abstract · web viewoil-like) and condensing gas drive mechanism (where rich-gas intermediate...

Impact of variation in multicomponent diffusion coefficients and salinity in

CO2-EOR: A numerical study using molecular dynamics simulation

Masoud Babaei*1, Junju Mu1, Andrew Masters1

1School of Chemical Engineering and Analytical Science, the University of Manchester, M13 9PL, Manchester,

UK

Abstract

CO2 injection in depleted or partially depleted oil reservoirs entails a three phase flow system

governed by physical processes such as molecular diffusion and solubility. Using numerical

modelling, the aims of this paper are two-fold. (i) We investigate the impact of variations in the

magnitude of diffusion of CO2 into oil on dissolution of CO2 in brine, and quantify the sensitivity of

the simulation outputs (recovery factor and amount of CO2 stored in water and oil phases) by use of

different sets of diffusion coefficients throughout the simulation based on the variations in the

compositions of the fluids. (ii) We investigate whether CO2 dissolution in brine in a water-flooded

system will be a competing or limiting factor for enhanced oil recovery by molecular diffusion of CO 2

into oil. To this end, we use molecular dynamics (MD) simulation to determine composition-

dependent diffusion coefficients for a multicomponent fluid system in a synthetic fractured reservoir

that undergoes CO2 injection. In total we consider 5 components interacting in the reservoir model,

namely, CO2, CH4, C4H10, C6H14 and C10H22. The fracture-matrix interaction is simplified with the

dual-porosity assumption. Our results show that (i) molecular diffusion not only enhances oil recovery

but also enhances CO2 dissolution in water. The enhancement, nevertheless, depends on the values of

the multicomponent diffusion coefficients and may exhibit an optimal condition for dissolution due to

the impact of CO2 diffusion and entrapment into matrix oil. (ii) The amount of CO2 stored in oil is

strongly affected by variation in molecular diffusion coefficients (we observe up to %13 difference).

(iii) The results show that there is 4% discrepancy between estimates of the recovery factor for

simulation cases that are run with different values of diffusion coefficients. Therefore it is important * Corresponding author: Telephone +44 (0)161 306 4554, email: [email protected]

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to account for compositions-dependent diffusion coefficients in simulation of CO2-enhanced oil

recovery processes.

1 – Background and introduction

Recent two-decade long attention to anthropogenic climatic change and greenhouse gas emissions is

rendering the CO2 injection into oil fields a “two birds one stone” operation: both to improve oil

production, i.e., CO2-enhanced oil recovery (CO2-EOR), and to sequestrate large amounts of CO2 and

offset the extra cost of storage. CO2-EOR is the second most common EOR process after thermal

methods (Espie, 2005). Holtz et al., (2001) investigated the possibilities of CO2 sequestration within

oil reservoirs in Texas, US. Screening more than 3,000 oil reservoirs, the authors found that there is

technical and economic potential in Texas for capture and sequestration of CO2 emitted from existing

fossil fuel-fired plants and using the CO2 for enhanced oil recovery. Other worldwide examples of

CO2-EOR feasibility studies include Weyburn CO2-EOR project in Canada (Whittaker et al., 2011),

North Sea (Lindeberg and Holt, 1994; Mendelevitch, 2014), China (Su et al., 2013). Ever since the

first commercial CO2 injection for enhanced oil recovery was conducted at SACROC Unit in Texas,

1972 (Brock and Bryan, 1989), understanding the mechanisms of enhanced oil recovery by CO2

injection has been the focus of continuous attention in the community of petroleum engineering.

Laboratory and field studies have established that CO2 can be an efficient agent featuring different

mechanisms by which it can displace oil from porous media, including oil swelling, interfacial tension

and viscosity reduction, increasing the injectivity index due to solubility of CO 2 in water and

subsequent reaction of carbonic acid with minerals (Alipour Tabrizy, 2014). An underlying physical

processes for these mechanisms of oil recovery is molecular diffusion. Molecular (interphase and

intraphase) diffusion is responsible for mixing of CO2 into oil at the pore level through a rate-

controlling mechanism that governs the gas-oil miscibility (Grogan et al., 1988). In secondary

recovery, the molecular diffusion is responsible for multiple contact miscibility achieved through

vaporising gas drive mechanism (where gas vaporises intermediate components from oil and becomes

oil-like) and condensing gas drive mechanism (where rich-gas intermediate components condensate

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into in-place oil and oil becomes gas like) (Stalkup, 1987). In tertiary recovery, the molecular

diffusion leads to mobilisation of waterflood residual oil by swelling of residual oil blobs when CO 2

diffuses through a blocking water phase (Grogan and Pinczewski, 1987). For a wide range of

conventional and unconventional reservoirs, e.g. heavy oil extraction in VAPEX (Yang and Gu, 2005)

or oil extraction by solution-gas-drive (Li and Yortsos, 1995), diffusion can act as an important

transport process.

