Modeling of Difusion in Fractured Reservoirs SPE-141937 -MS-P

8/13/2019 Modeling of Difusion in Fractured Reservoirs SPE-141937 -MS-P

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SPE 141937

Proper Modeling of Diffusion in Fractured ReservoirsHussein Hoteit, SPE, ConocoPhillips

Copyright 2011, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Reservoir Simulation Symposium held in The Woodlands, Texas, USA, 2123 February 2011.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by theSociety of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronicreproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not morethan 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

Diffusion may play a key role in a number of oil recovery processes such as heavy oil and naturally fractured reservoirs. In

fractured media, several laboratory experiments and numerical studies showed that CO2 injection can improve recovery.Molecular diffusion, gravity drainage, and oil swelling are the main contributing mechanisms. Proper modeling of diffusion

of hydrocarbon mixtures at the reservoir PVT and geological conditions is not a trivial task. The challenge is in computing

the diffusion coefficients for the non-ideal multicomponent mixtures in oil and gas phases, and in physically accuratemodeling of the diffusion driving force. One common approach in most simulators is to use the classical Ficks law which

simplifies the multicomponent diffusion fluxes by only considering the main-diffusion (diagonal) terms and neglecting the

cross-diffusion (off-diagonal) terms. The diffusion fluxes are assumed independent and the diffusion driving force of each

component is proportional to the component self concentration gradient. In this work, we demonstrate analytically and

numerically that this simplified approach may have a major inconsistency related the flux balance constraint and, in someapplications, it may fail to capture the right direction of diffusion as a result of neglecting the dragging effect. We propose an

alternative model based on the generalized Ficks law in which diffusion coefficients are calculated as a function of

temperature, pressure, and composition. The proposed approach can be seen equivalent to the Maxwell-Stephan model inwhich the diffusion driving force is the chemical potential instead of the composition gradient. We also tackle another

problem that may occur in fractured media when fractures get fully saturated with gas in an under-saturated oil surrounding.

Intra-phase gas and oil diffusions will not be initiated due to the discontinuity of phases between the fracture and the rock

matrix. The proposed approach in the literature that allows for direct gas-to-oil diffusion may not have a sound bases for

issues related to the driving force and the estimation of the mass transfer coefficients. We provide a solution for the cross-phase diffusion flux based on the assumption of having chemical equilibrium at the gas-oil contact. Several numerical

examples are provided.

Introduction

Predicting mass transfer rates in gas-oil systems is crucial in many industrial processes such as evaporation, condensation,

absorption and distillation. Proper modeling of molecular diffusion mechanisms is a key. Significant efforts have been

devoted to understand these mechanisms in the area of chemical and petrochemical engineering (Cussler 1976; Taylor and

Krishna 1993; Wesselingh and Krishna 2000). In reservoir engineering, diffusion has less significance and is often ignored

for mainly two reasons: 1) convection is the predominate flow mechanism in primary, secondary, and most tertiary recoveryprocesses, 2) artificial dispersion produced by numerical methods in most modeling apporaches could be high enough to

compensate for molecular diffusion and physical dispersion (Coats 1980, Hoteit and Firoozabadi 2006). In some recoveryprocesses, however, molecular diffusion may have a vital role in the recovery schemes involving vapor hydrocarbon solventsin heavy oil reservoirs and miscible gas injection in naturally fractured reservoirs.

In solvent-based schemes in heavy oil reservoirs, the gas solvent mixes with the heavy oil and results in viscosity reduction.

The mixing process involves dispersion and mass transfer via molecular diffusion of the solvent into the sand oil (Guerrero-

Aconcha and Kantzas 2009; Salama and Kantzas 2005). In gas-flooding schemes in naturally fractured reservoirs, viscous

forces, gravity drainage and diffusion are the main flow mechanisms that would mobilize oil from the matrix blocks towardsthe producers. The dominate recovery mechanisms depend on the reservoir heterogeneity, fluid properties and PVT

conditions. In case of low matrix permeability, thin matrix blocks, or insignificant density difference between the oil and theinjected gas, viscous forces and gravity drainage become inefficient. In such cases, molecular diffusion, strongly influenced

by fracture intensity, matrix block size, and the magnitude of diffusion coefficients, controls the mass cross-flow rates


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SPE 141937 2

between the matrix and fracture (Hoteit and Firoozabadi 2006). The fracture intensity determines the specific gas-oil contact

surface and the matrix size determines the characteristic length of diffusion. Several experimental and field observations

concluded that molecular diffusion can be an efficient recovery mechanism (McKay 1971; Thomas et al. 1991; Le Romanceret al. 1994; Lagalaye et al. 2002; Darvish et al. 2006a, 2006b, Karimaie et al. 2010).

The governing mechanisms in fractured reservoirs under gas flooding (viscous forces, gravity drainage, molecular diffusion

and mechanical dispersion) are usually interacting, rate dependant, and therefore may occur with varying levels of

significance in different parts of the reservoir. We refer to Chordia and Trivedi (2010) for a comprehensive review aboutthese mechanisms in naturally fracture reservoirs. Modeling diffusion and mechanical dispersion at the reservoir conditions is

associated with several uncertainties and challenges:

i) Diffusion flux models that properly describes the diffusion behavior from the driving force that captures thedragging effect due to species interactions.

ii) Measuring and predicting diffusion coefficients for multicomponent hydrocarbon mixtures at reservoir conditionsthat takes into consideration the thermodynamic non-ideality of the mixture.

iii) Gas-oil mass transfer (cross-phase interaction) at the fracture-matrix interface.iv) Mechanical dispersion due to the tortuosity factor of the porous media and the multiphase interaction effect.v) Water saturation and rock wettability impact on hydrocarbon phase continuity and gas-oil mass transfer.vi) Fracture network characterization: fracture aperture, intensity, orientation and matrix block sizes.vii) Choice of simulation model: dual porosity/dual permeability versus discrete fracture models.

In this work, we focus on the first three items, that is, diffusion flux, diffusion coefficients, and mass transfer at the gas-oil

interface. The other items are of vital importance but are not included in the scope of this discussion.

There are three commonly used models to describe molecular diffusion flux for multicomponent mixtures. The most popular

is based on the classical Ficks first law, the second is the Maxwell-Stephan (MS) model, and the third is the generalized

Ficks law originated form the irreversible thermodynamics (Krishna 1987; Taylor and Krishna 1993; Wesselingh andKrishna 2000). The classical Ficks law for multicomponent mixtures assumes that each component in the mixture transfers

independently and does not interact with the other components. The driving force for a given component is the self

concentration gradient multiplied by a diffusion coefficient. The diffusion coefficient is often constant and assumed

independent of composition and PVT conditions. The second and third models are equivalent under certain conditions. They

can be seen as a generalization of the classical Ficks law. Unlike Ficks law, the flux driving force is proportional to the

chemical potential gradient. Furthermore, the thermodynamic non-ideality and the dragging effect due to species interactionare taken into account.

The classical Ficks is the mostly utilized model in reservoir engineering literature and in commercial and academic reservoirsimulators (Coats 1989; Shrivastava et al. 2005; Darvish et al. 2006b; Kazemi and Jamialahmadi 2009; Alavian and Whitson

2010). Current practice is to use this model in the context of effective diffusivity where diffusion in multicomponent mixtures

is assumed to behave as a pseudo-binary (da Silva and Belery 1989). This model is elegant and simple from computational

point of view and may provide reasonable results for many applications. However, it may not honor the equimolar condition

that states that the total diffusion flux must be zero. This issue will be clarified later. Furthermore, in some cases, this modelmay fail to provide even qualitatively correct description of diffusion behavior (Duncan and Toor 1962). The failure of the

classical Ficks law is a result of neglecting the dragging effect that is described by the off-diagonal elements in the diffusion-

coefficient matrix (Krishna and Wesselingh 1997). Krishna and Standart (1979) argued that the classical Ficks law could beonly valid for cases with special conditions, such as, ideal binary mixtures and ideal multicomponent mixtures having

diffusion coefficients that can be regarded as equal. Nevertheless, these conditions that specify the validity range for the

classical Ficks law are sufficient but may not be necessary. The bottom line is that it is difficult to anticipate when Ficks

law does and does not work. To overcome this issue, we use the generalized Ficks law. The fundamental difference is in theflux driving force that is based on the chemical potential gradient instead of the intrinsic concentration gradient. Later, we

show that for nc-component mixtures, there are (nc-1) independent diffusion fluxes expressed by the generalized Fick's law

and (nc-1)2diffusion coefficients known as the Fickian diffusion coefficients. These coefficients are different but related to

the MS diffusion coefficients (Cussler 1976; Taylor and Krishna 1993).

