the resultsindicated that the capillary pressure effect cannot beneglected in nanopores. MMP of CO2 injection decreaseswith the capillary pressure effect, resulting in the suppres-sion of the resistance of fluid transport. Li and Sheng [24]performed the phase equilibrium of Wolfcamp shale reser-voir by coupling Peng-Robinson equation of state (PR-EOS) with capillary pressure and shifted critical properties,and results presented that nanopore confinement narrowedthe two-phase region and decreased the interfacial tension.The confined space in tight oil reservoirs causes the molecu-lar radius to be comparable to the pore size, and the interac-tion between the fluid molecular and pore wall strengthensto a point that cannot be ignored. Yang et al. [25] recently

25

20

15

10

5

02000 2010 20102020 20202030 20302040 20402050 2050

60

50

40

30

20

10

02000

2019 2019

History Projections History Projections

Crude oil productionmillion barrels per day

Dry natural gas production by typetillion cubic feet

Tight/shalgasOtherlower48onshoreLower 48offshoreOther

40

30

20

10

0

Tight oil

Alaska

Gulf of mexicoOther

Figure 1: Historical and projected sources of crude oil and natural gas in the US (EIA, 2020).

2 Geofluids

revealed the size effect of the solid-liquid interface energyand found that the mechanical action of the solid-liquidinterface plays an important role in capillary condensationunder nanometer or subnanometer scale, rather than thegas-liquid interface, which is generally believed to play adominant role. In order to consider the influence of theinteraction between the fluid and pore wall on the phaseequilibrium, Travalloni et al. [26] proposed a modified PR-EOS considering both molecule–molecule and molecule-wall interaction and claimed that the molecule-wall interac-tion cannot be negligible. Yang et al. [27] extended PR-EOSby introducing a new term representing the molecule-wallinteraction and showed that the molecule-wall interactioncauses a significant alteration of the two-phase region.Adsorption has been an important factor in studying thefluid phase behavior under nanopore confinement. Donget al. [28] coupled the multicomponent potential theory ofadsorption with PR-EOS to investigate the fluid phasebehavior of pure hydrocarbons and their mixtures in organicslit-like and cylindrical nanopores and showed that adsorp-tion played an important role in the fluid phase behavior.Cui et al. [29] improved PR-EOS by reducing mole numberof fluids caused by adsorption. Sandoval et al. [30] exploredthe adsorption effect on the fluid phase behavior in nano-pores and incorporated the adsorption film thickness intothe calculation of the effective capillary radius. Song et al.[31] introduced a novel method for describing fluid adsorp-tion in nanopores by modifying the molar volume term inPR-EOS and showed that adsorption induced critical shiftsof confined fluids in nanopores.

As mentioned above, nanopore confinement effectsincluding capillary pressure, fluid-wall interaction, and mol-ecule adsorption cannot be ignored in porous media withpore diameters less than 50nm and greater than 2nm[32–34]. Although numerous models have been proposedto explore the phase behavior in nanopores, they focusedon one or both aspects of nanoscale confinement [23–31],and there still lack an accurate model that takes into accountcapillary pressure, fluid-wall interaction, and adsorptioneffect simultaneously. Additionally, despite methods beingthere to investigate nanopore confinement effect, only afew of them have studies the influence of nanopore confine-ment effect on CO2 injection recovery mechanisms in tightoil reservoirs. Motivated by these points, a modified PR-EOS model is established to study the fluid phase behaviorand recovery mechanisms in tight oil reservoirs for CO2injection.

The rest of the paper is organized as follows. In Section2, the methodology section illustrates the procedures ofmodel development by coupling fluid-wall interaction andadsorption effect in the EOS and capillary pressure in theflash calculation. Subsequently, the proposed model is vali-dated with CMG-Winprop and experimental results withbulk and confined fluids, and then, we performed to analyzeone case study from the Bakken tight oil reservoir at variouspore sizes in Section 3. In Section 4, based on the investiga-tion of the phase behavior of tight oil with CO2 injectionunder different scenarios, the recovery mechanisms affectedby minimum miscible pressure (MMP), solution gas-oil

ratio, oil volume expansion, viscosity reduction, extractionof lighter hydrocarbons, and molecular diffusion are studiedwith respect to the confinement effect. At the end, summaryand conclusions are provided in Section 5 to give some sug-gestions. The developed novel model provides a betterunderstanding of confinement effect on the phase behaviorin tight reservoirs and even nanoscale porous media. Thefindings of this study can also help for better understandingof the flow mechanism of tight oil reservoirs especially in thecase of CO2 injection for enhancing oil recovery.

2. Materials and Methods

The fluid phase behavior in tight oil reservoirs is governedby the interactions of fluid-fluid and fluid-wall interactionswithin the confining geometry [11]. In the pore networks,fluid molecules are usually adsorbed onto the pore wall[35–37]. At large pore sizes, the number of moleculesadsorbed is negligible compared to the volume of the liquid.When the pore size decreases further (<10nm), the interac-tion between molecules and pore walls of porous media issignificant [32]. Adsorption will be significant and greatlyreduces the number of fluid molecules in the free state whichthen will affect the molecular molar volume. In the nano-pores, the larger capillary pressure, van der Waals forces,fluid-wall interaction, and adsorption effect lead to the devi-ation of physical properties in the bulk fluid.

In this section, the methodology is introduced todescribe the nanopore confinement effect on the phasebehavior, and the PR-EOS and Rachford-Rice flash calcula-tion are modified considering capillary pressure, fluid-wallinteraction, and adsorption effect.

2.1. Fluid-Wall Interaction. The original PR-EOS consists ofa repulsion pressure PR and an attraction pressure PA asfollows [38]:

P = PR + PA =RT

Vm − b−

aVm Vm + bð Þ + b Vm − bð Þ , ð1Þ

PR =RT

Vm − b, ð2Þ

PA = −a

Vm Vm + bð Þ + b Vm − bð Þ , ð3Þ

where P is the system pressure, R is the gas constant, T is thesystem temperature, Vm is the molar volume, a represents“attraction” parameter, and b is the van der Waals co-vol-ume, which represent “repulsion” parameter.

In nanopores, in order to consider fluid-wall interactioneffect, we introduce a molecule-wall interaction pressureterm PFW which plays a role of diminishing the attractivecomponent [25] into Equation (1) as follows:

3Geofluids

P =RT

Vm − b−

aVm Vm + bð Þ + b Vm − bð Þ +

cVm Vm + bð Þ + b Vm − bð Þ ,

PFW =c

Vm Vm + bð Þ + b Vm − bð Þ ,

ð4Þ

where c is the fluid-wall interaction effect coefficient.

2.2. Adsorption Effect. There are many models describing theadsorption effect, such as ideal adsorbed solution theory,multicomponent potential theory of adsorption, and Lang-muir isothermal adsorption model [39–41]. For the sake ofincorporating the adsorption effect on nanopore confine-ment conveniently, in this paper, the fluid moleculesadsorbed in the organic matters and pore walls are assumedstationary, and adsorption leads to the reduction number ofmovable fluid molecules, which in turn increases the effec-tive mole volume of fluid molecules in the bulk phase [29].An adsorption effect coefficient α is introduced into Equa-tion (1) as follows:

P =RT

αVm − b−

aαVm αVm + bð Þ + b αVm − bð Þ : ð5Þ

2.3. The Modified PR-EOS. The original PR-EOS could bemodified by two parameters representing separately fluid-wall interaction and adsorption effect. Then, the PR-EOS ismodified as follows:

P =RT

αVm − b−

a − cαVm αVm + bð Þ + b αVm − bð Þ : ð6Þ

From Equation (6), it can be seen that when α = 1 andc = 0, that is to say, fluid-wall interaction and adsorptioneffect are not taken into account, the modified PR-EOS canbe reduced to original PR-EOS.

For the PR-EOS, the isotherm merely has a horizontaltangent and inflection point at the critical point in the typi-cal pressure-volume diagram [42]. This can be expressedmathematically that the first and second derivatives of pres-sure with respect to volume at a constant temperature areequal to 0.