In fractured reservoirs, the dispersive and segregated flux through fractures tends to accentuate

compositional differences between matrix and fracture hydrocarbons (da Silva and Belery, 1989) and

as a result the incremental oil recovery from CO2 injection processes in fractured reservoir are more

influenced by molecular diffusion. Extensive computational and experimental studies are available in

literature that evaluate the diffusion effects on hydrocarbon recovery from fractured reservoirs when

diffusion is a controlling mechanism (da Silva and Belery, 1989; Ghorayeb and Firoozabadi, 2000;

Darvish et al., 2006; Hoteit and Firoozabadi, 2009; Yanze and Clemens, 2012; Moortgat and Firoozabadi, 2013a; Wan et al., 2014; Trivedi

and Babadagli, 2009; Kazemi and Jamialahmadi, 2009; Zuloaga-Molero et al., 2016). As oil remains in matrix blocks in fractured

reservoir after primary recovery, the gravity drainage mechanism provides initial recovery of oil. The

density difference between gas in the fracture and oil in the matrix causes production of oil until

gravitational forces are equalised by capillary forces (Kazemi and Jamialahmadi, 2009). In low

permeability matrix the dominant mechanism is molecular diffusion of oil and gas (Kazemi and

Jamialahmadi, 2009). In small size matrix blocks and high capillary pressure, gravity drainage is very

low or ineffective. Injection of dry gas causes mass transfer between the gas in the fracture and the

gas/oil system saturating the matrix blocks (Kazemi and Jamialahmadi, 2009). The process leads to

horizontal movement of CO2 in addition to gravitational drainage.

When the rate of oil recovery during secondary or tertiary oil displacement by injection gas is

significantly affected by diffusion, multicomponent molecular diffusion coefficients are important

parameters to be determined. There are numerous experimental analyses in the past displaying the

range of composition-dependency of the diffusion coefficients for multicomponent systems, such as,

CO2/CH4/N2-rich gas-crude oil systems (Guo et al., 2009), crude oil-CO2 systems (Yang and Gu,

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2008)(Li and Dong, 2009), CO2/N2-water systems (Cadogan et al., 2014), C3H8–nC4H10 CO2-heavy oil

systems (Zheng and Yang, 2016), CO2 n-decane systems (Liu et al., 2016), CO2-heavy oil systems

(Zheng and Yang, 2017). Numerical examples include the work by da Silva and Berry (da Silva and

Belery, 1989) that predicted multicomponent molecular diffusion based on each component pair in a

hypothetical "average mixture" corresponded to the final equilibrium state of the two fluids in contact.

They assumed equal amounts of moles from each fluid are mixed to form the average mixture and

then they used the composition of the average mixture and binary diffusion coefficients calculated

through density-diffusivity correlation (extended Sigmund’s correlation) to calculate an effective

diffusion coefficient. As a drawback of this approach, since the diffusion inside the gas phase

(vapour-vapour diffusion) is usually tenfold faster than the liquid phase (vapour-liquid and liquid-

liquid diffusion), the effective diffusion coefficient will be unphysically closer to the gaseous phase

instead of an average mixture or liquid phase. Several researchers (e.g., Hoteit and Firoozabadi, 2009;

Moortgat and Firoozabadi, 2013a; Leahy-Dios and Firoozabadi, 2007) used composition-dependent matrix of

diffusion coefficients based on Stefan-Maxwell binary coefficients (described later)─ in which

gradients in chemical potential are the driving force for Fickian diffusion in fractured reservoirs. They

showed that unlike phase compositions-derived diffusion, chemical potentials do not require phase

identification and the gradient can be computed self-consistently across the phase boundaries. In their

work, however, they did not consider a three-phase CO2-oil-water system.

The two most important performance indicators for CO2-EOR is the oil recovery factor (Rf) and

amount or volume of CO2 stored ( , or , i.e., number of moles, volume or mass of

CO2) in the reservoir fluid by dissolution, or trapped in its own phase by capillary hysteresis or in

stratigraphic entrapments. EOR and EGR (enhanced gas recovery) operations are reported to have the

lowest capacity of all options for geological CO2 sequestration (Bachu et al., 2004). However there is

a potential to utilize at least some parts of the existing infrastructure (Kovscek, 2002). A crucial factor

to be explored for different geological structures and storage sites is the amount of CO2 that is “lost”

to water through dissolution that may not be accessible to mobilise the oil. Therefore, CO2-EOR and

CO2 storage objectives may not be aligned. This potential conflict often ends in favour of the EOR

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objective because the tangible economic benefits of EOR outweigh that of the storage (Kovscek and

Cakici, 2005; Leach et al., 2011; Ettehadtavakkol et al., 2014; Ampomah et al., 2016b). In order to determine the level of

competition between water and oil in absorbing CO2, dissolution and diffusion have to be taken into

account.

Using the capabilities of molecular dynamics simulation and numerical modelling of CO2-EOR

processes, in this work we investigate the effects of concurrent dissolution of CO2 into water and its

diffusion into remaining oil in a fractured reservoir. We account for diffusion by calculating the

multicomponent diffusion coefficients and account for dissolution using the correlations for

dissolution of CO2 into variably saline water. This is a novel application of molecular dynamics

simulation in the context of CO2-EOR. We aim to answer the following questions in this article using

three phase CO2-oil-water system:

1 – What is the susceptibility of the simulation results towards the range of variation in the

multicomponent diffusion coefficients and to their method of representation (concentration

gradient-based or chemical potential-based)?

2 – What is the interplay between diffusion of CO2 into oil and its dissolution in brine in the context

of CO2-EOR performance metrics?

Outline

In Section 2 we briefly introduce the formulation of diffusive flux, in Section 3 we define the metrics

for CO2-EOR and the methodology to extract data from the simulator to determine amount of CO 2

stored in water. In Section 4 we describe the geological model and fluid properties used in the

simulation, In Section 5 we describe our molecular dynamics simulations to obtain the molecular

diffusion coefficients. In Section 6 we describe the numerical simulation cases and the results of

simulation. We finish the article with conclusions in Section 7.