Accurate diffusion coefficient measurements or prediction is crucial for diffusion flux calculation. Diffusion coefficients are

difficult to measure and experiments are usually limited to binary or ternary mixtures (Cullinan and Toor 1965; Sigmund

1976a, 1976b; Renner 1988; Leahy-Dios et al. 2005). For non-ideal mixtures, diffusion coefficients are function of pressure,

temperature and composition. It was observed, for example, that the diffusion coefficient for binary mixtures vanishes whenapproaching the critical point conditions (Hasse 1972; Chang and Myerson 1986). McKay (1971) investigated the diffusion

behavior for a gas cap miscible hydrocarbon solvent in a diffusion cell at high temperature and pressure. He concluded that

diffusion coefficients are strong function of concentration and that the multicomponent system cannot be approximated by a


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pseudo-binary. In this work, we use the model by Leahy-Dios and Firoozabadi (2007) to predict the diffusion coefficients for

gas and oil mixtures as a function of temperature, pressure and composition. This model, which is an improvement of an

earlier version by Ghorayeb and Firoozabadi (2000), is shown to be superior to Sigmund correlation (1976b).

The models discussed above describe molecular diffusion in single phases, that is, diffusion in gas phase and diffusion in oil

phase. The interaction among components in gas phase to those in oil phase is referred to as interface mass transfer rather

than diffusion. There are some approaches in the literature that suggest modeling gas-oil mass transfer by a Ficks law-type

model similarly to diffusion in a single phase. These approaches may not have a sound basis. Modeling mass transfer acrossphases is discussed extensively in chemical engineering literature (Krishna and Standart 1976; Taylor and Krishna 1993;

Bentez 2009). The commonly used approach, which is based on the Film theory, is to assume thermodynamic equilibrium at

the gas-oil contact and impose the continuity of the normal components of total fluxes across the interface. Haugen andFiroozabadi (2009) used this approach to model mass transfer at the gas-oil interface in a PVT cell. A dynamic gridding logic

is used to track the moving GOC. Mohammad et al. (2009) and Guo (2009) also used a similar but with neglected dragging

effect. In cases when a fracture gridblocks saturated in gas is adjacent to a matrix gridblock saturated with oil, the intra-phase

diffusion will not start because of phase discontinuity. The interface mass transfer is therefore needed to initiate diffusion. In

this work, we calculate the interface diffusion flux using the generalized Ficks law and the thermodynamic equilibrium atthe gas-oil contact. This method is more rigorous than the one introduced by da Silva and Belery (1989) that suggested

mixing equal proportions of gas and oil to calculate the interface fluid compositions.

The outline of the paper is as follows. We first review the three diffusion models and discuss the drawbacks of the classical

Ficks law. We then introduce the diffusion coefficient model. The interface mass transfer problem is then discussed. We

follow by several examples and end with a summary.

Diffusion flux modelsWe briefly review the diffusion flux models: Maxwell-Stephan, the generalized Ficks law, and the classical Ficks law. We

also show the relationship with the diffusion model from irreversible thermodynamic. The objective is to highlight the

consistency issue of the classical Ficks law. We first consider the molecular diffusion process in the bulk of a nc-component

mixture (free space). The impact of porous media on diffusion will be discussed later. In all cases, the system is assumed to

be isothermal. Before introducing the different flux models, we recall the following preliminary concepts.

Let , 1, ,i cx i n= denote the species mole fractions in the mixture that can be in oil or gas state. The convective and diffusive

molar flux that describes the moles of species ipassing through a unit area per unit time is given by :

, 1, ,i i i cN c u i n= =

, (1)

where i ic cx= is the molar density of species i , c is the overall molar density, and iu represents the velocity of the ith

species with respect to a fixed reference frame. The mixture total flux is obtained by summing the cn fluxes in Eq.(1):

1

cn

t i i

i

N c u cu

=

= = . (2)

In the above equation,

1

cn

i i

i

u x u

=

= is the molar average velocity of the mixture.

The diffusion process refers to the motion of a species i relative to the mixture as a whole. One common approach is to

express the molar diffusion fluxi

J of species i in terms of species velocityi

u and the average mixture velocity u (Taylor

and Krishna 1993), such that,

( ) , 1, , ,i i i cJ c u u i n= = (3)

Summing all the cn diffusion fluxes defined in the above equation and using Eqs.(1), (2), and the mole fraction

constraint1

1cn

ii

x=

= , one can readily show that the total diffusion flux is zero, that is,

1

0cn

i

i

J

=

= . (4)

This equation, which has particular interest in our discussion later, shows that not all the cn diffusion fluxes are independent.


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SPE 141937 4

Using Eq. (3), one can rewrite the molar flux given in Eq. (1) in terms of the diffusion flux, iJ , and the convection flux, ic u ,

that is,

, 1, ,i i i cN J c u i n= + = . (5)

We will use this flux expression when discussing the interfacial mass transfer between gas and oil phases.

Maxwell-Stephan Model

In an isothermal system and with the absence of external forces, the generalized Maxwell-Stephan (MS) formulation is based

on the hypothesis that there are two equally counterbalanced forces that control diffusion of a component i . The first is the

driving force, which is proportional to the chemical potential gradient, and the second is related to the friction (velocity) of

the molecules of component i with other molecules of component j in the mixture. We refer to Taylor and Krishna (1993)

and Krishna and Wesselingh (1997) in recent literature for more details. Writing the equality of the two forces, one gets:

( ),

1

, 1, , .cn

i j i jiT p i c

ijjj i

x x u uxi n

RT

=

= =

(6)

In the above equation, i is the chemical potential and , , 1, , ( )ij ci j n i j= are the MS diffusion coefficients, which

represent the mutual diffusivity for every pair of components in the mixture. Note that these coefficients are symmetric andthe diagonal elements, , 1, ,ii ci n= , do not exist. Therefore, there are only ( 1) / 2c cn n MS diffusion coefficients. At

constant temperature, T, and pressure, p , the following constraint holds due the Gibbs-Duhem equation.

,

1

0cn

i T p i

i

x

=

= (7)

The above equation, which is consistent with Eq.(4), shows that there are only ( 1)cn independent fluxes. The chemical

potential gradient can be written in terms of the fugacity, if , and the composition gradient as follows (Firoozabadi 1999).

( )1

,1

lncn

i

T p i jjj

f

RT xx

=

= . (8)

Replacing the molar diffusion flux defined in Eqs. (1)-(5) in the right-hand side of Eq. (6) and the chemical potential

gradients defined in the above equation in in the left-hand side of Eq. (6), one gets,

( ) ( )1

1 1

ln 1, 1, , 1.

c cn nj i i ji

i j cj ijj j

j i

x J x Jfx x i n

x c

= =

= =

(9)

Only ( 1)cn equations are written because of the constraint in Eq. (7). One notes the nonlinear relationship among the

diffusion fluxes, iJ , the diffusion coefficients, ij , and the composition gradients jx .

In matrix form, Eq. (9) can be written as,

c= BJ x, (10)

where,

1

, 1 1;

1 1

c

c

c

nci k

in ik kk i

ij iji j n

i

ij in

x xi j

B B

x i j

=

=

+ =

= =

B


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( ), 1 1

ln;

c

iij ij i

i j nj

fx

x=

= =

,

1 1ci i n

x=

= x , and 1 1ci i nJ

= = J .

The matrix B is a function of the inverse of the MS coefficients, and represents the thermodynamic non-ideality effect.

For ideal mixtures,

is the identity matrix. To get an explicit expression of the flux, Eq. (10) can be multiplied by theinverted matrix 1B , that is,

1c

= J B x. (11)

Not that in the above formulation we selected last component as a reference and therefore diffusion fluxcn

J is eliminated.

The generalized Ficks law

From the classical Ficks law, the diffusion flux of the first component in a binary mixture can be written as a linear function

of the mole fraction gradient of component 1, such that,

1 12 1J cD x= , (12)

where 12D is the mutual diffusion coefficient of components 1 and 2 . An analogous expression can also be written for the

second component.

2 21 2J cD x= . (13)

In an equimolar system, Eq. (4) holds, i.e., 1 2J J= . In view of Eqs. (12) and (13), one can readily show that the two

diffusion coefficients are identical.