∂P∂V

� �T=Tc

=∂2P∂V2

!T=Tc

= 0, ð7Þ

where Tc represents the critical temperature.Imposing Equation (7) on Equation (6) and parameters

a − c and b yields could be expressed by

a − c = 0:45724R2TC

2

Pc, ð8Þ

b = 0:07780RTc

αPc: ð9Þ

The details of “a − c ” and “b” calculations are specifiedin the Appendix. From Equations (6)–(9), the expressions

of critical pressure and critical temperature can be obtainedas follows:

PCC = 0:01324a − c

α2b2,

TCC = 0:17015a − cαbR

,ð10Þ

where PCC and TCC are the critical pressure and criticaltemperature determined by the modified PR-EOS.

The dimensionless shifts of critical pressure ΔP and crit-ical temperature ΔT are defined as follows:

ΔP =Pc − Pcc

Pc=aα2 − a + c

aα2, ð11Þ

ΔT =Tc − Tcc

Tc=aα − a + c

aα: ð12Þ

2.4. Correlation for Critical Pressure and CriticalTemperature. Equations (11) and (12) exhibit that fluid-wall interaction and adsorption effect influence the criticalpressure and critical temperature in nanopores. It has beenreported that the pore size rp and collision diameter σLJ(Lennard-Jones molecular size parameter) are the importantfactors on shifts of critical pressure and critical temperature[27, 31]. The confined fluid critical pressure shift and criticaltemperature shift for different components (CO2, CH4,C2H4, C2H6, C4H10, C8H18, and C10H22) are collected fromreferences [43–51]. Then, the correlations between the shiftsof critical properties and the dimensionless pore size (rp/σLJ )can be obtained and demonstrated in Figures 2(a) and 2(b).

As described in Figures 2(a) and 2(b), the correlations ofshifts of critical pressure and critical temperature withdimensionless pore size are

ΔP = 0:9793rpσLJ

� �−0:6366,

ΔT = 0:7597rpσLJ

� �−0:7708,

ð13Þ

where rp is the radius of pore throat and σLJ is the Lennard-Jones molecular size parameter.

From Equations (11) and (12), both fluid-wall interac-tion effect coefficient α and adsorption effect coefficient ccan be calculated by shifts of critical pressure and tempera-ture.

α =1−ΔT1−ΔP

=1 − 0:7597 rp/σLJ

� �−0:77081 − 0:9793 rp/σLJ

� �−0:6366 ,

c = 1 − α 1−ΔTð Þð Þa = 1 −1 − 0:7597 rp/σLJ

� �−0:7708� �21 − 0:9793 rp/σLJ

� �−0:63660B@

1CAa:

ð14Þ

4 Geofluids

2.5. Phase Equilibrium Calculation considering CapillaryPressure, Fluid-Wall Interaction, and Molecule Adsorption.According to the thermodynamic theory, for a system con-taining Nc components, the thermodynamic condition forthe phase equilibrium state is that when the temperatureand pressure of each phase are equal, the chemical potentialor fugacity of each component is equal.

f Li T , PL, xið Þ = f Vi T , PV , yið Þi = 1, 2,⋯,Nc,

f Li = xiφLi PL,

f Vi = yiφVi PV ,

ð15Þ

where f Li and f Vi are fugacity of component i in the liquid andvapor phases, respectively. xi and yi are the mole fraction ofcomponent i in the liquid and vapor phases, respectively. PLand PV are the liquid and vapor pressures, respectively. φL

iand φV

i are the fugacity coefficient of component i in the liquidand vapor phases. Nc is the number of components.

According to the mass balance equation, Rachford andRice [52] proposed an isothermal flash calculation methodto determine the equilibrium phase composition and molefraction of the component in the liquid and gas phases.The mass balance equation and Rachford-Rice equationare presented in

〠Nc

i=1xi = 〠

Nc

i=1yi = 1 i = 1, 2,⋯,Nc, ð16Þ

zi = 1 − nVð Þxi + nVyi, ð17Þ

〠Nc

i=1

Ki − 1ð Þzi1 + nV Ki − 1ð Þ = 0 i = 1, 2,⋯,Nc, ð18Þ

where zi is the overall mole fraction of component i. nV isoverall number of moles in vapor phase. Ki is phase equilib-rium ratio of component i.

The difference between liquid pressure and vapor pres-sure is defined as capillary pressure Pcap, which is calculatedby the Young-Laplace equation [53]:

Pcap = PV − PL =2σ cos θ

rp, ð19Þ

where θ is the contact angle. rp is the radius of pore throat. σis the interfacial tension which can be calculated byMacleod-Sugden correlation [54] as follows:

σ = 〠Nc

i=1ρLxi P½ �i − ρVyi P½ �i� �" #4

, ð20Þ

where ρL and ρV are the densities of liquid phase and vaporphase, respectively. ½P�i is the parachor of component i in theliquid or vapor phase.

The formula of the compressibility factor in nanopores ispresented as follows:

Z =αPVm

RT, ð21Þ

where Z is the compressibility factor.Hence, rearranging Equation (6) into the compressibility

factor form rewrites

Z3 − 1 − Bð ÞZ2 + A − 3B2 − 2B� �

Z − AB − B2 − B3� �= 0,ð22Þ

0.9

1.0

0.8

0.70.6

0.5

0.4

0.3

0.2

0.1

0.00 1 2 3 4 5 6 7 8 9 10 11 12 13

R2 = 0.8120

CO2

CH4

C2H6 Fitting curve

C4H10

C8H18

ΔP = 0.9793(rp/σLJ)–0.6366

rp/σLJ

ΔP

(a)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00 3 6 9 12 15 18 21 24 27 30

R2 = 0.9160

ΔT

CO2

CH4

C2H6

C2H4

Fitting curve

C4H10

C10H22

C8H18

ΔP = 0.7597(rp/σLJ)–0.7708

rp/σLJ

(b)

Figure 2: (a) Correlation of shift of critical pressure and (b) correlation of shift of critical temperature with the dimensionless pore size.

5Geofluids

where

A =a − cð ÞPRTð Þ2 ,

B =bPαRT

:

ð23Þ

Since the pressures of the liquid and vapor phases arenot equal, Equation (21) can be separately rewritten for liq-uid phase with the following:

ZL3 − 1 − BLð ÞZL

2 + AL − 3BL2 − 2BL

� �ZL − ALBL − BL

2 − BL3� �

= 0,

ð24Þ

where ZL is the compressibility factor of the liquid phase.Similarly, for vapor phase is as follows:

ZV3 − 1 − BVð ÞZV

2 + AV − 3BV2 − 2BV

� �ZV − AVBV − BV

2 − BV3� �

= 0,

ð25Þ

where ZV is the compressibility factor of the vapor phase.The fugacity coefficient for the liquid and vapor phases

are defined by the following expressions:

ln φLi = − ln ZL − BL½ � + biL

bLZL − 1ð Þ + AL

2ffiffiffi2

pBL

� 2∑Nci=1xjL 1 − kij

� � ffiffiffiffiffiffiffiffiffiffiffiaiLajLp

aL−biLbL

" #

lnZL + BL 1 +

ffiffiffi2

p� �ZL + BL 1 −

ffiffiffi2

p� �24

35,

ln φVi = − ln ZV − BV½ � + biV

bVZV − 1ð Þ + AV

2ffiffiffi2

pBV

� 2∑Nci=1xjV 1 − kij

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiaiVajVp

aV−biVbV

" #

lnZV + BV 1 +

ffiffiffi2

p� �ZV + BV 1 −

ffiffiffi2

p� �24

35: ð26Þ

Hence, the liquid-vapor phase equilibrium accountingfor the capillary pressure effect can be obtained using theaforementioned equations. Successive substitution andNewton-Raphson method are applied for solving nonlinearequations. Figure 3 demonstrates the flow chart for thephase equilibrium calculation with the nanopore confine-ment effect considering capillary pressure, fluid-wall interac-tion, and molecule adsorption.

3. Model Validation and Analysis

3.1. Model Validation. To validate the accuracy of modifiedmodel considering capillary pressure, fluid-wall interaction,and molecule adsorption effect, in this section, the developedmodel results are compared with CMG-Winprop results andexperimental results. The phase equilibrium ratio (K-value)of fluid components in Tahe Oilfield in China at T =124:4°C and P = 20:78MPa is first calculated. The fluid typein Tahe Oilfield is bulk fluids.