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2 – Formulation of diffusive flux

To represent diffusion flux, we can use two formulations for diffusion: (i) diffusion driven by

concentration:

Eq. 1

and (ii) diffusion driven by the chemical potential:

Eq. 2

where is the molar flux of component i per unit time, is the total molar concentration and

is the total volume of the mixture, is the normal or concentration-based diffusion coefficient

of component i, is the activity-corrected diffusion coefficient of component i, is the thermal

diffusion coefficient of component i (which is assumed zero for all components in this study), is

the mole fraction of component i, is the gradient in the direction of flow, is the molecular

weight of component i, is the acceleration due to gravity, is the height, is the reference

height, is the temperature, is the gas universal constant. The chemical potential of component i

is , where is the reference chemical potential, and is the component

fugacity. For a horizontal flow in isothermal systems, Eq. 2, can be written as:

Eq. 3

where , Comparing Eq. 1 and Eq. 2, one can find that

Eq. 4

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In the matrix form, several researchers (Moortgat and Firoozabadi, 2013a; Leahy-Dios and

Firoozabadi, 2007) formulated the diffusive flux as:

Eq. 5

where , and by using Stefan-Maxwell binary diffusion

coefficients ( ), the matrix of activity-corrected composition-dependent diffusion coefficients (

) can be written as:

Eq. 6

where is the number of components and is the mole fraction of mixture.

In this study we calculate normal diffusion coefficients ( ) of each component in liquid mixture by

molecular dynamics (MD) simulation using the GROMACS 4.6.7 package (Bekker et al., 1993;

Berendsen et al., 1995; van der Spoel et al., 2005). In order to make comparison, we use Eq. 4 and

model the CO2-EOR process with molecular diffusion driven by chemical potential gradient as well.

The term is a thermodynamic factor of the liquid mixture and is calculated

analytically by Peng-Robinson EoS extended for multicomponent mixtures from analytical

formulation derived for binary mixtures (Tuan et al., 1999). The formulation is given in Appendix A.

Using above formulation we combine molecular dynamics simulation-based normal diffusion

coefficients ( ) of liquid or gas, with EoS-based thermodynamics factor ( ). Procedurally, we need

certain mixtures of fluid at different pressures. We carry out flash calculations on these mixtures to

calculate Z-factor and phase molar compositions at equilibrium, from which the thermodynamic

factor for each component is computed. We develop a flash calculation code based on the

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combination of the successive substitution method and Powell’s method (in case of poor convergence)

to solve Rachford-Rice isothermal multicomponent flash equation based on Peng-Robinson EoS. The

procedure is fully described elsewhere (Nghiem et al., 1983).

We use Schlumberger ECLIPSE E300 (Schlumberger, 2010) as an industry-standard software tool for

modelling compositional three phase CO2-water-oil systems (with CO2SOL option enabled) in dual

porosity-dual permeability setting. The dual porosity-dual permeability model is an oversimplification

to discrete fractured and matrix (DFM) models, replacing a complex set of fractures with upscaled

orthogonal fracture media surrounding the matrix media. Unfortunately the computational expense of

simulating flow over DFM’s means that petroleum engineering modelling relies heavily on use of

dual porosity-dual permeability models. Oda method (Oda, 1985) built in Schlumberger is used for

upscaling DFM to dual grid. The Oda permeability upscaling method is based on the statistical

calculation of fracture geometry and distribution in each cell. The method is described in (Dershowitz

et al., 1998). Oda’s solution does not require flow simulations, therefore it does not take fracture size

and connectivity into account and is limited to well-connected fracture networks. More advanced

methods of upscaling DFM is presented in (Matthai and Nick, 2009; Nick and Matthäi, 2011; Correia

et al., 2015).

The software, is also unable to account for the variations in the multicomponent diffusion coefficients

due to compositional changes. Therefore we will run various cases of simulation with constant

diffusion coefficients and provide a range of variations in CO2-EOR metrics. Stored amounts of CO2

in water and oil are calculated by the following from Schlumberger ECLIPSE E300 (Schlumberger,

2010) outputs:

(a) In order to calculate CO2 stored in water we use two dynamic outputs of the simulation:

water moles per volume of gridblock j, , and aqueous component mole fraction

:

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Eq. 7

Eq. 8

where is the pore volume of the gridblock j (that can be either matrix or fracture), and the

strikethrough variables show that the water is not allowed to vaporise, and CH4, C4H10,

C6H14 and C10H22 are not dissolved into aqueous phase. Eq. 7 and 8 are used to calculate

amount of CO2 stored in aqueous phase, .

(b) To calculate the amount of CO2 dissolved/stored in oil we use:

Eq. 9

Where , and are molar fraction of CO2 in oil, oil density and oil saturation in

gridblock j, respectively. With above formulations, we can determine the contribution of

matrix and fracture continua in storing CO2.

3 – Geological model and fluid properties

In this paper we model a 3D dual porosity system with information reported in Table 1. Position of

the injection and production wells and an illustration of the dual continua are shown in Figure 2. The

injection and production schedule consists of injecting water with the rate of Qinj = 16 standard

condition m3/day (sm3/day) to the initially fully oil saturated reservoir for first 10 years and then

injection of CO2 with Qinj = 31.8 s m3/day for another 10 years. The oil production is constrained with

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Qo = 16 sm3/day on both stages of injection. For fluid Pressure-Volume-Temperature (PVT)

properties we use information reported in Table 2.

In CO2SOL, CO2 component is allowed to exist in all three phases, it uses modified Peng Robinson

EoS to describe the state of the fluid and the interaction between oil and gas. Data required for water

include CO2 solubility in water, water formation volume factor, water compressibility, and water

viscosity. They are entered as a function of pressure at the reservoir temperature (Schlumberger,

2010).