12 21D D= . (14)

In a multicomponent non-ideal mixture, the generalized expression of the Fickian diffusion flux is written as:

1

1, 1, , 1

cn

i ij j c

j

J c D x i n

== =

(15)

In the above expression, only ( 1)cn independent diffusion fluxes appear. The diffusion flux of the last component can be

calculated form Eq. (4). In matrix from, Eq. (15) simplifies to:

c= J D x, (16)

Where, D is a ( 1) ( 1)c cn n matrix known as the Fickian diffusion coefficient matrix. The diagonal entries are the main

diffusion coefficients and the off-diagonal entities are the cross or coupling diffusion coefficients, which are generally

nonzero and not symmetric (that is, , , ;i j j iD D i j ).

Comparing the MS and the Fickian flux expressions given in Eqs. (11) and (16) leads to the following relationship:

1=D B (17)

The Maxwell-Stephan and the generalized Ficks law are therefore equivalent in this context. The MS diffusion coefficients

are generally different from the Fickian diffusion coefficients. One exception is for ideal binary mixtures where both

coefficients become identical.

The following items summarize the cons and pros of the expressions of the two models and their diffusion coefficients.

- The MS coefficients can be physically interpreted and independent of the reference frame. The Fickian coefficientsmay not have physical interpretations and they depend on the numbering of the components. This can be seen form


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Eq. (17) where the resulting matrix of the product1

B depends on the ordering of the components.

- There are ( 1) / 2c cn n MS coefficients versus2( 1)cn Fickian coefficients.

- The MS coefficients are weak function of composition compared to the Fickian coefficients and therefore aregenerally easier to measure. For liquid and gas fluids at high pressure, the Fickian coefficients are strong function of

composition due to .- The MS model splits the diffusion coefficients (B ) from the non-ideality correction factor .

- The generalized Ficks model is more convenient from numerical point of view since the flux is explicitly

expressed as a linear combination of the concentration gradients.

The approach we opted is to use the diffusion coefficient model by Leahy-Dios and Firoozabadi (2007) to estimate the MS

diffusion coefficients. These coefficients are then converted via Eq. (17) to the Fickian expression to be used in the flowmodel. More details will follow.

The classical Ficks law

The classical Ficks law is often used in the context of effective diffusivity model. In a multicomponent mixture, diffusion

processes of different components are assumed independent and the driving force is the self mole fraction gradient multipliedby an effectivediffusion coefficient. The effective diffusion coefficient is often considered independent of composition

regardless of the thermodynamic ideality of the mixture. Even though this model is empirical, it may provide reasonable

results for many applications but unfortunately one cannot predict when and where it fails. Most of the existing reservoir

simulators support this model where the diffusion flux is defined by:

, 1, ,effi i i cJ cD x i n= = . (18)

The coefficienteffiD in the above equation denotes the effective diffusion coefficient of component i in the mixture.

Different mixing techniques have been proposed in the literature to estimate these coefficients (Wilke 1950; Sigmund 1976).

This model may suffers form some drawbacks as discussed afterwards.

Model inconsistency

Summing the cn diffusion fluxes in Eq.(18) and using the constraint given in Eq.(4) (also Eq. (7)), one gets:

1 1

0c cn n

effi i i

i i

J c D x

= =

= =

. (19)

Since the sum of the mole fraction gradient is zero (1

0cn

ii

x=

= ), one can eliminate the gradient of the last component inEq. (19) to obtain:

( )1

1

0c

c

n

effeffi in

i

D D x

=

= . (20)

The above constraint must hold for every composition. The only possible solution is when all the diffusion coefficients are

identical (Krishna and Standart 1976), that is,

, 1, , 1c

effeff

i cnD D i n= = . (21)

Therefore, for multicomponent mixtures, the effective diffusion model will always violate the molar balance constraint (Eqs.

(4) and (7)) unless all the diffusion coefficient are taken to be equal. In other words, the bottleneck issue is that this model

cannot account for different diffusion coefficients in multicomponent mixtures without violating the molar balanceconstraint. The provided examples 1aand 1bdiscusses the numerical consequences from violating Eq. (4).

Model limitations

The first limitation of this model is in neglecting the off-diagonal entities of the diffusion matrix and, therefore, assuming

that the diffusion driving force is the self concentration gradient of each component. The off-diagonal entities describe the

dragging or coupling effect among the components. Duncan and Toor (1962) observed three diffusion phenomena that do notobey the classical Ficks law:


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SPE 141937 7

1) Osmotic diffusion: nonzero diffusion flux for a species in the presence of zero concentration gradient.

2)Diffusion barrier: zero diffusion flux in the presence of nonzero concentration gradient.

3)Reverse or uphill diffusion: diffusion of a species in a direction opposing the sign of the concentration gradient.

Later, we provide two numerical examples, 2aand 2b,to highlight this issue.

The second limitation is when using constant diffusion coefficients. As discussed previously, the Fickian diffusion

coefficients being incorporating the thermodynamic non-ideality effect are expected to be strong functions of composition at

reservoir conditions. In a binary mixture for example, is null at the critical point because of the condition of criticality, thatis, the derivative of the chemical potential with respect to composition is null (Firoozabadi 1999). Therefore, the Fickian

coefficient form Eq. (17) should vanish when approaching the critical point conditions. This phenomenon is an extreme casebut it has been observed in the lab, as discussed in the introduction. Therefore, the assumption of constant diffusion

coefficient may not be always valid.

Irreversible thermodynamic model

The theory of irreversible thermodynamic enables to express explicitly the chemical potential gradient as the driving force for

diffusion (De Groot and Mazur 1984, Ghorayeb and Firoozabadi 2000). Therefore, diffusion occurs between two fluids withdifferent chemical potentials and vanishes when the fluids become at chemical equilibrium (that is, equal chemical

potentials). This concept is consistent with the common numerical approach that neglects diffusion at the simulation

gridblock level as a result of local chemical equilibrium assumption. Under isothermal and isobaric conditions, the

( 1)cn independentdiffusion fluxes are often expressed by:1

1

, 1, , 1,c

n

i ik k c

k

J L Y i n

=

= = (22)

where,1

,

1

1, 1, , 1

cn

k

k jk T p j c

j nc

xY k n

T x

=

= + =

. (23)

The coefficientik

L in Eq. (22), known as the phenomenological or Onsagar coefficients, are symmetric. This flux expression

can be shown equivalent to the Maxwell-Stephan model (Krishna and Standart 1979).

We note that in some simulators, the diffusion flux in Eq. (22) is simplified such that the diffusion driving force of

component i is its own chemical potential gradient multiplied by a diffusion coefficient, that is,

,, 1, ,eff

i i T p i cJ cD i n= = . (24)

Furthermore, the chemical potential gradient in the above equation is simplified without justification such that:

,

1

cn

i i

T p i j i

j j i

x xx x

=

=

. (25)

The diffusion flux in Eq. (24) becomes:

, 1, ,eff

i i i cJ cD x i n= = , (26)

where ( )eff eff i i i iD D x= .

Comparing the above expression with Eq. (18), one notices that this model is not fundamentally different from the classicalFicks law. It just provides an approximate correction term, which is a function of T, p and composition. The accuracy of

this approach is unknown and it suffers from similar limitations as the classical Ficks.

Estimation of diffusion coefficientsSeveral models have been proposed in the literature to predict diffusion coefficients of multicomponent mixtures (Krishna

1993; Wesselingh and Krishna 2000). Most models are associated with uncertainty in predicting diffusion coefficients formixtures with more than three components. The reason is related to the limited experimental data available in the literature

for three and more component mixtures. Furthermore, diffusion experiments are generally difficult to run and could be

sensitive to other effects such as gravity, capillary forces and mechanical dispersion driven by diffusion if measured inporous media. Therefore, accurate prediction of diffusion coefficients including the off-diagonal elements is a key issue.


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SPE 141937 8

The MS diffusion coefficients are easier to predict thanks to of the weak composition dependency as compared to the Fickian

coefficients. In a binary mixture, as the mole fraction of one component approaches one, the ideality factor approaches one

too. Therefore, a common technique in the literature is to estimate the MS diffusion coefficients from binary infinite dilution

coefficients. Note that the infinite dilution coefficient ijD

is defined to be the MS diffusion coefficient of component i

infinitely diluted in component j . A widely used correlation is by Vignes (1966) that suggests interpolating the mutual MS

diffusion coefficient in a binary mixture from the infinite dilution coefficients such that:

1 1112 12 21( ) ( )

x xD D

= . (27)

In a binary mixture, this correlations implies that the logarithmic of 12 should be a linear function of 1x (Krishna and

Wesselingh 1997). In multicomponent mixture, the Vignes correlation can be generalized to (Kooijman and Taylor 1991;

Leahy-Dios and Firoozabadi 2007):

( )/2

1,

( ) ( ) , , 1, , ;c

kj i

nxx x

ij ij ji ik jk c

kk i j

D D D D i j n i j

=

= = . (28)

The remaining item to be discussed is how to estimate the infinite dilution coefficients, ijD . In this work, we used the model

by Leahy-Dios and Firoozabadi (2007) who developed a correlation to estimate the infinite dilution coefficients based on an

extensive set of experimental data. We refer to their paper for details. After computing the infinite dilution coefficients ijD

,

Eq. (28) is used to calculate the ( 1) / 2c cn n MS diffusion coefficients expressed in the matrix B . The thermodynamic

effect in is approximated from Peng-Robenson EOS (Peng and Robinson 1976). The Fickian diffusion coefficients are

then calculated form Eq. (17) .