The predicted results and CMG-Winprop results aresummarized in Table 1. Furthermore, we calculate the phaseequilibrium ratio (K-value) of fluid components at T = 344:8K and P = 61:8psi. The data from experimental studies arereported in the literature [10], and the fluid type is confinedfluids. The predicted results and experimental results aresummarized in Table 2. The compared analysis indicates agood agreement between the developed model data withCMG-Winprop and experimental results. The average devi-ation between the predicted K-value and CMG-Winpropresults is 2.71%, indicating that the modified model pro-posed in this study can effectively predict the phase behaviorof fluids in bulk. The average deviation between the pre-dicted K-value and experimental results is 1.02%. Comparedwith the model only considering capillary pressure whoseaverage deviation is 2.78%, the error is more acceptable.

3.2. Nanopore Confinement Effect on Phase Behavior ofFluids in Bakken Tight Oil Reservoir. The Bakken tight oilreservoir is one of the largest tight oil resources in the world,with total daily production exceeding 19 × 104 t. In thisstudy, we investigate the nanopore confinement effect onthe phase behavior of a typical fluid in Bakken tight oil res-ervoirs. The composition and physical property parametersof crude oil are taken from Zhang and Yu et al. [32, 55],which are shown in Tables 3 and 4, respectively.

3.2.1. Phase Envelope. The modified model is used to deter-mine the phase envelope of the Bakken tight oil reservoirfor the pore radius ranging from 5nm to 50nm. Figure 4conducts the sensitivity analyses of pore size on the phaseenvelope by separately considering capillary pressure, fluid-wall interaction, and adsorption effect. As shown inFigure 4(a) where only capillary pressure is considered, thebubble point curve is lowered as the pore size decreases.After considering capillary pressure, the bubble point pres-sure is obviously suppressed when the pore size is less than50nm. The significant changes caused by the capillary pres-sure effect on bubble point pressures can reach as high as630 psi when the pore size reduces to 5 nm. As the systemgoes above the bubble point pressure, the fluids in the Bak-ken tight oil reservoir may remain in the single-phase region.When the pressure is lower than the bubble point pressure,the amount of light or medial hydrocarbons tend to be evap-orated to the form of gas bubble. Hence, the reduction in thebubble point pressure delays the process of vapor extractsout of the liquid and indicates that the fluids may remainin the liquid phase much longer. Therefore, consideringcapillary pressure has a positive influence on the tight oil

6 Geofluids

Estimate initial K–value by wilson equation

Flash calculation with rachford–rice equation

Modified PR – EOS for vaporModified PR – EOS for liquid

Solve laplace equation for Pc PL ≠ PV

NO

NO

YES

YES

pv = pL+PC

Input T, P, Nc, zi, rp, Pzi, Tci, MWi,ɷi, [P]i, binary coefficient

PL = P PV = P

Obtain nL, nv, xi, yi

Obtain f iL

Obtain xi, yi, nv, f iL, f iV, nL, nV

Obtain f iV

Update K-valueKi

n+1 = Kni (fiL/f v

i)

Σ (1–f iL/f iV)<𝜀 ?i = 1

NC

Figure 3: Flow char for phase equilibrium calculation with modified PR-EOS.

Table 1: Comparison of phase equilibrium ratio between predicted results and CMG-Winprop results.

Fluid types Components Molar fraction (%) CMG-Winprop results Simulation results Relative deviation (%)

Bulk fluids

C1 43.96 2.3186 2.3798 2.64

C2-C4 21.03 1.0071 0.9943 1.27

C5-C7 6.81 0.3447 0.3337 2.94

C8-C9 7.59 0.1897 0.1823 3.90

C10+ 18.71 0.0068 0.0070 3.13

CO2 1.90 1.6850 1.7255 2.40

Table 2: Comparison of phase equilibrium ratio between predicted results and experimental or simulation results.

Fluid types ComponentsMolar

fraction (%)Experimental

dataSimulation results

with Pc

Relativedeviation (%)

Simulation resultsthis paper

Relativedeviation (%)

Confinedfluids

i-C4 61.89 2.652 2.816 6.18 2.594 2.19

n-C4 18.11 1.885 1.904 1.01 1.894 0.48

C8 20.00 0.0524 0.053 1.15 0.0522 0.38

7Geofluids

production, as illustrated in Figure 4(b) where only fluid-wall interaction is considered, the entire phase envelope issuppressed as the pore size decreases. Especially, consideringfluid-wall interaction also reduces the critical point insteadof considering only the capillary pressure. As the pore sizedecreases, the critical point and bubble point pressurebecome smaller. The phase envelope at 50 nm is almost thesame as that of bulk phase, whereas the critical pressure at5 nm is suppressed by 8.3%. This is because when the poresize decreases, the interaction between fluid molecules andpore wall increases. As can be seen in Figure 4(c) where onlyadsorption effect is considered, it’s similar to just consider-ing fluid-wall interaction. It can be seen from theFigures 4(a)–4(c) that the effect of capillary pressure, fluid-wall interaction and adsorption effect on the phase behaviorin nanopores cannot be ignored.

By considering capillary pressure, fluid-wall interaction,and adsorption effect together, the P-T phase envelope ofthe Bakken tight oil reservoir at various pore sizes is plottedin Figure 4(d). It is illustrated that the phase envelope tendsto move downward with the decrease of pore size. The crit-ical point is reduced and the dew point curve shrinks com-pared with the phase envelope that only considers capillarypressure. The bubble point curve shrinks further than thephase envelope that considers only fluid-wall interaction oradsorption effect.

3.2.2. Bubble Point Pressure. The bubble point pressures atdifferent pore sizes are calculated. Figure 5 describes thebubble point pressure at the reservoir temperature of 230°Fand compares the nanopore confinement on bubble pointpressure with different pore sizes. As illustrated inFigure 6, the smaller the pore size, the more significant thenanopore confinement effect. When the pore size is smallerthan 10nm, significant changes can be observed. The bubblepoint pressure reduces to 1542 psi (19.1% smaller) when thepore size is 10 nm, and the bubble point pressure suppressesto 650 psi (65.9% smaller) as the pore size decreases to 2 nm.However, when the pore size is above 100nm, the bubble

point pressure approaches to the bulk fluids, and the nano-pore confinement effect can be neglected.

3.2.3. Interfacial Tension and Capillary Pressure. In this sec-tion, the interfacial tension (IFT) and capillary pressure (Pc)in different radii under different pressures at the reservoirtemperature of 230°F are calculated. As shown in Figure 6,as the pressure increases, the interfacial tension decreases.The interfacial tension is significantly affected by the exis-tence of nanopore confinement. The smaller the pore size,the greater the interfacial tension decreases. As describedby Figure 7, when the pressure increases, the capillary pres-sure also decreases. The smaller the pore size, the greater thecapillary pressure. When the pressure is 1500 psi, the capil-lary pressure with pore size of 5 nm is more than 5 times thatof the bulk phase. As the pressure is smaller, the multipleincreases. Therefore, the nanopore confinement effect willlead to high capillary pressure which cannot be ignored intight oil reservoirs.

4. Recovery Mechanisms of CO2 Injection

The accepted recovery mechanisms of CO2-EOR in conven-tional reservoirs are as follows: (1) reduction of the vaporand liquid phase interfacial tension towards achieving misci-bility with the crude oil, (2) CO2 dissolves into crude oil,leading to oil volume expansion and crude oil viscosityreduction, (3) CO2 extraction of lighter hydrocarbons fromthe liquid phase, (4) CO2 molecular diffusion, and (5) thesweep efficiency is improved, thereby enhancing the ultimateoil recovery [56, 57]. Because of these recovery mechanisms,CO2 flooding can greatly improve oil recovery. However,CO2 injection in unconventional reservoirs such as tight oilreservoirs has not attracted enough attention. A betterunderstanding of the effect of nanopore confinement effecton recovery mechanisms will help to optimize the designstrategy for CO2 injection in tight oil reservoirs.

4.1. Minimum Miscible Pressure (MMP). Minimum misciblepressure (MMP) is an important factor to decide miscibleflooding. It is defined as the minimum pressure where theinjected gas and the oil phase have no obvious interfaceand then become miscible with each other [58]. A series ofPVT and core flooding tests have validated that CO2 is mucheasier to become miscible with crude oil than other gasesincluding flue gas, natural gas, and nitrogen [59]. Based onthe calculation of phase equilibrium, we obtain the MMPat different CO2 injection ratios. As observed in Figure 8,the MMP decreases gradually with the increase of CO2

Table 3: The composition and physical property parameters of crude oil for the Bakken formation.