For solubility of CO2 in water as a function of salinity of water, the decreased solubility of CO 2 in

brine is accounted for empirically (Chang et al., 1996), by the following factor correlated to the

weight percent of dissolved solid:

Eq. 10

Where is CO2 solubility in standard m3 of CO2 per standard m3 of brine, is CO2 solubility in

standard m3 of CO2 per standard m3 of distilled water (itself correlated with pressure and temperature

(Chang et al., 1996)), is the salinity of brine in weight percent of solid, and is temperature (°F).

Eq. 7 matches the CO2 solubility data in NaCl solution within ≈ %18 sm3/sm3 (Chang et al., 1996).

Figure 1 shows the comparison between measured and calculated solubility curves with respect to

pressure. The error of Eq. 7 can clearly increase for high pressure systems.

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0 100 200 300 400 500 600 700 800 900 10000

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Exp. Data distilled waterExp. Data distilled waterExp. Data distilled waterExp. Data for S = 10 wt%Exp. Data for S = 26 wt%Calculated for S = 10 wt%

Pressure (bar)

Rs (s

m3/

sm3)

Figure 1 – Comparison of experimental data and calculated values for solubility of CO2. The

experimental data for distilled water is for various temperatures (313.15 K, 323.15 K and 373.15 K)

Wiebe (Wiebe, 1941). The experimental data for NaCl brine are from McRee (McRee, 1977).

For formation volume of water saturated with gas at the specified pressures, first the density of pure

water is calculated (Kell and Whalley, 1975), then Ezrokhi’s method is used to calculate the effect of

salt and CO2 (Zaytsev and Aseyev, 1992). For water compressibility and viscosity we use cw =

4.41⨯10─5 bar─1 and 0.31 cp, respectively.

Table 1 – Geometrical and geological properties of the simulation domain.

Properties Values Description

Lx, Ly, Lz1825.8 m, 30.48 m, 18.288 m Length in x, y and z directions

dx, dy, dz 30.48 m, 30.48 m, 3.048 m Block dimensions in x, y and z directionsZ 2,133.6 m Depth of top of the reservoir

0.1 Matrix porosity

0.005 Fracture porosity

, and 1 mD Matrix permeability

, and 100 mD Fracture permeability

917,465 m3 Rock volume of matrix continuum1,014,309 m3 Rock volume of fracture continuum101,904 m3 Pore volume of matrix continuum

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5,095 m3 Pore volume of fracture continuum

1 m-2 Multiplier in the construction of the matrix-fracture coupling transmissibilities

S 100,000 ppm (10 wt%) and 260,000 ppm (26 wt%) Salinity of water considered in two cases

T0 345 K Constant reservoir temperaturep0 200 bar Initial pressure at 2133.6 m

Figure 2 – The illustration of dual porosity dual permeability division of the domain (red shows

fracture while blue shows matrix). We note that the vertical direction is exaggerated 10-fold.

Table 2 – Fluid PVT properties of the reservoir and injection stream, BIC stands for binary

interaction coefficients.

Com

pone

nts

Mol

ecul

ar w

eigh

t

Crit

ical

te

mpe

ratu

re (K

)

Crit

ical

pre

ssur

e (b

ar)

Crit

ical

mol

ar

volu

me

(m3 /k

gmol

)

Crit

ical

Z-f

acto

r

Ace

ntric

fact

or

Initi

al o

vera

ll m

ole

com

posi

tion

Inje

ctio

n co

mpo

sitio

nB

IC w

ith C

O2

BIC

with

CH

4

BIC

with

C4H

10

BIC

with

C6H

14

CO2 44.01 304.7 73.865 0.094 0.274 0.225 0 1.0 - 0.1 0.1 0.1CH4 16.043 190.6 46.042 0.098 0.284 0.013 0.2 0 0.1 - 0 0.027

9C4H10 58.124 419.5 37.469 0.258 0.277 0.1956 0.06 0 0.1 0 - 0C6H14 84 507.5 30.103 0.351 0.250 0.299 0.14 0 0.1 0.0279 0 -C10H22 134 626 24.196 0.534 0.248 0.385 0.6 0 0.1 0.0409

20 0

For the three-phase relative permeabilities we use linear functions for oil-water and oil-gas relative

permeability curves in fractures and use quadratic functions for oil-water and oil-gas relative

permeability curves in matrices. The connate water saturation, residual gas saturation and residual oil

saturation are all set to zero. Also we ignore capillarity. (Moortgat and Firoozabadi, 2013b) have

studied the impact of capillarity on fractured media. In compositional multiphase flow, capillarity

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considerably complicates the problem because of the high degree of additional nonlinearity caused by

the strong saturation and composition dependence of the capillary pressures (Moortgat and

Firoozabadi, 2013b). In conclusion, while capillary pressure gradients at the fracture/matrix interface

trap water in the matrix, oil drains from the matrix blocks that result in higher recovery (Moortgat and

Firoozabadi, 2013b).