Diffusion in f racture mediaThe governing equations for three-phase compositional flow are given by the species balance equations in gas and oil phases,

water mass balance, Darcys law, and the thermodynamic equilibrium between the two hydrocarbon phases. The aqueous

phase and hydrocarbon phases are assumed immiscible. The species balance for component i in a cn -component

hydrocarbon mixture is given by the following convection-diffusion equation:

( ), ,. 0, 1, ,i o i o g i g i o i g ccz

c x c y J J i n

t

+ + + + = =

. (29)

The velocity for each phase is described by Darcys law,

( ) , ,rk

p o g

= =k

g . (30)

In the above equations, and k are, respectively, the porosity and permeability of the porous medium, iz is the overall

mole fraction of component i , and, rk , S , , , and p are the relative permeability, saturation, viscosity, mass

density and pressure of phase , respectively. The gravitational vector is denoted by g .

The diffusion fluxes are modified from those defined in open space (Eq.(15)) to account for the porous media effect. In gas

phase, the diffusion flux of component i becomes:

1

,

1

, 1, , 1cn

gi g g g ij j c

j

J S c D y i n

=

= = , (31)

where the multiplier ( gS ) is introduced to mimic the surface area open to diffusion flux in gas phase. Diffusion coefficients

is porous media are usually smaller than those measured in open space. In porous media, the diffusion coefficient gijD can be

estimated from the Fickian coefficientsgijD :


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SPE 141937 9

gijg

ij

DD

= . (32)

The correction factor 1/ , which is a function of the tortuosity, porosity, and constrictions of the porous media, varies

commonly in the range between 0.15 and 0.7 (Brigham et al. 1961; Donaldson et al 1976). Diffusion flux in oil phase ,i oJ is

modeled similarly to Eqs. (31)-(32).

Summing Eq. (29) over all species and using the mole fraction constraints and the molar balance constraint given in Eq. (4),

one gets the overall balance equation, that is,

( ). 0o o g gc

c ct

+ + =

(33)

Our numerical model is formulated from the above equation and the first ( 1)cn species balance equation in Eq. (29). Note

that the diffusion fluxes disappears from the above equation because of Eq. (4). Therefore, with this approach the molar

balance constraint is naturally honored. In fractured media with dual-porosity model, the diffusion flux within a single phase

is written proportional to the shape facture developed by Gilman and Kazemi (1988).

Mass transfer at gas-oil interfaceThe discussion has been so far about intra-phase diffusion, that is, diffusion occurring in gas phase and in oil phase. Masstransfer across enter-phase that occurs at the gas-oil contact under non-equilibrium thermodynamic conditions is important

mechanism that requires further attention. This mechanism has been marginally discussed in the reservoir simulation

literature.

During gas injection in naturally fracture reservoirs, gas has the tendency to flow through fractures. The injected gas maytherefore push by viscous forces or vaporize all the initial oil in the fracture before any significant penetration into the matrix.

A modeling problem occurs if the fracture gets saturated with gas while under-saturated oil resides in the surrounding matrix

blocks. The intra-phase gas and oil diffusion fluxes will be zero (see Eq. (31)) because of the phase discontinuity at thefracture-matrix interface. Therefore, gas and oil diffusions cannot be initiated. Several techniques have been proposed to

circumstance this issue. Hoteit and Firoozabadi (2009) suggested using the cross-phase equilibrium at the matrix-fracture

interface in discrete fracture models. The concept assumes that the fracture gas is in thermodynamic equilibrium with matrixoil within a thin region adjacent the fracture. With this approach, a thin two-phase gridblock is introduced between the

fracture blocks and the matrix blocks as shown in Fig. 1. Therefore, gas-gas and oil-oil diffusions between the matrix and the

fracture blocks can occur via the two-phase region. The introduced region should be thin enough to reduce the girdding

effect. It was found that at the reservoir scale with fracture aperture around 1mm, for example, the two-phase gridblock

thickness should be in the range of 10cm(Hoteit and Firoozabadi, 2009). This approach provides reasonable accuracy and issimple to apply in single-porosity and discrete-fracture models. However, it is not clear how to apply it in the context of dual-

porosity models.

Other techniques suggested circumventing this issue by introducing a gas-liquid diffusion flux of the form:

( ),,i og

i og i i

DJ c x y

l= , (34)

where ,i ogD is a diffusion coefficient and l denotes the distance. Later, we show that the validity of this approach is limited.

Mass transfer at the interface between two phases under non-equilibrium conditions has been thoroughly discussed in the

chemical engineering community. The most common approach is based on the Film theory, which has broad applications

including mass transfer at gas-oil interface. In an isothermal system, we assume that there is an infinitely small region at thegas-oil contact where the two fluids totally mix and consequently are in chemical equilibrium. If no reaction occurring at the

interface, the continuity of molar fluxes across the interface of each component holds.

Consider the situation where a fracture gridblock saturated with gas is adjacent to a matrix gridblock with undersaturated oil.A sketch of the two blocks is shown in Fig. 2. We note that the gas-oil interface coincides with the matrix-fracture interface

or gridblock boundary. Letf( , 1, , )i c

x i n= denote the gas composition in the fracture block and m( , 1, , )i cy i n= denote

the oil composition in the matrix block. At the matrix-fracture boundary, a two-phase fluid appears as a result of chemical

equilibrium, as previously discussed. We denote byI

( , 1, , )i cz i n= ,I

( , 1, , )i cx i n= , andI

( , 1, , )i cy i n= the overall


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SPE 141937 10

composition, the oil composition and the gas composition at the matrix-fracture interface, respectively (see Fig. 2). The

fracture and matrix bulk compositionsf

ix andm

iy are known, whereas the fluid compositions at the interfacesI

iz ,I

ix ,

and Iiy are unknown.

The governing equations of the mass transfer at the gas-oil interface become:

Chemical equilibrium:

( ) ( )I I I I I I I, 1 2 1 , 1 2 1, , , , , , , , , , , 1, ,c cI

i o n i g n cT p x x x T p y y y i n = = . (35)

Continuity of the molar fluxes defined from both sides of the interface (see Eqs.(5)):

, 1, ,i i cN N i n= = f m . (36)

The molar flux in the above equation includes both the diffusion and the convection fluxes (Eq. (5)). The convection molar

flux given by ( )g i gc y is defined by Darcys velocity multiplied by the component mobility in a given phase (see Eq. (29)).In the finite difference (FD) method, the mobility at the interface between two gridblocks is often determined by an upstream

weighting technique, that is, the fluid properties from the upstream gridblock are used to calculate the component mobilities

at the interface. This implicitly imposes a unique definition (i.e. continuity) of the convective flux from both sides of theinterface. In this context, the flux continuity equations given in Eq. (36) imply the continuity of the diffusion flux, that is,

, 1, ,i i cJ J i n= = f m (37)

The system given in Eqs.(35) and (36) represents the governing equations to calculate the gas-oil mass transfer. The solutionmethod requires calculating the interfacial composition. The general framework of this method has been discussed in the

literature with various levels of simplifications (Taylor and Krishna 1993, Haugen and Firoozabadi 2009). The solution

method is introduced as follows.

Solution method:Using a first order FD approximation of the gradients, the diffusion flux defined by the generalized Ficks law in the

fracture block across the interface becomes (see Eq. (15)):

1

1

( ), 1, , 1

cn

i ii ij c

j

y yJ c D i n

l

=

= =

I ff f f

f. (38)

Similarly, the diffusion flux in the matrix block across the interface can be approximated by :

1

1

( ), 1, , 1

cn

i ii ij c

j

x xJ c D i n

l

=

= =

m Im m m

m. (39)

In the above equations, l f and lm denote the distances between the centers of the fracture and matrix gridblocks and the

interface, respectively. We also note that the gradients are calculated with the assumption that the positive direction isoriented downwards (see Fig. 2).