Components Tci (K) Pci (bar) Vc (L/mol) MWi (g/mol) ωi Parachor zi (%)

C1 190.60 45.40 0.0990 16.04 0.0080 77.0 25.06

C2-C4 363.30 42.54 0.1970 42.82 0.1432 145.2 22.00

C5-C7 511.56 33.76 0.3338 83.74 0.2474 250.0 20.00

C8-C9 579.34 30.91 0.4062 105.91 0.2861 306.0 13.00

C10+ 788.74 21.58 0.9208 200.00 0.6869 686.3 19.94

Table 4: Binary interaction parameters for oil components.

Components C1 C2-C4 C5-C7 C8-C9 C10+

C1 0 0.0078 0.0242 0.0324 0.0779

C2-C4 0.0078 0 0.0046 0.0087 0.0384

C5-C7 0.0242 0.0046 0 0.0006 0.0169

C8-C9 0.0324 0.0087 0.0006 0 0.0111

C10+ 0.0779 0.0384 0.0169 0.0111 0

8 Geofluids

injection. When the gas injection ratio is 100%, the MMPdecreased by 9.7% compared with the CO2 injection ratiois 10%. In order to explore the influence of nanopore con-finement effect on the MMP in tight oil reservoir, we alsoevaluated the MMP of 100% CO2 injection at various poresizes. As illustrated in Figure 9, the MMP tends to be low-ered as the pore size decreases. When the pore size is 5 nm,the MMP decreased by 16.2% compared with the pore sizeis 100nm.

4.2. CO2 Solution Gas-Oil Ratio (GOR). The solution gas-oilratio (GOR) is defined as the volume of gas at standard con-dition that evolves from the oil divided by the volume of oilat standard condition. CO2 could dissolve into crude oil,leading to oil volume expansion and crude oil viscosity

reduction. GOR is an important factor in evaluating thedegree of CO2 dissolution. We calculate the CO2 solutiongas-oil ratio of Bakken tight oil versus pressure at variousCO2 injections at T = 230°F as shown in Figure 10. It is obvi-ous that the CO2 solution gas-oil ratio increases with theincrease of gas injection. When the gas injection ratio is100%, the CO2 solution gas-oil ratio is about 3 times thatof the gas injection ratio is 30%. In order to explore theinfluence of nanopore confinement effect on the CO2 solu-tion gas-oil ratio in tight oil reservoir, we also evaluatedthe CO2 solution gas-oil ratio of 100% CO2 injection at var-ious pore sizes. As illustrated in Figure 11, before CO2 iscompletely dissolved into crude oil, the CO2 solution gas-oil ratio increases with the decrease of pore size. It indicatesthat considering the influence of nanopore confinement

20nm

2500225020001750150012501000

750500250

00 200 300 400 500 600 700 800

Pres

sure

, psi

5 nm10nm

50nmBulk

Temperature, °F

100

cp

(a)

Critical point of bulkCritical point of 50nmCritical point of 20nmCritical point of 10nmCritical point of 5nm

2500225020001750150012501000

750500250

00 200 300 400 500 600 700 800

Pres

sure

, psi

Temperature, °F

100

20nm

5 nm10nm

50nmBulk

(b)

2500225020001750150012501000

750500250

00 200 300 400 500 600 700 800

Pres

sure

, psi

Temperature, °F

100

Critical point of bulkCritical point of 50nmCritical point of 20nmCritical point of 10nmCritical point of 5nm

20nm

5 nm10nm

50nmBulk

(c)

2500225020001750150012501000

750500250

00 200 300 400 500 600 700 800

Pres

sure

, psi

Temperature, °F

100

Critical point of bulkCritical point of 50nmCritical point of 20nmCritical point of 10nmCritical point of 5nm

20nm

5 nm10nm

50nmBulk

(d)

Figure 4: P-T phase envelope for Bakken tight oil reservoir at various pore sizes considering (a) capillary pressure, (b) fluid-wall interaction,(c) adsorption effect, and (d) nanopore confinement effect.

9Geofluids

effect, the amount of gas dissolved into crude oil increases,which will lead to the increase of the oil volume expansionand the decrease of the viscosity of crude oil, as shown inthe following two sections. When the reservoir pressure ishigher than the bubble point pressure, all the gas will be dis-solved into the crude oil without free gas, and the value ofsolution gas-oil ratio will remain unchanged.

4.3. Oil Volume Expansion. CO2 dissolves into the oil phasethat causes the oil volume increase. The volumetric expan-sion capacity of crude oil can be characterized by the oil for-mation volume factor. The oil formation volume factordefines as the oil volume at reservoir condition divided bythe oil volume at standard condition.

Bo =nL Vmð ÞL� �

RCnL Vmð ÞL� �

STDð27Þ

where nL represents liquid phase mole fraction, ðVmÞL repre-sents liquid phase molar volume, RC represents reservoircondition, and STD represents standard condition.

Based on the phase equilibrium calculation, we calculatethe oil formation volume factor of Bakken tight oil versuspressure at various CO2 injections at T = 230°F. As shownin Figure 12, it is obvious that the oil formation volume fac-tor increases with the increase of gas injection. When theCO2 injection ratio is 100%, the oil formation volume factorincreased by 31.9% compared with that without CO2 injec-tion at pressure 4351.2 psi. This can be also explained bythe variation trend of CO2 solution gas-oil ratio in the uppersection. In order to explore the influence of nanopore con-finement effect on the oil formation volume in tight oil

2000

1800

1600

1400

1200

1000

800

600

1 10 100 1000

Bubb

le–p

oint

pre

ssur

e, ps

i

Pore radius, nm

With nanopore confinement

Without nanopore confinement

Figure 5: Bubble point pressure of the Bakken tight oil at variouspore radii at T = 230°F.

30.027.525.022.520.017.515.012.510.0

7.55.02.50.0

200 400 600 800 1000 1200 1400 1600 1800 2000

50 nm100 nmBulk

Inte

rfaci

al te

nsio

n, d

yne/

cm

Pressure, psi

10 nm20 nm

5 nm

Figure 6: Interfacial tension of the Bakken tight oil versus pressureat various pore radii at T = 230°F.

850800750700650600550500450400350300250200150100

500

200 400 600 800 1000 1200 1400 1600 1800 2000

Pressure,psi

Capi

llary

pre

ssur

e,psi

50 nmBulk

5 nm

20 nm10 nm

Figure 7: Capillary pressure of the Bakken tight oil versus pressureat various pore radii at T = 230°F.

3650

3600

3550

3500

3450

3400

3350

33000 10 20 30 40 50 60 70 80 90 100 110

Min

imum

misc

ible

pre

ssur

e, ps

i

CO2 injection ratio, %

Figure 8: Minimum miscible pressure of the Bakken tight oilversus CO2 injection.

10 Geofluids

reservoir, we also evaluate the oil formation volume factor ofBakken tight oil versus pressure at various pore radii at T= 230°F. As illustrated in Figure 13, the oil formation vol-ume factor of the oil increases as the pore size decreases.When the pore size decreases from infinity to 50 nm,20 nm, 10nm, and 5nm, the increment of oil formation vol-ume factor at pressure 4351.2 psi is almost 1.5%, 5.9%, 9.2%,and 14.7%.

4.4. Viscosity reduction. The oil viscosity is calculated by theJossi-Stiel-Thodos (JST) model [60, 61].

μ − μ∗ð Þ + 10−4 1/4 = a0 + a1ρr + a2ρ

2r + a3ρ

3r + a4ρ

4r , ð28Þ

where μ is the viscosity of crude oil under formation condi-tions, and the values of a0, a1, a2, a3, and a4 are 0.1023,0.023364, 0.058533, -0.040758, and 0.0093324, respectively.

ρr is defined as

ρr = ρL 〠Nc

i=1xiVci

" #1/α, ð29Þ

where Vci is a critical volume of component i, and the valueof α is 1.

3400

3300

3200

3100

3000

2900

2800

2700

26000 10 20 30 40 50 60 70 80 90 100 110

Pore radiu, nm

Min

imum

misc

ible

pre

ssur

e, ps

i

Figure 9: Minimum miscible pressure of the Bakken tight oilversus pore size.