4 – Molecular dynamics simulation

Molecular dynamics simulation offers a robust method for calculating multicomponent diffusion

coefficients and is becoming a routine service in petroleum engineering (Zabala et al., 2008; Garcia-

Rates et al., 2012; Wang et al., 2014; Stetsenko, 2015; Uddin et al., 2016a; Uddin et al., 2016b;

Yaseen and Mansoori, 2017). There are very few works that, strictly in the context of molecular

diffusion coefficients for multicomponent subsurface systems. Previously Zabala et al., (2008) used

molecular dynamics simulation to calculate diffusion coefficients in CO2/n-alkane binary liquid

mixtures. They made the interesting argument that molecular dynamics simulation can be employed

as a tool for the determination of Fick diffusivities in high pressure systems, like in oil reservoirs,

without the need to construct complicated and expensive experiments. Wang et al. (Wang et al., 2014)

extended these diffusion calculations to supercritical CO2/alkyl benzene binary mixtures emphasizing

the structural aspects. They also made a similar argument that molecular dynamics simulation

technique is a powerful way to predict diffusion coefficients of solutes in supercritical fluids. More

recently, Uddin et al., (2016a) and Uddin et al., (2016b) used molecular dynamics simulation as a

cost-effective method to generate physically reasonable oil and gas property parameters. This has

important economic benefits as the generation of all such properties via a strictly experimental

approach is unrealistic. This is especially true for advanced enhanced oil recovery process involving a

wide range of dynamically generated compositional variations (it would be impossible to do all

experiments required). Instead the utilization of selected experiments as reference points for a wider

ranged simulation study is envisioned.

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The strategy to calculate the multicomponent diffusion coefficients is as follows. First we simulate the

model with molecular diffusion disabled. We record the overall mole fractions (zi) of the reservoir

fluid at 3 locations evenly spaced across the domain in x-axis, and for 5 times throughout the second

stage of injection (post-water flooding CO2 injection), namely years 12, 14, 16, 18 and 20. From these

overall mole fractions, we calculate the thermodynamic factor in Eq. A1 and liquid mole fractions (xi)

and liquid molar density (ρ in kg.m-3). Some of the recordings are reported in Table 3.

Table 3 - The compositions of various liquid mixtures throughout the simulation under different

pressures, T = 345 K.

xi at year 12

xi at year 14

xi at year 16

xi at year 18

xi at year 20

Location 1 at 304 m

CO2 0.715 0.953 0.991 0.998 0.998CH4 0.047 0.007 0.001 0.000 0.000C4H10 0.016 0.003 0.001 0.000 0.000C6H14 0.040 0.007 0.001 0.000 0.000C10H22 0.182 0.030 0.006 0.002 0.002p (bar) 139.98 155.55 162.10 161.34 152.11ρ (kg.m-3) 690 557 521 507 474

Location 2 at 912 m


Location 3 at 1521 m


The self-diffusion constant of each coefficients (Di) in liquid mixtures that are shown in Table 3 were

calculated using MD simulations. MD simulation is a computational approach to study the physical

movement of microscopic particles such as molecules and atoms. The movement of the particles were

calculated by integrating the equations of motion according to the intramolecular and intermolecular

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interactions between particles, which are based on molecular models that are used. In this study, the

optimised OPLS-AA (Siu et al., 2012) and EPM2 (Harris and Yung, 1995) models are used for

alkanes and CO2, respectively; as they are known to best reproduce the thermodynamic properties of

these molecule species. For the OPLS-AA models the intramolecular interactions consist of bond

stretching, bending and torsions, while the intermolecular interactions consist of electrostatic and 12-6

Lennard-Jones potential. The EPM2 model uses the same intermolecular interactions as the OPLS-AA

models but uses rigid bonds for intramolecular interactions.

In order to simulate the systems in GROMACS standard, three-dimensional cubic periodic boundary

conditions are applied with a cut-off length of 1.2 nm (Allen and Tildesley, 1989). To accurately

incorporate system evolution as a function of time, the equations of motion are integrated using the

Leap Frog algorithm (Hockney et al., 1974) with a time step of 1 fs. All long-range electrostatic

forces are resolved using the smooth Particle-mesh Ewald (PME) approach (Essmann et al., 1995) and

the Lennard-Jones potentials are incorporated into the simulation using a Force-Switch function (van

der Spoel and van Maaren, 2006). The LINCS algorithm is used to constrain the bond lengths of all

the bonds that contain hydrogen atoms.

A total of 15 MD simulations are conducted, where each simulated system is corresponding to a

mixture that is shown in Table 3. Each of the simulated system contains a total number of 20,000

atoms, where the number of each species varies with the compositions. The initial configuration of

each simulation is generated by inserting each molecule species into the simulation box where the

density is fixed to the conditions in Table 3. The initial positions of all the molecules in the simulation

boxes are totally random. To generate a realistic initial energy distribution an energy minimization

algorithm known as the Steepest Descent (Peng et al., 1996) is applied until the maximum force is

below 1000 kJ mol─1nm─1. After the energy minimization has been implemented the system is

equilibrated over 5 ns during which the velocity-rescaling (Hoover, 1985) NVT ensemble are used to

stabilize the system at the specified temperature. A coupling constant of 0.1 ps is used for the

velocity-rescaling thermostat. This follows by a 1 ns production run using Nosé-Hoover NVT

(Hoover, 1985; Nose, 1984) ensemble where the data are accumulated and used for calculations. A

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coupling constant of 1 ps is used for the Nosé-Hoover thermostat. The temperature of all the

simulated systems are set as 347.039 K and the box length of each equilibrated system is about 6.7

nm.

The self-diffusion coefficients of the molecule species in each of the system are calculated using the

Einstein relation (Allen and Tildesley, 1989)

Eq. 11

where D is the self-diffusion coefficient of a molecule species, ri(t) is the centre of mass of a molecule

i at time t, the angle brackets denote ensemble averaging over all the molecules of the same species

and time origins. D is estimated by fitting a straight line to a plot of against t, in the

interval between 100 and 300 ps in the production run and dividing the gradient by 6. This time

interval is long enough for the molecules to de-correlate from their initial positions and short enough

to avoid the large statistical uncertainties experienced at longer time intervals (Williams and Carbone,

2015; Mu et al., 2016). The results are shown in Table 4.