It is evident that, the phase split composition ixI or iy

I is required to calculate the fluxes. The system is nonlinear because the

overall composition izIis unknown and therefore the flash calculation cannot be applied directly. On common approach in

the literature (da Silva and Belery 1989) is to determine izI by assume that the gas and oil mix with equal proportions at the

interface. This approximation is ad-hoc and does not guaranty the flux continuity constraint (Eq. (37)). A proper approach is

to simultaneously solve both equations by a linearization method. The system of equation is:


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SPE 141937 11

( ) ( )

1 1

1,

1 1

2, , 1 2 1 , 1 2 1

( ) ( ): 0, 1, , 1

: , , , , , , , , , , 0, 1, ,

c c

c c

n n

i i i ii ij ij c

j j

i i o n i g n c

x x y yF c D c D i n

l l

F f T p x x x f T p y y y i n

= =

+ = =

= =

m I I f m m f f

m f

I I I I I I I I

(40)

The second equation in Eq. (40) represents the fugacity equality. It is equivalent to Eq. (35). The variables are, ixI , iy

I for

1, , 1ci n= and V , where V denotes the gas phase mole fraction such that ,

I I I(1 ) , 1, ,i i i cz Vy V x i n= + = . (41)

The oil mole fraction, 1L V= , can also be selected instead of V . The system is therefore closed with (2 1)cn independent

equations and with equal number of variables. The system can be solved using a combined SSI-Newton method. Note that

Tand p are interpolated at the interface and the diffusion coefficient ijDm and ijD

f are calculated form the fluid properties in

the adjacent matrix and fracture blocks, respectively.

Before ending this section, lets investigate the accuracy of the model defined in Eq. (34) by considering a particular case

when 2cn = (binary mixture). The first equation in Eq.(40) becomes:

( )1 1 1 1( ) 0a x x a y y + =m m I f I f , (42)

where 1,1 /a c D l=m m m m and 1,1 /a c D l=

f f f f . Lets introduce the thermodynamic equilibrium constant (K-value), 1Kf such

that:

1 1 1y K x=I I I . (43)

Using the above equation in Eq.(42), one gets:

1 1

11

a x a y

x a K a

+

= +

m m f f I

m I f . (44)

The gas-oil mole transfer can then be expressed independently of 1xIby :

( )1 1 1 11

a aJ K x y

a K a=

+

m fm I m f

m I f (45)

Comparing Eq. (45) to Eq.(34), one notices that they are consistent when 1 1K =I

. Consequently, the direct gas-oil diffusion

model (Eq.(34)) may make sense only when all the K-values are equal to one, that is, at the critical point. The accuracy of

this model is therefore very limited.

Numerical examplesAs previously discussed, the classical Ficks law may provide reasonable results for many applications. In some cases,

however, the model may fail. We present here simple examples to highlight the drawbacks of this model.

Example 1a: Molar balance violation for binary mixturesWe consider a simple 1D diffusion problem in porous media. The objective here is to investigate the impact of violating the

molar balance constraint (Eq. (4)) by the classical Ficks law. We consider a conceptual diffusion problem for a binary

mixture composed of C1 and CO2. The domain can be seen as a 1D porous medium of length 100ft, where gases areinitialized such that C1saturates one half of the domain (left side) and CO2saturates the second half. A sketch of the domain

and the initial gas compositions is shown in Fig.3. Counter-current diffusion starts at the initial C1-CO2contact located at the

middle of the domain. The system has closed boundary condition with no sink/source terms.


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SPE 141937 12

To simplify the problem, we consider diffusion at standard conditions ( 1 , 60o

p atm T F= = ) so that C1and CO2behave as

ideal gases. Mixing from diffusion is therefore expected to be isobaric (occurs at constant pressure) and isothermal. This

problem has an analytical solution expressed by (Li 1972):

2

12 12

1 / 2 2 x / 2 2 x(x, ) 1 erf erf

2 4 4CO

k

L kL L kLy t

D t D t

=

+ + = +

, (46)

where2CO

y is the CO2mole fraction, 12D is the C1-CO2diffusion coefficient, L is the domain length, and x , and tare the

space and time variables, respectively. For this particular case, this solution serves us as a reference.

As previously discussed, for a binary mixture there is one diffusion coefficient (see Eq.(14)). However, some existing

reservoir simulators support the effective diffusivity model given in Eq.(18), and therefore allows for cn independent

diffusion coefficients, that is, two diffusion coefficients in this case. We evaluated this approach using two commercialsimulators. The results are shown from one of them because of the symmetry in their behavior. We checked the composition

and pressure solutions in the domain by considering three cases:

Case 1: 1 2/ 1D D = ,

Case 2:1 2/ 0.2

D D = ,

Case 3: 1 2/ 0.5D D = .

In all cases, the CO2diffusion coefficient is2

1 1 /D ft day= . This problem is isobaric and therefore convection is irrelevant.

We first deactivate convection in the numerical model by taking infinitely low medium permeability. The objective is to

detect any pressure variations from diffusion. Fig. 4 and Fig. 5 show the CO2mole fracture and pressure profiles versus time

for the three cases. In Case 1, the analytical and numerical solutions of CO2mole fractions are almost identical, as appears in

Fig. 4. The pressure is invariant in this case, as expected. In Cases 2 and 3, CO2mole fractions are different (Fig. 4) and thepressure in not constant (Fig. 5). The variation in pressure is due to the violation of the molar balance constraint in Eq.(4),

which results in a non-equimolar condition and therefore unbalanced pressure.

Activating convection by considering a permeably medium will indeed mask the pressure imbalance for Cases 2 and 3 asshown in Fig. 7. However, the solution is now a result of diffusion and also convection caused by artificial pressure

variations (Fig. 6). In Case 1, the solution is the same with and without convection as expected.

The bottomline here is that taking different diffusion coefficients does not make sense for binary mixtures. Formulticomponent mixtures, the problem is more complicated as discussed in Example 1b.

Example 1b: Molar balance violation for multicomponent mixtures

In this example, we consider a ternary mixture composed of CO2, C1, and N2. The problem is similar to the one in Example

1a. The gases, CO2, C1, and N2are initialized with mole fractions (0.1,0.7,0.2) in one half (the left side) of the domain and(0.9,0.1,0.0) in the right side. We first consider diffusion at standard conditions so that the system is isobaric and isothermal.

The results are again from the same commercial simulator where we checked the composition and pressure behavior by

considering three cases:

Case 1: 1 2/ 1D D = ,

Case 2: 1 2/ 0.2D D = ,

Case 3: 1 2/ 0.5D D = .

In all cases, the CO2and N2diffusion coefficients are assumed identical, such that,2

1 3 1 /D D ft day= = . Plots for C1mole

fraction for the three cases and the pressure profiles are shown in Fig. 8 and Fig. 9, respectively. Only for Case 1, when all

diffusion coefficients are identical, the pressure stays invariant. In the two other cases, when not all coefficients are identical,the molar balance constraint is violated and therefore the model cannot honor the molar balance constraint.

We tested the generalized Ficks law (see Eq. (15)) which we implemented in our in-house reservoir simulator Psim. Withthis approach, there are two diffusion fluxes and two diagonal diffusion coefficients. The off-diagonal diffusion coefficients

are neglected in this example. The third diffusion flux is computed such that the molar balance constraint is honored, that is,


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SPE 141937 13

1

1

c

c

n

n i

i

J J

=

= . (47)

Fig. 10 and Fig. 11 show the C1 mole fractions and pressures versus distance for the three cases. Note that this modelcoincides with the classical Ficks law model when all diffusion coefficients are equal (Case 1). The pressure is invariant for

all cases (Fig. 11). This is expected because the molar balance constraint is honored by definition (Eq.(47)).

The error in pressure shown in the previous cases may look modest. However, at higher initial pressures, violating the molar

balance constraint has more dramatic. Fig. 12 shows the pressure profiles for the three cases using the commercial simulator.

The initial pressure is 300psi. This problem is not isobaric anymore because of the system non-ideality effect. This explains

the minor variation in pressure in Case 1, as appears in Fig. 12. However, the pressure error for Cases 2 and 3 where thebalance constraint is violated is more dramatic (around 100psi). This is because the system is less compressible and thereforeless tolerant for material misbalance. This problem is often undetectable in most applications when convection is active

because any pressure variation will create convection that will eventually regain pressure balance. The numerical solution

will therefore be a result of diffusion balanced with artificial convection. Therefore, the reliability of such a solution is likely

to be problem dependent.