220200180160140120100

80604020

00 500 1000 1500 2000 2500 3000 3500 4000 4500

CO2 s

olut

ion

gas–

oil r

atio

, m3 /

m3

Pressure, psi

30% CO2 injection50% CO2 injection100% CO2 injection

Figure 10: CO2 solution gas-oil ratio of the Bakken tight oil versuspressure at various CO2 injections at T = 230°F.

0 500 1000 1500 2000 2500 3000 3500 4000 45000

2040

60

80

100120140160

180

200220

Pressure, psi

50 nmBulk

CO2

solu

tion

gas–

oil r

atio

, m3 /m

3

5 nm10 nm20 nm

Figure 11: CO2 solution gas-oil ratio of the Bakken tight oil versuspressure at various pore radii at T = 230°F.

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Pressure, psi

0.7

0.80.91.01.11.21.31.41.51.61.71.81.92.0

Oil

volu

me f

acto

r, bb

/STB

50% CO2 injectionNo CO2 injection30% CO2 injection 100% CO2 injection

Figure 12: Oil volume factor of the Bakken tight oil versus pressureat various CO2 injections at T = 230°F.

11Geofluids

The viscosity parameters of the mixture ξ are calculatedby the following formula:

ξ = 〠Nc

i=1xiTci

!1/6

〠Nc

i=1xiMi

!−1/2

〠Nc

i=1xiPci

!−2/3

:

ð30Þ

μ∗ is defined as

μ∗ =∑Nc

i=1 xiμ∗i M

1/2i

� �∑Nc

i=1 xiM1/2i

� � : ð31Þ

μ∗i can be calculated by the Stiel-Thodos formula:

μ∗i ξi =4:610T0:618

ri − 2:04e−0:449Tri

+1:94e−4:058Tri + 0:1

" #× 10−4, ð32Þ

where ξi = T1/6ci M

1/2i P2/3

ci , Tri = T/Tci.Based on the phase equilibrium calculation, we calculate

oil viscosity of the Bakken tight oil versus pressure at variousCO2 injections at T = 230°F. As shown in Figure 14, it isobvious that the viscosity decreases with the increase of gasinjection. When the CO2 injection ratio is 100%, the oil vis-cosity decreased by 62.4% compared with that without CO2injection at pressure 4351.2 psi. This can be explained by thevariation trend of CO2 solution gas-oil ratio in the uppersection. Hence, CO2 has obvious viscosity reduction effect.In order to explore the influence of nanopore confinementeffect on the oil viscosity in tight oil reservoir, we also eval-uate the oil viscosity of Bakken tight oil versus pressure atvarious pore radius at T = 230°F. As illustrated inFigure 15, the viscosity of the oil reduces as the pore sizedecreases. When the pore size decreases from infinity to50 nm, 20nm, 10 nm, and 5nm, the suppression of oil vis-cosity at corresponding bubble point pressure are almost13.4%, 24.8%, 33.8%, and 42.8%. It indicates that consider-ing the influence of nanopore confinement effect, the oilmobility could be increased.

0 500 1000 1500 2000 2500 3000 3500 4000 4500

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Oil

volu

me f

acto

r,bb/

STB

Pressure, psi

50 nm5 nm10 nm20 nm

Bulk

Figure 13: Oil volume factor of the Bakken tight oil versus pressureat various pore radii at T = 230°F.

0 500 1000 1500 2000 2500 3000 3500 4000 45000.0

0.2

0.4

0.6

0.8

1.4

1.2

1.0

Liqu

id v

iscos

ity,cp

Pressure, psi

50% CO2 injectionNo CO2 injection30% CO2 injection 100% CO2 injection

Figure 14: Oil viscosity of the Bakken tight oil versus pressure atvarious CO2 injections at T = 230°F.

0 2000500 1000 1500 2500 3000 3500 4000 45000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Liqu

id v

iscos

ity,cp

Pressure, psi

50 nm5 nm10 nm20 nm

Bulk

Figure 15: Oil viscosity of the Bakken tight oil versus pressure atvarious pore radii at T = 230°F.

12 Geofluids

4.5. Extraction of Lighter Hydrocarbons. Many experimentsand simulations have shown that the content of light com-ponents in the oil produced after CO2 injection increaseswhile the content of heavy components decreases. The masstransfer between CO2 and formation crude oil results in sub-stantial physical changes of the system.

The extraction coefficient βe of lighter hydrocarbon isdefined as

βe =nVyizi

: ð33Þ

Based on the phase equilibrium calculation, we calculatethe extraction coefficient of lighter hydrocarbon of the Bak-ken tight oil versus pressure at various CO2 injections at T= 230°F. As shown in Figure 16, it is obvious that the extrac-tion coefficient of lighter hydrocarbon increases with theincrease of gas injection. This can be explained by the fol-lowing mechanisms: Carbon dioxide dissolves and the liquidphase is relatively light, which contributes to the evaporationof light components and the enrichment of gas phase.Because the gas phase is enriched and the liquid phase is rel-atively light, the difference of components in the gas and

0 500 1000 1500 2000 2500 30000.0

0.2

0.1

0.3

0.4

0.6

0.5

0.7

0.8

0.9

1.0

Extr

actio

n co

effici

ent

Pressure, psi

No CO2 injection30% CO2 injection

50% CO2 injection100% CO2 injection

Figure 16: Extraction coefficient of lighter hydrocarbons for theBakken tight oil versus pressure at various CO2 injections at T = 230°F.

0 500 1000 1500 2000 2500 30000.0

0.1

0.2

0.3

0.5

0.4

0.6

0.7

0.8

0.9

1.0

Pressure, psi

50 nm5 nm10 nm20 nm

Bulk

Extr

actio

n co

effici

ent

Figure 17: Extraction coefficient of lighter hydrocarbons for theBakken tight oil versus pressure at various pore radii at T = 230°F.

00.00

0.02

0.04

0.06

0.08

0.10

0.12

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Pressure,psi

Co 2

diff

usio

n co

ffici

ent o

f liq

uid

phas

e

30% CO2 injection50% CO2 injection

100% CO2 injection

Figure 18: CO2 diffusion coefficient of liquid phase for the Bakkentight oil versus pressure at various CO2 injection at T = 230°F.

0 500 1000 1500 2000 2500

Pressure,psi

Co 2

diff

usio

n co

ffici

ent o

f vap

or p

hase

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

30% CO2 injection50% CO2 injection100% CO2 injection

Figure 19: CO2 diffusion coefficient of vapor phase for the Bakkentight oil versus pressure at various CO2 injections at T = 230°F.

13Geofluids

liquid phase is relatively small. According to the similarcompatibility principle, the mass transfer capacity of gas-liquid system is enhanced, and the result is that the enrichedgas further extracts the light hydrocarbon and the intermedi-ate hydrocarbon components of crude oil to form the richhydrocarbon phase. Therefore, the CO2 extraction of lighterhydrocarbons is conducive to improving the properties ofcrude oil and enhancing oil recovery.

In order to explore the influence of nanopore confine-ment effect on the oil formation volume in tight oil reservoir,we also evaluate the extraction coefficient of lighter hydro-

carbon of the Bakken tight oil versus pressure at variouspore radii at T = 230°F. As illustrated in Figure 17, theextraction coefficient of lighter hydrocarbon slightly reducesas the pore size decreases. It indicates that considering nano-pore confinement effect seems to have a slightly reducedeffect on extraction of lighter hydrocarbons.

4.6. Molecular Diffusion. Molecular diffusion is one of favor-able mechanism factors for oil recovery, especially for tightoil reservoirs. The flow velocity is low in tight oil reservoirswith low matrix permeability that the relative contributionof molecular diffusion becomes more significant comparedwith molecular diffusion in conventional reservoirs. Twoempirical correlations frequently applied are by the Wilke-Chang correlation [62] and the Sigmund correlation [63].In this study, we employed Wilke -Chang equation to esti-mate the CO2 diffusion coefficient as

Dik =7:4 × 108 ϕMið Þ1/2T

μkV0:6kb

, ð34Þ

where Dik represents the diffusion coefficient of component iin phase k, MCO2 is the molar mass of component i, T is thetemperature of system, μk is viscosity of phase k, Vkb is thecritical volume of phase k under the bubble point, and ϕ isassociation factor, which depends on the properties of thesolvent itself. If ethanol is the solvent, it has a value of 1.5.However, the calculation error is larger for nonassociativesystems, and Sun and Chen [64] deduced the relationshipbetween the association factor and temperature throughthe free volume theory as follows:

ϕ Tð Þ = 3:97 − 3:92 × 10−3T: ð35Þ

Based on the phase equilibrium calculation, we evaluatethe CO2 diffusion coefficient for the liquid and vapor phasesat various CO2 injections at T = 230°F. As shown inFigures 18 and 19, CO2 diffusion coefficient of the liquidphase increases with the increase of gas injection while.