Table 4 - Calculated multicomponent diffusion coefficients of different components of liquid under

pressure, temperature and compositions given in Table 3.

Di (×10-5 cm2 s-1)Mixture 1 Mixture 2 Mixture 3 Mixture 4 Mixture 5

Location 1 at 304 m

CO2 17.968 ± 0.052 38.161 ± 0.016 47.693 ± 0.027 52.834 ± 0.073 56.733 ± 0.092C1H4 24.630 ± 0.612 55.880 ± 1.179 54.455 ± 1.677 77.133 ± 2.069 78.681 ± 7.752C4H10 11.167 ± 0.758 25.153 ± 0.752 32.322 ± 4.897 37.629 ± 7.685 41.886 ± 6.885C6H14 9.940 ± 0.451 23.971 ± 0.508 29.964 ± 1.029 35.829 ± 3.934 38.579 ± 1.663C10H22 7.088 ± 0.038 16.130 ± 0.437 18.985 ± 0.131 23.092 ± 0.765 25.984 ± 0.524

Location 2 at 912 m

CO2 7.236 ± 0.234 15.575 ± 0.067 33.998 ± 0.069 37.833 ± 0.030 57.723 ± 0.133C1H4 7.746 ± 0.068 20.052 ± 0.724 50.875 ± 1.672 55.533 ± 0.45 73.095 ± 4.472C4H10 3.302 ± 0.083 10.581 ± 0.665 21.517 ± 1.671 27.173 ± 0.432 41.068 ± 5.341C6H14 2.972 ± 0.095 8.058 ± 0.203 19.717 ± 0.008 22.058 ± 0.065 34.343 ± 1.329C10H22 2.279 ± 0.183 6.008 ± 0.010 14.710 ± 0.054 16.005 ± 0.146 23.035 ± 0.734

Location 3 at 1521 m

CO2 N/A 8.059 ± 1.233 8.612 ± 0.010 19.808 ± 0.008 41.125 ± 0.150C1H4 7.623 ± 0.094 7.306 ± 0.037 9.651 ± 0.044 25.850 ± 0.293 59.820 ± 1.917C4H10 3.287 ± 0.002 3.168 ± 0.001 4.281 ± 0.020 12.949 ± 0.100 30.191 ± 0.248C6H14 2.941 ± 0.011 2.661 ± 0.146 3.731 ± 0.084 10.081 ± 0.328 22.068 ± 0.051C10H22 2.220 ± 0.160 2.186 ± 0.213 2.798 ± 0.207 7.814 ± 0.290 17.643 ± 0.355

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Using the results in Table 4 and plotting diffusion coefficient of component i (Di) with respect to its

liquid mole fraction (xi), Figure 3 shows exponential behaviour of all 5 components in the systems for

from molecular dynamics simulations conducted. This shows that the diffusion coefficients can be

correlated with the molar fractions only, and without the need to input pressure. Another observation

is that the diffusion coefficient increases only for CO2 with increase in mole fraction, and the

diffusion coefficients of other components undergo exponential decay with increase in mole fraction.

Our explanation is that the diffusion coefficients of all the species increase with the decrease of the

concentration of long chain alkanes. This is because that the long chain alkanes stop the other

molecules from diffusing as they are heavy, sluggish and occupy large volumes. The fewer the long

chain alkanes in the system, the less possibility the small molecules being prevented from diffusing.

Figure 3 – The exponential dependency of the MDS-based multicomponent diffusion coefficients to

the mole fractions.

5 – Results of numerical simulations

A (potentially major) shortcoming in ECLIPSE E300 is the lack of a mechanism that the user can

input into the simulator compositionally-variable multicomponent diffusion coefficients throughout

the simulation. This can be a potential problem if the output metrics display large changes with

respect to varying multicomponent diffusion coefficients. To this end, we define a range of simulation

cases with different diffusion coefficients (normal and activity-corrected) and for two cases of salinity

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(10 wt% and 26 wt%). The cases are reported in Table 5. In order to be able to extract conclusions in

a more straightforward manner, and isolate the effect of molecular diffusion of oil and salinity of

water in incremental oil recovery and CO2 storage, we discard diffusion effects in gas and water

phases in our simulations. Also we discard cross-phase diffusion.

Table 5 – Simulation cases defined in this study.

Case no. Diffusion condition Salinity

Case 1 Molecular diffusion disabled S = 10 wt%

Case 2 Di’s for liquid-liquid diffusion from Figure 3 for lowest CO2 concentration:D = [8.06, 73.09, 34.34, 72.07, 12.95 × 10-9m2/sec) S = 10 wt%

Case 3 Di’s from Figure 3 for highest CO2 concentrationD = [57.72, 7.62, 3.29, 2.66, 2.22 × 10-9m2/sec) S = 10 wt%

Case 4 Di’s from Figure 3 for lowest CO2 concentrationD = [8.06, 73.09, 34.34, 72.07, 12.95 × 10-9m2/sec) S = 26 wt%

Case 5 ’s for highest CO2 concentrationDa = D/Γ = [28.43, 6.58, 4.84, 3.47, 1.83 × 10-9m2/sec)

S = 10 wt%

Overall, in 10 years of CO2 injection, 872,508 kgmol (38,390 tonnes) of CO2 are injected into the

reservoir.