Example 2a: Loschmidt tube

Our objective here is to validate our proposed diffusion model and to highlight another limitation related to the classical

Ficks law. Arnold and Toor (1967) used Loschmidt tube to study the diffusion behavior of a ternary mixture composed ofmethane, argon, and hydrogen. A sketch of the experiment set-up is shown in Fig. 13. The composition of (C 1,Ar,H2) in the

top and bottom parts of the tube were (0.,0.509,0.491) and (0.515,0.485,0.0), respectively. The pressure and temperature were

1atm and 34oC. At time 0, the two fluids were allowed to inter-diffuse and average concentrations in the top and bottom tubeswere measured at different times. The mole factions of C1and Ar in the top and bottom tubes are shown versus time in Fig.

14a and 14b, respectively. Fig. 14a shows that C1in the top tube diffuses towards the bottom tube consistently with the C1

composition gradient. However, the diffusion of argon follows initially a direction opposite to its concentration gradient. This

non-Fickian diffusion behavior is known as uphill or reverse diffusion. Fig. 14 also shows the numerical solutions from theclassical Ficks law. One notices that the model can be tuned to match the C1composition but not the Ar composition. This is

expected as the Argon reverse diffusion behavior is a result of the dragging effect described by the diffusion off-diagonal

coefficients which are neglected in the classical Ficks law.

The Maxwell-Stephan model that can also be described by the generalized Ficks law is therefore indispensable for this

problem. Fig. 15a and 15b show an excellent agreement between the solutions form the generalized Ficks law (Maxwell-

Stephan) and the observed data. The diffusion coefficients including the off-diagonal elements are calculated using the modelby Leahy-Dios and Firoozabadi (2007). The diffusion coefficients are calculated as a function PVT conditions. Fig. 16

demonstrates these coefficients versus time. One notices that their variations are insignificant. This is a result of the mixture

weak non-ideality at experiment conditions. In all cases, we used 20 simulation gridblocks which we found fine enough to

provide solution with negligible gridding effect.

It should be noted that we did minor tuning for the diffusion coefficients from our model. They were shifted by around 40%

to get the best match with the observed data. The solution with unmodified diffusion coefficients as provided by the model

are shown in Fig. 17. The model provides a reasonable match without any tuning. We have found that to best control the

diffusion model, two multiplies 1M i and 2M iper diffusion flux iJ are needed. The multiplier 1M i is used to modify the main

diffusion term, and 2M i to modify the dragging effect represented by the off-diagonal terms. Therefore, the generalized Ficks

diffusion flux given in Eq. (15) becomes:

( )1

1 2

1

, 1, , 1M Mcn

i i ii i i ij j c

jj i

J cD x c D x i n

=

= + =

. (48)

Note that the molar balance constraint in Eq. (4) is always honored. The classical Ficks law can be simply obtained by

setting all 2M i to zero and the flux for the last component is calculated from (47).

Example 2b: Two-bulb diffusion cellThis experiment conducted by Ducan and Toor (1962) to study diffusion of a ternary mixture in a two-bulb diffusion device.

The device is composed of two bulbs connected by means of a capillary tube as sketched in Fig. 18a. The radii of bulb 1 and

bulb 2 are 26.49mm and 26.58mm, respectively. The joining capillary tube length is 85.9mm and diameter 2.08mm. Ducan


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SPE 141937 14

and Toor studied the diffusion of H2, N2, and CO2at room temperature and 1atm. The mixture has ideal thermodynamicbehavior. The initial compositions in the bulbs at t=0 are given by :

Bulb 1 (H2,N2,CO2): 0.0, 0.501, 0.499

Bulb2 (H2,N2,CO2) : 0.501, 0.499, 0.0

The gas components were allowed to inter-diffuse through the capillary tube and the average concentrations were measured

at different times. There are two methods to create a simulation model to describe the two-bulb device. The first is to simplyuse a 1D corner-point grid to describe the capillary tube and the bulbs. However, we found that it is not enough to use a

single simulation gridblock for each bulb. As a result, this complicates a bit the construction the grid. A different simplifiedmethod is to use the analysis by Duncan and Toor (1962) (see also Taylor and Krishna 1992) who showed that diffusion in

this device can be seen equivalent to diffusion in a Loschmidt-type tube with dimension / 0.038A mm= , as shown in Fig.18b. Therefore, the simulation grid can be taken similarly to Example 2a. We used the classical Ficks law and thegeneralized Ficks law to simulate this experiment. The results are shown in Figs. 19 and 20. Similarly to the discussion in

Example 2a, the classical Ficks law does not hold for this experiment either. The proposed model based on the generalized

Ficks law is in good agreement with the observed data (Fig. 20). The diffusion coefficients calculated as a function of

composition, temperature and composition were shifted by around 30% for best fit.

Summery and conclusionsMolecular diffusion could be an efficient recovery mechanism in many applications in reservoir engineering. In fractured

reservoirs, molecular diffusion allows to produce trapped oil in the matrix blocks by creating counter-current material transferbetween the fracture and the matrix.

In most reservoir simulators molecular diffusion is often approximated by the effective diffusivity model which is used in thecontext of the classical Ficks law. The classical Ficks law assumes that diffusion of each component in a mixture occurs

independently of the other components and the driving force is proportional to the own concentration gradient. In this work,

we highlighted two issues about the classical Ficks law:

1-The classical Ficks law has an inherit inconsistency because it assumes cn independent diffusion coefficients in cn -

component mixtures. Therefore, the equimolar condition cannot be honored for multicomponent mixtures unless all the

diffusion coefficients are taken identical. We showed with simple examples that violating the molar balance constraint

may create artificially pressure gradient and therefore convention. This problem is unnoticeable in most applicationsbecause convection from Darcys law leads to regain pressure balance and therefore the variation artifacts in pressure

will be masked. Therefore the accuracy of the solution is unknown because of possible significant impact of the artificialconvection.

2-The classical Ficks law neglects the off-diagonal diffusion coefficients that describe the component interaction and thedragging effect in multicomponent mixture. Neglecting the dragging effect may have major consequences in someapplications that cannot be modeled accurately with the classical Ficks law.

We showed that the Maxwell-Stephan model described by the generalized Ficks law is indispensable when the off-diagonalcoefficients are important compared to the main diffusion coefficients. We highlighted the relationship between the Fickian

diffusion coefficients and the Maxwell-Stephan diffusion coefficients. The Fickian diffusion coefficients incorporate the

thermodynamic non-ideality effect. These coefficients are generally strong function of composition, temperature andpressure. The diffusion model by Leahy-Dios and Firoozabadi (2007) provided satisfactory results for the cases we tested

without any tuning. Shifting the coefficients by 30% to 40% enabled to get very good agreement with the observed data. We

have also noted that to best control the diffusion model, two multipliers are needed per diffusion flux. One multiplier is used

for shift the main diffusion coefficients and one for the off-diagonal coefficients. The molar balance constraint is notimpacted by the multipliers.

We have discussed the gas-oil transfer at the matrix-fracture interface. This mechanism is needed for situations where intra-

phase diffusion cannot be initiated due to phase discontinuity. We provided a solution method that uses the generalizedFicks law and assumes chemical equilibrium at the gas-oil contact. We also showed that modeling gas-oil transfer by using a

Fick-type model may not be valid.

We should note that this work does not suggest dropping the classical Ficks law in favor of the Maxwell-Stephan or the

generalized Ficks law. The classical Ficks law is practical and simple. However, one should be aware of its limitations. The

classical Ficks law could be tuned if possible so that it is consistent with the generalized Ficks law. Therefore, thegeneralized Ficks law need to be supported by reservoir simulators and the classical Ficks law should not be used in

applications involving significant cross-diffusion effects.