CO2 diffusion coefficient of vapor phase decreases withthe increase of gas injection. In order to explore the influ-ence of nanopore confinement effect on the CO2 diffusioncoefficient in tight oil reservoir, we also evaluate the CO2diffusion coefficient of Bakken tight oil versus pressure atvarious pore radii at T = 230°F. As illustrated in Figures 20and 21, the CO2 diffusion coefficient is also different forthe liquid phase and vapor phase under nanopore confine-ment effect. CO2 diffusion coefficient for the liquid phase islarger in smaller pores whereas CO2 diffusion coefficientfor the vapor phase is smaller.

5. Summary and Conclusions

(1) An efficient model is proposed to calculate phasebehavior in tight oil reservoir with capillary pressure,fluid-wall interaction and adsorption effect in thiswork. Fluid-wall interaction and adsorption effectare introduced to modify the PR-EOS, and capillary

0 500

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.001000 1500 2000 2500 3000 3500 4000 4500 5000

Pressure,psi

Co 2

diff

usio

n co

ffici

ent o

f liq

uid

phas

e

50 nm5 nm

20 nm10 nm Bulk

Figure 20: CO2 diffusion coefficient of liquid phase for Bakkentight oil versus pressure at various pore radii at T = 230°F.

Pressure, psi

0 500 1000 1500 2000 2500 30000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Co 2

diff

usio

n co

ffici

ent o

f vap

or p

hase

5 nm10 nm20 nm

50 nmBulk

Figure 21: CO2 diffusion coefficient of vapor phase for Bakkentight oil versus pressure at various pore radii at T = 230°F.

14 Geofluids

pressure is used to modify the flash calculation forphase equilibrium calculation. The model has beenverified against commercial software (CMG-Win-prop) and experimental data with bulk fluids andconfined fluids. The results show that the nanoporeconfinement effects including capillary pressure,fluid-wall interaction, and adsorption effect are sig-nificant when the pore radius reduces to the ordersof nanometers

(2) For the Bakken tight oil reservoir, nanopore confine-ment effects impose an overall shrinkage to the P-Tphase envelope. Capillary pressure lowers bubblepoint curve as the pore size decreases. Fluid-wallinteraction and adsorption effect suppress the entirephase envelope as the pore size decreases. Especially,they reduce the critical point instead of capillarypressure. The bubble point pressure of Bakken oilis suppressed by 19.1% when the pore size is 10 nmunder nanopore confinement effect compared withthe counterpart case without pore proximity. It isreduced by 65.9% when the pore size is 2 nm. Asthe pore size decreases, the IFT decreases whereasthe capillary pressure increases obviously

(3) The recovery mechanisms for CO2 injection are alsoinvestigated in the Bakken tight oil reservoir fromdifferent respects. Results show that the MMPdecreases gradually with the increase of CO2 injec-tion. When the gas injection ratio is 100%, theMMP decreased by 9.7% compared with the CO2injection ratio is 10%. On this basis, the effect ofnanopore confinement on the MMP for CO2 injec-tion is also calculated. Results indicate that underthe nanopore confinement effect, the MMP willdecrease, resulting in the suppression of the resis-tance of fluid transport. Hence, it will be easier toreach miscibility and improve the well performance.Besides, as the increase of CO2 injection, the CO2solubility will increase, the volume of crude oilexpands, and the oil viscosity will decrease. Withthe nanopore confinement effect, the CO2 solutiongas-oil ratio and the oil formation volume factor ofthe oil increase with the decrease of pore size. Inturn, the oil viscosity reduces as the pore sizedecreases. Furthermore, the extraction coefficient oflighter hydrocarbon increases with the increase ofgas injection. With the nanopore confinement effect,the extraction coefficient of lighter hydrocarbonslightly reduces as the pore size decreases. It indi-cates that considering nanopore confinement effectseems to have a slightly reduced effect on extractionof lighter hydrocarbons. The CO2 diffusion coeffi-cients for liquid phase and vapor phase are differentin that CO2 diffusion coefficients for liquid phaseincrease with the increase of gas injection whileCO2 diffusion coefficient of vapor phase decreases.On this basis, we also evaluate the effect of nanoporeconfinement on the CO2 diffusion. Results show thatCO2 diffusion coefficient are also different for liquid

phase and vapor phase under nanopore confinementeffect. CO2 diffusion coefficient for liquid phase islarger in smaller pores whereas CO2 diffusion coeffi-cient for vapor phase is smaller

(4) An efficient model is proposed to calculate phasebehavior in tight oil reservoir with capillary pressure,fluid-wall interaction, and adsorption effect simulta-neously. However, the model does not take the effectof water on phase behavior into account, and thepore size distribution is also not considered. Furtherextensive research on these will continue toinvestigate

Appendix

The modified PR-EOS is given as follows:

P =RT

Vm′ − b−

a − c

Vm′ Vm′ + b� �

+ b Vm′ − b� � : ðA:1Þ

The first and second derivatives of pressure with respectto volume at critical point are yielded:

∂P∂V

� �T=Tc

= −αRTc

αVc − bð Þ2+

2α a − cð Þ αVc + bð ÞαVc αVc + bð Þ + b αVc − bð Þ½ �2

= 0,

ðA:2Þ

∂2P∂V2

!T=Tc

=2α2RTc

αVc − bð Þ3

+2α2 a − cð Þ b αVc − bð Þ − αVc + bð Þ 3αVc + 4bð Þ½ �

αVc αVc + bð Þ + b αVc − bð Þ½ �3 = 0:

ðA:3ÞFrom Equation (A.2), it is given as

RTc

αVc − bð Þ3 =2 a − cð Þ αVc + bð Þ

αVc αVc + bð Þ + b αVc − bð Þ½ �2 αVc − bð Þ :

ðA:4Þ

Imposing Equation (A.4) on Equation (A.3), it is yieldedas

2 αVc + bð ÞαVc − b

=αVc + bð Þ 3αVc + 4bð Þ − b αVc − bð Þ

αVc αVc + bð Þ + b αVc − bð Þð Þ : ðA:5Þ

It can be rewritten as

− αVc + bð Þ αVc − bð Þ 3αVc + 2bð Þ + 2αVc αVc + bð Þ2 + b αVc − bð Þ2 = 0:

ðA:6Þ

We assume b = kαVc, and Equation (A.6) can be rewrit-ten as

3k3 + 3k2 + 3k − 1 = 0: ðA:7Þ

15Geofluids

Solving for k could obtain k = 0:25308.

b = 0:25308αVc: ðA:8Þ

Imposing Equation (A.8) on Equation (A.2), it is writtenas

−RTc

1 − kð Þ2+

2 a − cð Þ 1 + kð ÞαVcð Þ 1 + kð Þ + k 1 − kð Þ½ �2

= 0: ðA:9Þ

Thus,

a − c =RTc αVcð Þ 1 + kð Þ + k 1 − kð Þ½ �2

2 1 + kð Þ 1 − kð Þ2 = 1:48742αRTcVc:

ðA:10Þ

Applying Equation (A.1) to the critical point andsubstituting Equations (A.8) and (A.10) into Equation(A.1), it is obtained as

PcVc

RTc=0:30740

α: ðA:11Þ

b is the van der Waals covolume, and the relationshipbetween b and Vc should be obtained from the traditionalPR-EOS,

b = 0:25308Vc: ðA:12Þ

From Equations (A.10), (A.11), and (A.12)), the param-eters a and b are given as follows:

a − c = 0:45724R2TC

2

Pc,

b = 0:07780RTc

αPc:

ðA:13Þ

Nomenclature

P: System pressureT : System temperatureR: Gas constantVm: Molar volumea: “Attraction” parameterb: “Repulsion” parameterc: Fluid-wall interaction effect coefficientα: Adsorption effect coefficientPCC: The critical pressure determined by the modified PR-

EOSTCC: The critical temperature determined by the modified

PR-EOSΔP: The dimensionless shifts of critical pressureΔT : The dimensionless shifts of critical temperaturerp: Pore sizeσLJ : Lennard-Jones molecular size parameterNc: The number of componentsf Li : Fugacity of component i in the liquid phase

f Vi : Fugacity of component i in the vapor phasexi: Mole fraction of component i in the liquid phaseyi: Mole fraction of component i in the vapor phasezi: Overall mole fraction of component iφLi : Fugacity coefficient of component i in the liquid

phaseφVi : Fugacity coefficient of component i in the vapor phase

nL: Overall number of moles in liquid phasenV : Overall number of moles in vapor phaseKi: Phase equilibrium ratio of component iPcap: Capillary pressureσ: Interfacial tensionθ: Contact angleρL: Density of liquid phaseρV : Density of vapor phase½P�i: The parachor of component iZ: The compressibility factorBo: Oil formation volume factorμ: Viscosity of crude oilVci: Critical volume of component iβe: The extraction coefficientDik: The diffusion coefficient of component i in phase kRC: Reservoir conditionSTD: Standard condition.