Figure 4(a) (amount of CO2 that are dissolved in water at the end of simulation) shows that:

Although diffusion in water phase is disabled, the amount of CO2 dissolving in water phase

when molecular diffusion is enabled (Case 2) is almost 3 times higher than no-diffusion case

(Case 1).

By increasing CO2 diffusion in Case 3 or using activity-corrected coefficients in Case 5, the

amount of CO2 in water actually decreases. This will be discussed later. The main message

here is that, although molecular diffusion benefits CO2 sequestration in water by providing

more contact of CO2 with water in the matrix continuum, it may have a certain threshold of

diffusion after which the sequestration capacity in water actually decreases.

As expected, increase in salinity of water considerably decreases the dissolution of CO2 in

water (Case 4 vs. Case 2).

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The sensitivity of simulation results to the diffusion coefficients are ranging between ~1,000

tonnes and ~1,200 tonnes. This means that (mass of CO2 stored in water) is only 2.6 to

3.1 % of CO2 injected. For larger reservoirs with more water contacts, higher fractions are

expected.

Activity-corrected coefficients (Case 5) produce comparable results with normal diffusion

coefficients (Case 3).

From Figure 4(b) (amount of CO2 that are found in oil at the end of simulation) we can observe that:

Molecular diffusion significantly enhances the storage of CO2 in oil phase (Case 2 vs. Case

1), and this transfer has increased by an increase in molecular diffusion of CO2 (Case 3 vs

Case 2). Nevertheless as we will discuss next, this enhancement is not due to increase in

diffusion of CO2. As a consequence of using two series of values for diffusion coefficients,

the amount of CO2 in oil varies between ~14,000 to ~19,000 tonnes, equivalent to ~13%

difference between the highest and lowest percentages of CO2 injection stored in oil. This

significant difference implies that multicomponent diffusion coefficients and their

composition-dependent values remarkably impact the estimates of .

Increase in salinity of water used in water flooding stage has no significant effect on the

amount of CO2 in oil phase (Case 4 vs Case 2),

Activity-corrected coefficients and use of chemical potential driven diffusion (Case 5)

produce fluctuating variations in the results, nevertheless the estimate lies within the range of

lowest to highest estimates of by concentration driven diffusion.

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(a) (b)Figure 4 – (a) Amount of CO2 dissolved in water for various simulation cases. (b) Amount of CO2 in

oil for various simulation cases.

Now we focus on CO2 production molar rate and inter-regional fluxes to elucidate the impact of

molecular diffusion. Although, the simulations suffer from convergence error within a couple of first

years of CO2 injection, the numerical instabilities disappear later. Unfortunately despite various

strategies, the poor convergence of the model (around year 2026 when CO2 injection starts) was not

resolved. Figure 5(a) shows that, ignoring the oscillations that appear in 2026 and assuming that these

oscillations have no impact on modelling results of later years, CO2 molar rate of transfer from

fracture-to-matrix is comparable between all cases with diffusion enabled (Cases 2, 3, 4 and 5).

Therefore increase in diffusion coefficient of CO2 has not considerably increased diffusion of CO2

from fracture to matrix. However, for matrix-to-fracture molar rate of CO2 transfer in Figure 5(b),

there is actually “a decrease” for Case 3 vs. Case 2. This means that, CO2 is trapped in matrix, and as

shown in Figure 4(b) this entrapment has contributed to storage of CO2 in oil. As a result, the molar

rate of CO2 production as well as molar rate of CO2 in produced oil decreases in Case 3 vs. Case 2 as

shown in Figure 5(c) and Figure 5(d). In summary for this figure:

In no-diffusion case, CO2 is back-produced ─ without being considerably stored in the system

─ by a constant rate.

Case 2 and Case 4 produce similar profiles of production rate (no impact of salinity on CO2

dynamism in oil phase).

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A delay is observed in back-production of CO2 in Case 3 (increased CO2 diffusion) and Case

5 (activity-corrected diffusion), thanks to increase in diffusion of CO2 in oil left-in-place.

This figure clearly illustrates the sensitivity of simulation results to the composition-

dependent diffusion coefficients.

The main question unanswered is: why does CO2 remain in matrix even when its diffusion coefficient

has increased (Case 3 vs. Case 2)?

Figure 6(a) shows that due to decrease in diffusion of n-decane from Case 3 to Case 2, the molar rate

of n-decane transfer from matrix to fracture has decreased, therefore oil that has contained CO 2,

remains in the matrix and as shown in Figure 6(b) n-decane production rate decreases. In contrast,

water rate from matrix to fracture increases in Case 3 vs. Case 2 as shown in Figure 6(c). Case 3

provides a more favourable condition for water flux and there is a higher cumulative amount of water

produced from the reservoir in Case 3 compared to other simulation cases as shown in Figure 6(d).

Loss of water from production well translates into decreased CO2 storage capacity in water as shown

in Figure 4(a). This means that the diffusion coefficient of CO2 is not the only determining factor in

fate and transport of CO2, but diffusion coefficients of other components have a considerable impact

on CO2 storage through change in oil and water transfer from matrix and fracture continua.

Figure 7(a) and Figure 7(b) show the oil production rate and recovery factor from different simulation

cases, respectively. Increase in CO2 diffusion coefficient and decrease in other components’ diffusion

coefficients in Case 3, translate into higher storage capacity in oil phase (Figure 4b) but lower storage

in water (as water was lost to production) and lower recovery efficiency (~0.90 for Case 3 vs. ~0.94

for Case 2). Therefore there is around 4% difference between estimates of oil recovery from the

synthetic reservoir under study due to variation in the values of multicomponent diffusion

coefficients.