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SPE 141937 15

NomenclatureB = Maxwell-Stephan diffusion matrix

ij = Maxwell-Stephan diffusion coefficient

ijD = Fickian diffusion coefficient

ic = molar density of component i

c = overall molar density

ijD

= infinite dilution coefficient

effiD

= effective diffusion coefficient

iJ = diffusion flux of component i

J = diffusion flux vector

1M i = multiplier for diffusion diagonal coefficients

2M i = multiplier for off-diffusion diagonal coefficients

iN = molar flux of component i

tN = total molar flux of all components

cn = number of components in the mixture

p = pressure

R = universal gas constant

T = temperaturet = time

iu = velocity of component i

u = average velocity of the phase

ix = mole fraction of component i in oil phase

iy = mole fraction of component i in gas phase

= matrix for thermodynamic non-ideality factor

i = chemical potential

ij = Kronecher delta

i = fugacity coefficient

ReferencesAlavian S.A. and Whitson C.H. 2010. Modeling CO2 Injection Including Diffusion in a Fractured-Chalk Experiment. Paper SPE135339

presented at the SPE Annual Technical Conference and Exhibition, Florence, 19-22 Sept.Bentez J. 2009.Principles and modern applications of mass transfer operations. New Jersy, John Wily & Sons.Brigham W., Philip W. and John N. 1961. Experiments on Mixing During Miscible Displacement in Porous Media, SPE J.1(1):1-8Chang, Y.C. and Myerson, A.S. 1986. Diffusivity of glycine in concentrated, saturated and supersaturated aqueous solution. AIChE J.,

32: 1567-1569.Chordia M. and Trivedi J. 2010. Diffusion in Naturally Fractured Reservoirs - A Review. Paper SPE 77339 presented at the SPE Asia

Pacific Oil and Gas Conference and Exhibition, Brisbane, 18-20 October.Coats K.,1989. Implicit compositional simulation of single-porosity and dual-porosity reservoirs. Paper SPE 18427 presented at the SPE

Symposium on Reservoir Simulation, Houston. 6-8 Feb.Coats, K. 1980. An equation of state compositional model, SPE J. 20(5): 363-376.Cullinan H. and Toor H., 1965. Diffusion in the three-component liquid system Acetone-Benzene-Carbon Tetrachloride. J. Phys. Chem.

69:3941.Cussler E. 1976.Multicomponent diffusion,Elsevier Sci. Co.da Silva F. and Belery P. 1989. Molecular diffusion in naturally fractured reservoirs: A decisive recovery mechanism. Paper SPE19672

presented at the SPE Annual Technical Conference and Exhibition, San Antonio, 8-11 Oct.Darvish G., Lindeberg E., Holt T. and Utne S. 2006a. Laboratory Experiments of Tertiary CO2 Injection Into a Fractured Core. Paper

SPE 99649 presented at the SPE/DOE Symposium on Improved Oil Recovery, Tulsa, 22-26 April.Darvish G., Lindeberg E., Holt T., Utne S., and Kleppe J. 2006b. Reservoir conditions laboratory experiments of CO2 injection into

fractured cores. Paper SPE 99650 presented at SPE Europec/EAGE Annual Conference and Exhibition, Vienna, 12-15 June.

De Groot S. and P. Mazur P. 1984.Non-Equilibrium Thermodynamics. Toronto: Gen. Publishing Co.Donaldson, E., Kendall R.and Manning, F. 1976. Dispersion and Tortuosity in Sandstones. Paper SPE 6190 presented at SPE Annual Fall

Technical Conference and Exhibition, New Orleans, 3-6 Oct.Duncan J. and Toor H. 1962. An experimental study of three component gas diffusion. 8(1):38-41

Firoozabadi A. 1999. Thermodynamics of Hydrocarbon reservoirs. New York: McGaw-Hill.


16/22

SPE 141937 16

Ghorayeb, K., and A. Firoozabadi. 2000. Molecular, Pressure, and Thermal Diffusion in Nonideal Multicomponent Mixtures. AIChE J.46(5): 883.

Gilman J. and Kazemi H. 1988. Improved Calculations for Viscous and Gravity Displacement in Matrix Blocks in Dual-Porosity

Simulators. SPE J.40(1):60-70.Grogan, A.T., Pinczewski, V.W., Ruskauff, G. J., and Orr Jr. 1988. Diffusion of CO2 at Reservoir Conditions: Models and

Measurements.SPE J. 3(1): 93-102.

Guerrero-Aconcha U. and Kantzas A. 2009. Diffusion of Hydrocarbon Gases in Heavy Oil and Bitumen. Paper SPE 122783 presented atthe Latin American and Caribbean Petroleum Engineering Conference, Cartagena de Indias, 31 May-3 June.

Guo P., Wang Z., Shen P. and Du J. 2009. Molecular Diffusion Coefficients of the Multicomponent Gas

Crude Oil Systems under HighTemperature and Pressure.Ind. Eng. Chem. Res.48(19):90239027.

Hasse R. 1972. Diffusion and Thermal Diffusion in the Critical Region of Binary Fluid Systems. Berichte der Bunsengesellschaft frphysikalische Chemie. 76(3): 256259

Haugen K. and Firoozabadi A. 2009. Composition at the Interface Between Multicomponent Non-Equilibrium Phases. J. Chem. Phys.130(6):64707-64707.

Hayduk W. and Minhas. B.,1982. Correlations for predictions of molecular diffusions in Liquids. Can. J. Chem. Eng.60:295.Hoteit H. and Firoozabadi A. 2006. Compositional Modeling by the Combined Discontinuous Galerkin and Mixed Methods, SPE J.

11(1): 19-34.Hoteit H. and Firoozabadi A., 2009.Numerical Modeling of Diffusion in Fractured Media for Gas-Injection and -Recycling Schemes.

SPE J.14(2): 323-337.Karimaie H. and Torster O. 2010. CO2 and C1 Gas Injection for Enhanced Oil Recovery in Fractured Reservoirs. Paper SPE 77339

presented at the SPE International Conference on CO2 Capture, Storage, and Utilization, New Orleans 10-12 Nov.Kazemi A. and Jamialahmadi M. 2009. The Effect of Oil and Gas Molecular Diffusion in Production of Fractured Reservoir During

Gravity Drainage Mechansim by CO2 Injection. Paper SPE 120894 presented at the EUROPEC/EAGE Conference and Exhibition,

Amsterdam, 8-11 June.Kooijman H.A. and Taylor R. 1991.Estimation of Diffusion Coefficients in Multicomponent Liquid Systems, Ind. Eng. Chem. Res.

30(6):1217-1222.

Krishna R. 1987. A unified theory of separation processes based on Irreversible Thermodynamics. Chem. Engng Commun., 59:33-64.Krishna R. and Standart G. 1976. Determination of mass and energy fluxes for multicomponent vapour-liquid systems, Letts Heat &

Mass Transfer, 3:173-182.

Krishna R. and Wesselingh J.A.,1997. The Maxwell-Stefan approach to mass transfer. Chem. Eng. Sci.,52:861-911Krishna, R. and Standart, G. 1979. Mass and energy transfer in multicomponent systems, Chem. Engng Commun. 3:201-275.Lagalaye Y., Nectoux A. and James N. 2002. Characterization of Acid Gas Diffusion in a Carbonate Fractured Reservoir Through

Experimental Studies, Numerical Simulation and Field Pilots. Paper SPE 77339 presented at the SPE Annual Technical Conference

and Exhibition, San Antonio, 29 September-2 October.Le Romancer, J-F.X., Defives D., and Fernandes G. 1994. Mechanism of Oil Recovery by Gas Diffusion in Fractured Reservoir in

Presence of Water. Paper SPE 27746 presented at the SPE/DOE Improved Oil Recovery Symposium, Tulsa, 17-20 April.Leahy-Dios A. and Firoozabadi A. 2007. Unified Model for Non-Ideal Multicomponent Molecular Diffusion Coefficients. AIChE J,

53(11):2932-2939.Leahy-Dios A., Bou-Ali M., and Firoozabadi A. 2005. Measurements of molecular and thermal diffusion coefficients in ternary mixtures.

J. of Chemical Physics. 122(23) 4502.Li W. 1972.Differential Equations of Hydraulic Transients, Dispersion, and Groundwater flow. Englewood Cliffs, Prentice-Hall.

McKay W. N. 1971. Experiments Concerning Diffusion of Multicomponent Systems At Reservoir Conditions.JCPT10(2):25-32Mohammad I., Boozarjomehry B., Pishvaie R Reza, M., Tavasoli A. 2009.A Study on the effects of thermodynamic non-ideality and

mass transfer on multi-phase hydrodynamics using CFD methods. World Academy of Science, Engin. and Tech. 58.Peng D. and Robinson D. 1976. A new two-constant equation of state. Ind. Eng. Chem Fundam. 15:59-64.

Renner T. 1988. Measurement and correlation of diffusion coefficients for CO2 and rich gas applications. SPE Res. Engr.3:517-523Salama D. and Kantzas A. 2005. Monitoring of Diffusion of Heavy Oils with Hydrocarbon Solvents in the Presence of Sand. Paper SPE

97855 presented at the SPE/PS-CIM/CHOA International Thermal Operations and Heavy Oil Symposium, Calgary, 1-3 Nov.

Shrivastava V., Nghiem L.X., Moore R., and Okazawa T. 2005. Modelling Physical Dispersion in Miscible Displacement-Part 1: Theoryand the Proposed Numerical Scheme.JCPT 44(2):25-33.

Sigmund Ph. 1976a. Prediction of molecular diffusion at reservoir conditions. Part 1- Measurements and prediction of binary dense gas

diffusion coefficients. JCPT15(2).