Data Availability

All the data supporting this study can be found in themanuscript.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

Acknowledgments

This work was supported by the National Nature ScienceFoundation of China (51674010). We would like toacknowledge Computer Modeling Group Ltd. for providingthe CMG software for this study of example validation.

References

[1] BP, BP Energy Outlook, BP, 2019.[2] EIA, “US energy information administration,” 2019, https://

www.eia.gov/.[3] L. Wang, E. Parsa, Y. Gao, J. T. Ok, K. Neeves, and X. Yin,

“Experimental study and modeling of the effect of nanocon-finement on hydrocarbon phase behavior in un- conventionalreservoirs,” in Presented at the SPE western North Americanand rocky mountain joint regional meeting, pp. 1–8, Denver,CO, USA, 2014.

[4] S. Salahshoor, M. Fahes, and C. Teodoriu, “A review on theeffect of confinement on phase behavior in tight formations,”Journal of Natural Gas Science and Engineering, vol. 51,pp. 89–103, 2018.

[5] Q. Feng, S. Xu, X. Xing, W. Zhang, and S. Wang, “Advancesand challenges in shale oil development: a critical review,”

16 Geofluids

Advances in Geo-Energy Research, vol. 4, no. 4, pp. 406–418,2020.

[6] S. Sun and T. Zhang, “A 6M digital twin for modelling andsimulation for subsurface reservoirs,” Advances in Geo-Energy Research, vol. 4, pp. 349–351, 2020.

[7] S. Roy, R. Raju, H. Chuang, B. Cruden, and M. Meyyappan,“Modeling gas flow through microchannels and nanopores,”Journal of Applied Physics, vol. 93, no. 8, pp. 4870–4879, 2003.

[8] P. H. Nelson, “Pore-throat sizes in sandstones, tight sand-stones, and shales,” AAPG Bulletin, vol. 93, no. 3, pp. 329–340, 2009.

[9] J. Wang, S. Wu, Q. Li, J. Zhang, and Q. Guo, “Characterizationof the pore-throat size of tight oil reservoirs and its control onreservoir physical properties: a case study of the Triassic tightsandstone of the sediment gravity flow in the Ordos Basin,China,” Journal of Petroleum Science and Engineering,vol. 186, article 106701, 2019.

[10] Y. Wang, B. Yan, and J. Killough, “Compositional modeling oftight oil using dynamic nanopore properties,” in Paper SPE166267 presented at SPE Annual Technical Conference andExhibition, pp. 1–13, New Orleans, LA, USA, 2013.

[11] B. Nojabaei, R. T. Johns, and L. Chu, “Effect of capillary pres-sure on phase behavior in tight rocks and shales,” in Paper SPE159258 presented at the SPE Annual Technical Conference andExhibition, pp. 281–289, San Antonio, TX, USA, 2012.

[12] S. Luo, H. Nasrabadi, and J. L. Lutkenhaus, “Effect of confine-ment on the bubble points of hydrocarbons in nanoporousmedia,” AICHE Journal, vol. 62, no. 5, pp. 1772–1780, 2016.

[13] S. Luo, J. L. Lutkenhaus, and H. Nasrabadi, “Experimentalstudy of pore size distribution effect on phase transitions ofhydrocarbons in nanoporous media,” Fluid Phase Equilibria,vol. 487, p. 8015, 2019.

[14] B. Pinho, S. Girardon, F. Bazer-Bachi, G. Bergeot, S. Marre,and C. Aymonier, “A microfluidic approach for investigatingmulticomponent system thermodynamics at high pressuresand temperatures,” Lab on a Chip, vol. 14, no. 19, pp. 3843–3849, 2014.

[15] V. R. Choudhary and K. Mantri, “Temperature-programmeddesorption of benzene on mesoporous Si-MCM-41, Na-AlSi-MCM-41, and H-AlSi-MCM-41,” Langmuir, vol. 16,pp. 8024–8030, 2020.

[16] H. Xu, “Probing nanopore structure and confined fluid behav-ior in shale matrix: a review on small-angle neutron scatteringstudies,” International Journal of Coal Geology, vol. 217, article103325, 2020.

[17] H. Cho, M. H. Bartl, andM. Deo, “Bubble point measurementsof hydrocarbon mixtures in mesoporous media,” Energy Fuels,vol. 31, no. 4, pp. 3436–3444, 2017.

[18] G. Günther, J. Prass, O. Paris, and M. Schoen, “Novel insightsinto nanopore deformation caused by capillary condensation,”Physical Review Letters, vol. 101, no. 8, article 086104, 2008.

[19] J. Zhong, S. Talebi, Y. Xu, Y. Pang, F. Mostowfi, and D. Sinton,“Fluorescence in sub-10 nm channels with an optical enhance-ment layer,” Lab on a Chip, vol. 18, no. 4, pp. 568–573, 2018.

[20] Y. Liu, X. Dong, Z. Chen, Y. Hou, Q. Luo, and Y. Chen, “Anovel experimental investigation on the occurrence state offluids in microscale pores of tight reservoirs,” Journal ofPetroleum Science and Engineering, vol. 196, article 107656,2020.

[21] S. Wang, Q. Feng, M. Zha et al., “Molecular dynamics simula-tion of liquid alkane occurrence state in pores and slits of shale

organic matter,” Petroleum Exploration and Development,vol. 42, no. 6, pp. 844–851, 2015.

[22] Z. Jin and A. Firoozabadi, “Thermodynamic modeling ofphase behavior in shale media,” SPE Journal, vol. 21, no. 1,pp. 190–207, 2016.

[23] Y. Zhang, W. Yu, K. Sepehrnoori, and Y. Di, “Investigation ofnanopore confinement on fluid flow in tight reservoirs,” Jour-nal of Petroleum Science and Engineering, vol. 150, pp. 265–271, 2017.

[24] L. Li and J. J. Sheng, “Nanopore confinement effects on phasebehavior and capillary pressure in a Wolfcamp shale reser-voir,” Journal of Taiwan Institute of Chemical Engineers,vol. 78, pp. 317–328, 2017.

[25] Q. Yang, P. Sun, L. Fumagalli et al., “Capillary condensationunder atomic-scale confinement,” Nature, vol. 588, no. 7837,pp. 250–253, 2020.

[26] L. Travalloni, M. Castier, F. W. Tavares, and S. I. Sandler,“Thermodynamic modeling of confined fluids using an exten-sion of the generalized van der Waals theory,” Chemical Engi-neering Science, vol. 65, no. 10, pp. 3088–3099, 2010.

[27] G. Yang, Z. Fan, and X. Li, “Determination of confined fluidphase behavior using extended Peng-Robinson equation ofstate,” Chemical Engineering Journal, vol. 378, pp. 122–132,2019.

[28] X. Dong, H. Liu, K. Wu et al., “Confined behavior of CO2/hy-drocarbon system in nanopores of tight and shale rocks,” inUnconventional Resources Technology Conference. Society ofExploration Geophysicists, pp. 1–14, Denver, Colorado, USA,2019.

[29] X. Cui, E. Yang, K. Song, and J. Huang, “Phase equilibrium ofhydrocarbons confined in nanopores from a modified Peng-Robinson equation of state,” in SPE Annual Technical Confer-ence and Exhibition, Society of Petroleum Engineers, pp. 1–21,Dallas, Texas, USA, 2018.