Finally Figure 8 shows the distribution of CO2 mass and gas saturation across the domain for both

fracture and matrix continua. The distribution of CO2 mass across the domain for Case 3 shows that

CO2 remains in the reservoir and does not push the oil out of the domain as it does for Cases 2, 4 and

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5. This is also visible in terms of the gas saturation front which is retarded in Case 3 compared to

Cases 2, 4 and 5. There is no major disparity between Case 2 and 4 as concluded from the previous

figures.

(a) (b)

(c) (d)Figure 5 – (a) CO2 molar rate of transfer from fracture continuum to matrix continuum, (b) CO2 molar

rate of transfer from matrix continuum to fracture continuum, (c) CO2 molar rate of production, and

(d) produced gas production volume.

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(a) (b)

(c) (d)Figure 6 – (a) C10H22 molar rate of transfer from matrix continuum to fracture continuum, (b) C 10H22

molar production rate, (c) Water molar rate of transfer from matrix continuum to fracture continuum,

and (d) Water production cumulative volume. There are oscillations in results in year 2026 due to

convergence issues.

(a) (b)Figure 7 – (a) Oil production rate and (b) recovery factor calculated from different simulation cases.

a(1)

a(2)

a(3)

a(4)

a(5)

b(1)

Sg

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b(2)

b(3)

b(4)

b(5)Figure 8 – a(1) to a(5): Number of moles CO2 per reservoir gridblock volume for Case 1 to Case 5,

and b(1) to b(5): gas saturation distribution after 20 years for Case 1 to Case 5.

6 – Conclusion and future works

The main conclusions of this work are the following:

- Accounting for molecular diffusion in three-phase CO2-EOR analysis is not only important

for EOR estimates but it is also important to estimate CO2 storage either in water or oil phase.

For a synthetic reservoir, flooded with water initially (Sw reaches ~ 0.43 by end of year 10),

we found that diffusion will lead to more CO2 trapped in water by providing further contact of

CO2 with water in matrix.

- The multicomponent diffusion coefficients that are varying throughout the simulation have

considerable impact on CO2 storage in oil phase. In a synthetic reservoir we found ~13%

difference between the highest and lowest percentages of CO2 injection stored in oil. This is

crucial to estimate the amount of CO2 stored in oil, as this is found to be the predominant

mechanism for storing CO2 within depleted oil reservoir (Ampomah et al., 2016a). This

difference pronounces the need for incorporating the changes in diffusion coefficients in a

multicomponent fluid system undergoing compositional changes.

- We found there are complex interactions between oil, water and CO2 due to diffusion of

multiple components in a matrix-fracture system: increasing CO2 diffusion may not directly

increase CO2 storage in water and oil recovery factor, and the influences of other components

on the matrix-fracture transfer should be carefully examined. For example procedures of

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upscaling matrix-fracture transfer function (e.g., (Correia et al., 2016)) should be revisited to

take into account the multicomponent molecular diffusion. We found that the lower diffusion

of hydrocarbon components translate into entrapment of CO2 in matrix within oil, lowering

oil mobility and increased water production. Therefore there are conflicting objectives in

sequestration and enhanced oil recovery due to the variations in multicomponent diffusion

coefficients.

- We have found no direct impact of salinity of water on dynamics of molecular diffusion of

CO2 into oil. Nevertheless the salinity has considerable implications on storage of CO 2 in

water.

- Molecular Dynamics simulation is a computationally viable method (compared to

experimental apparatus) to determine compositionally varying molecular diffusion coefficient

throughout the simulation.

We recommend developing a rigorous compositional modelling for three-phase CO2-EOR processed

that can incorporate multicomponent diffusion coefficients derived directly from molecular dynamics.

It is important to estimate CO2 storage in water and oil phases at large scale reservoir models by this

compositional simulators. For example, most recently, (Moortgat and Firoozabadi, 2013a) have not

considered the important interaction of CO2-oil-water systems. Also the impact of multiscale

heterogeneity in fractured media (Correia et al., 2015; Hardebol et al., 2015), the distribution of

fracture network properties (as shown e.g., in Bisdom et al., 2017; Bisdom et al., 2016a; Bisdom et

al., 2016b), the use of discrete fracture networks (e.g., Bisdom et al., 2016c) as opposed to dual

porosity assumptions and simplifications within the context of CO2-EOR processes with strong

fracture-matrix interactions and molecular diffusion need further investigations to quantify the

interplay of diffusion and dissolution.

Acknowledgements

The computational resources were provided by the University of Manchester EPS Teaching and

Research Fund. The main author would like to thank this institution.

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Appendix A

The thermodynamic factor ( ) in Eq. 4 is obtained through the following analytical solution:

Eq. A1

where

,

Eq. A2

2 21 1

, ,c cN N

ij mij m i j ij

i i

a P a PA A a x x aRT RT

,

Eq. A3

Eq. A4

Eq. A5

The derivative terms in Eq. 5 are calculated as:

Eq. A6

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Eq. A7

Eq. A8

Finally, the Z-factor is calculated by solving the following dimensionless cubic equation:

Eq. A9

where Nc is the number of component in the mixture, , and are respectively the critical

temperature critical pressure and acentric factors of the component i in mixture and is the reduced

temperature of component i.

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abstract · web viewoil-like) and condensing gas drive mechanism (where rich-gas intermediate...

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