Sigmund Ph. 1976b. Prediction of molecular diffusion at reservoir conditions. Part 2- Estimating the effect of molecular diffusion andconvective mixing in multicomponent systems,JCPT15(3).Taylor R., and Krishna R. 1993.Multicomponent mass transfer, New York: John Wily & Sons.

Thomas L.K., Dixon T., and Pierson R. 1991. Ekofisk nitrogen injection. SPE J. 6(2):151-160.Vignes, A. 1966. Diffusion in Binary Solutions.Ind. Eng. Chem. Fundam. 5(2): 189-199.Warren J., and Root P, "The behavior of naturally fractured reservoirs," SPE J. (Sept. 1963) 245.Wesselingh J. and Krishna R. 2000.Mass Transfer in Multicomponent Mixtures, The Netherlands: Delft University Press.Wilke C.R. 1950. Diffusional Properties of Multicomponent Gases. Chem. Engng. Prog.46:90-104.


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SPE 141937 17

Fracture

2-phase region

Matrix

Fracture

2-phase region

Matrix

Fig. 1-Cross-flow equilibrium concept: gas-gas and oil-oil diffusions occur across a two-phase gridblock.

Fracture (gas)

Matrix (oi l)

f

iy

m

ix

, ,I I I

i i iz x y

GOC

Mass

transfer

m

iJ

f

iJ

Fracture (gas)

Matrix (oi l)

f

iy

m

ix

, ,I I I

i i iz x y

GOC

Mass

transfer

m

iJ

f

iJ

Fig. 2-Mass transfer at the gas-oil phase: gas-gas and oil-oil diffusion occur across an infinitely small film at the interface.

C1 : 100%CO2: 0%

C1 : 0%CO2: 100%

50 ft 50 ft

DiffusionC1 : 100%CO2: 0%

C1 : 0%CO2: 100%

50 ft 50 ft

Diffusion

Fig. 3-Intra-diffusion set-up for a binary mixture.


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SPE 141937 18

0.00

0.25

0.50

0.75

1.00

0 25 50 75 100

Distance (ft)

CO2mo

lefraction

Exact solution

Case 1: D1/D2=1

Case 2: D1/D2=0.5

Case 3: D1/D2=0.2

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

0 25 50 75 100

Distance (ft)

Pressure(psi)

Case 1:D1/D2=1

Case 2:D1/D2=0.5

Case 3:D1/D2=0.2

Fig. 4: CO2 mole fraction vs. distance for 3 cases and analyticalsolution after 100days, no convection, Example 1a.

Fig. 5: Pressure vs. distance for 3 cases after 100days, noconvection, Example 1a.

0.00

0.25

0.50

0.75

1.00

0 25 50 75 100

Distance (ft)

CO2molefraction

Exact solution

Case 1: D1/D2=1

Case 2: D1/D2=0.5

Case 3: D1/D2=0.2

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

0 25 50 75 100

Distance (ft)

Pressure(psi)

Case 1:D1/D2=1

Case 2:D1/D2=0.5

Case 3:D1/D2=0.2

Fig. 6: CO2 mole fraction vs. distance for 3 cases and analyticalsolution after 100days, with convection, Example 1a.

Fig. 7: Pressure vs. distance for 3 cases after 100days, withconvection, Example 1a.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 25 50 75 100

Distance (ft)

C1molefraction

Case 1: D1/D2=1

Case 2: D1/D2=0.5

Case 3: D1/D2=0.2

10.0

12.0

14.0

16.0

18.0

20.0

0 25 50 75 100

Distance (ft)

Press

ure(psi)

Case 1:D1/D2=1

Case 2:D1/D2=0.5

Case 3:D1/D2=0.2

Fig. 8: C1 mole fraction vs. distance after 200 days, results from acommercial simulator, no convection, Example 1b.

Fig. 9: Pressure vs. distance after 200 days, results from acommercial simulator, no convection, Example 1b.


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SPE 141937 19

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 25 50 75 100

Distance (ft)

C1mol

efraction

Case 1: D1/D2=1

Case 2: D1/D2=0.5

Case 3: D1/D2=0.2

10.0

12.0

14.0

16.0

18.0

20.0

0 25 50 75 100

Distance ( ft)

Pressure(psi)

Case 1:D1/D2=1

Case 2:D1/D2=0.5

Case 3:D1/D2=0.2

Fig. 10: C1 mole fraction vs. distance after 200 days, results from Psim,no convection, Example 1b.

Fig. 11: Pressure vs. distance after 200 days, results from Psim,no convection, Example 1b.

150.0

200.0

250.0

300.0

350.0

400.0

450.0

0 25 50 75 100

Distance (ft)

Pressure(psi)

Case 1:D1/D2=1

Case 2:D1/D2=0.5

Case 3:D1/D2=0.2

Fig. 12: Pressure vs. distance after 200 days, results from a commercial simulator, no convection, Example 1b.

C1 : 0.0

Ar : 0.509

N2 : 0.491

C1 : 0.515

Ar : 0.485

N2 : 0.0

40.5cm

40.5cm

Diffusion

C1 : 0.0

Ar : 0.509

N2 : 0.491

C1 : 0.515

Ar : 0.485

N2 : 0.0

40.5cm

40.5cm

Diffusion

Fig. 13: Loschmidt tube set-up, Example 2a.


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SPE 141937 20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 20 40 60 80 100 120

Time (mins)

C1molefraction

Calc( bottom)

Calc (top)

Data (botto m)

Data (top)

a) Methane mole fraction

0.40

0.45

0.50

0.55

0.60

0 20 40 60 80 100 120

Time (mins)

Armol

efraction

Calc (bottom)

Calc (top)

Data (bottom )

Data (top)

b) Argon mole fraction

Fig. 14: Observed data in the top and bottom with numerical solution based on the classical Ficks law, Example 2a.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 20 40 60 80 100 120Time (mins)

C1molefraction

Calc( bottom)

Calc (top)

Data (bottom)

Data (top)


0.40

0.45

0.50

0.55

0.60

0 20 40 60 80 100 120

Time (mins)

Armolefraction

Calc (bottom)

Calc (top)

Data (bottom)

Data (top)


Fig. 15: Observed data in the top and bottom with numerical solution based on the generalized Ficks law, best fit, Example 2a.

0.E+00

1.E-05

2.E-05

3.E-05

4.E-05

5.E-05

6.E-05

7.E-05

8.E-05

9.E-05

0 20 40 60 80 100 120

Time (mins)

DiffusionCo

efs.(m2/s)

D11 D21 D12 D22

D22

D11

D21

D21

Fig. 16: Fickian diffusion coefficients. vs. time for 1-C1 and 2-Ar, Example 2a.


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SPE 141937 21

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 20 40 60 80 100 120

Time (mins)

C1mole

fraction

Calc( bottom)

Calc (top)

Data (botto m)

Data (top)


0.40

0.45

0.50

0.55

0.60

0 20 40 60 80 100 120

Time (mins)

Armolefraction

Calc (bottom)

Calc (top)

Data (bottom )

Data (top)


Fig. 17: Solution based on the generalized Ficks law with unmodified diffusion coefficients, Example 2a.

A

Diffusion

Bulb1 Bulb2

Bulb1

Bulb2 2/

A

Diffusion

Bulb1 Bulb2

Bulb1

Bulb2 2/

a)Two-bulb diffusion cell b)Loschmidt tube

Fig. 18: Diffusion in a two-bulb cell that can be seen equivalent to a Loschmidt tube, Example 2b.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20

Time (hr)

N2molefraction

Calc(bulb 1)

Calc (bulb 2)

Data (bulb 1)

Data (bulb 2)

a) Nitrogen mole fraction

0.40

0.45

0.50

0.55

0.60

0 5 10 15 20

Time (hr)

H2molefraction

Calc (bulb 1)

Calc (bulb 2)

Data (bulb 2)

Data (bulb 1)

b) Hydrogen mole fraction

Fig. 19: Solution based on the classical Ficks law, Example 2b.


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SPE 141937 22

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20

Time (hr)

N2molefraction

Calc(bulb 1)

Calc (bulb 2)

Data (bulb 1)

Data (bulb 2)

a) Nitrogen mole fraction

0.40

0.45

0.50

0.55

0.60

0 5 10 15 20

Time (hr)

H2molefract

ion

Calc (bulb 1)Calc (bulb 2)

Data (bulb 2)

Data (bulb 1)

b) Hydrogen mole fraction

Fig. 20: Solution based on the generalized Ficks law, Example 2b.

Documents

Modeling of Difusion in Fractured Reservoirs SPE-141937 -MS-P