[30] D. R. Sandoval, W. Yan, M. L. Michelsen, and E. H. Stenby,“Influence of adsorption and capillary pressure on phase equi-libria inside shale reservoirs,” Energy & Fuels, vol. 32, no. 3,pp. 2819–2833, 2018.

[31] Z. Song, Y. Song, J. Guo, Z. Zhang, and J. Hou, “Adsorptioninduced critical shifts of confined fluids in shale nanopores,”Chemical Engineering Journal, vol. 385, article 123837, 2020.

[32] Y. Zhang, H. Lashgari, Y. di, and K. Sepehrnoori, “Capillarypressure effect on phase behavior of CO2/hydrocarbons inunconventional reservoirs,” Fuel, vol. 197, pp. 575–582, 2017.

[33] X. Zhang and W. Wang, “Square-well fluids in confined spacewith discretely attractive wall-fluid potentials: critical pointshift,” Physical Review E, vol. 74, no. 6, article 062601, 2006.

[34] K. Wu, Z. Chen, X. Li, and X. Dong, “Methane storage innanoporous material at supercritical temperature over a widerange of pressures,” Scientific Reports, vol. 6, no. 1, article33461, 2016.

[35] Z. Jiang, L. Zhao, and D. Zhang, “Study of adsorption behaviorin shale reservoirs under high pressure,” Journal of NaturalGas Science and Engineering, vol. 49, pp. 275–285, 2018.

[36] K. Zhang, N. Jia, and L. Liu, “Adsorption thicknesses of con-fined pure and mixing fluids in nanopores,” Langmuir,vol. 34, no. 43, pp. 12815–12826, 2018.

[37] A. Taghavinejad, M. Sharifi, E. Heidaryan, K. Liu, andM. Ostadhassan, “Flow modeling in shale gas reservoirs: acomprehensive review,” Journal of Natural Gas Science andEngineering, vol. 83, article 103535, 2020.

17Geofluids

[38] D. Y. Peng and D. B. Robinson, “A new two-constant equationof state,” Industrial and Engineering Chemistry Fundamentals,vol. 15, no. 1, pp. 59–64, 1976.

[39] M. Fechtner and A. Kienle, “Efficient simulation and equilib-rium theory for adsorption processes with implicit adsorptionisotherms - ideal adsorbed solution theory,” Chemical Engi-neering Science, vol. 177, pp. 284–292, 2018.

[40] A. A. Shapiro and E. H. Stenby, “Potential theory of multicom-ponent adsorption,” Journal of Colloid and Interface Science,vol. 201, no. 2, pp. 146–157, 1998.

[41] R. S. Myong, “Gaseous slip models based on the Langmuiradsorption isotherm,” Physics of Fluids, vol. 16, no. 1,pp. 104–117, 2004.

[42] A. Tarek, Equation of State and PVT Analysis: Application forImproved Reservoir Modeling, Gulf Publishing Company, 2ndedition, 2016.

[43] C. G. Burgess, D. H. Everett, and S. Nuttall, “Adsorption ofcarbon dioxide and xenon by porous glass over a wide rangeof temperature and pressure-applicability of the Langmuircase VI equation,” Langmuir, vol. 6, no. 12, pp. 1734–1738,1990.

[44] A. Vishnyakov, E. M. Piotrovskaya, E. N. Brodskaya, E. V.Votyakov, and Y. K. Tovbin, “Critical properties of Lennard-Jones fluids in narrow slit-shaped pores,” Langmuir, vol. 17,no. 14, pp. 4451–4458, 2001.

[45] B. R. Didar and I. Y. Akkutlu, “Pore-size dependence of fluidphase behavior and properties in organic-rich shale reser-voirs,” in SPE International Symposium on Oilfield Chemistry.Society of Petroleum Engineers, pp. 1–19, Woodlands, Texas,USA, 2013.

[46] T. Pitakbunkate, P. Balbuena, G. J. Moridis, and T. A. Blasin-game, “Effect of confinement on PVT properties of hydrocar-bons in shale reservoirs,” in SPE Annual TechnicalConference and Exhibition. Society of Petroleum Engineers,pp. 1–16, Amsterdam, Netherlands, 2014.

[47] B. Jin and H. Nasrabadi, “Phase behavior of multi-componenthydrocarbon systems in nano-pores using gauge-GCMCmolecular simulation,” Fluid Phase Equilibria, vol. 425,pp. 324–334, 2016.

[48] S. Luo, J. L. Lutkenhaus, and H. Nasrabadi, “Use of differentialscanning calorimetry to study phase behavior of hydrocarbonmixtures in nano-scale porous media,” Journal of PetroleumScience and Engineering, vol. 163, pp. 731–738, 2018.

[49] S. K. Singh, A. Sinha, G. Deo, and J. K. Singh, “Vapor-liquidphase coexistence, critical properties, and surface tension ofconfined alkanes,” Journal of Physical Chemistry C, vol. 113,no. 17, pp. 7170–7180, 2009.

[50] S. P. Tan, X. Qiu, M. Dejam, and H. Adidharma, “Criticalpoint of fluid confined in nanopores: experimental detectionand measurement,” Journal of Physical Chemistry C, vol. 123,no. 15, pp. 9824–9830, 2019.

[51] A. W. Islam, T. W. Patzek, and A. Y. Sun, “Thermodynamicsphase changes of nanopore fluids,” Journal of Natural Gas Sci-ence and Engineering, vol. 25, pp. 134–139, 2015.

[52] H. H. Rachford and J. D. Rice, “Procedure for use of electronicdigital computers in calculating flash vaporization hydrocar-bon equilibrium,” Journal of Petroleum Technology, vol. 4,no. 10, pp. 19–30, 1952.

[53] A. W. Adamson, Physical Chemistry of Surfaces, John Wiley &Sons, 5th edition, 1990.

[54] K. S. Pedersen and P. L. Christensen, Phase Behavior of Petro-leum Reservoir Fluids, CRC Press, 2007.

[55] W. Yu, H. R. Lashgari, K. Wu, and K. Sepehrnoori, “CO2 injec-tion for enhanced oil recovery in Bakken tight oil reservoirs,”Fuel, vol. 159, pp. 354–363, 2015.

[56] B. Jia, J. S. Tsau, and R. Barati, “A review of the current prog-ress of CO2 injection EOR and carbon storage in shale oil res-ervoirs,” Fuel, vol. 236, pp. 404–427, 2019.

[57] Z. Song, Y. Song, Y. Li, B. Bai, K. Song, and J. Hou, “A criticalreview of CO2 enhanced oil recovery in tight oil reservoirs ofNorth America and China,” Fuel, vol. 276, article 118006,2020.

[58] O. O. Adekunle and B. T. Hoffman, “Minimum miscibilitypressure studies in the Bakken,” in Paper SPE 169077 presentedat the SPE Improved Oil Recovery Symposium, pp. 1–16, Tulsa,OK, USA, 2014.

[59] S. Wu, Z. Li, and H. K. Sarma, “Influence of confinement effecton recovery mechanisms of CO2-enhanced tight-oil recoveryprocess considering critical properties shift, capillarity andadsorption,” Fuel, vol. 262, article 116569, 2020.

[60] P. Yoonm and G. Thodos, “Viscosity of nonpolar gaseous mix-tures at normal pressures,” AICHE Journal, vol. 16, no. 2,pp. 300–304, 1970.

[61] D. K. Fong and L. X. Nghiem, “A viscosity model for reservoirfluid,” Computer Modeling Group Research Report, 1980.

[62] C. R. Wilke and P. Chang, “Correlation of diffusion coeffi-cients in dilute solutions,” AICHE Journal, vol. 1, no. 2,pp. 264–270, 1955.

[63] P. M. Sigmund, “Prediction of molecular diffusion at reservoirconditions. Part 1-measurement and prediction of binarydense gas diffusion coefficients,” Journal of Canadian Petro-leum Technology, vol. 15, pp. 48–57, 1976.

[64] C. K. J. Sun and S. H. Chen, “Tracer diffusion in dense ethanol:a generalized correlation for nonpolar and hydrogen-bondedsolvents,” AICHE Journal, vol. 32, no. 8, pp. 1367–1371, 1986.

18 Geofluids