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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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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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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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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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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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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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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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( ) ( )
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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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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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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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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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
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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,
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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.
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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.
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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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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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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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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)
a) Methane mole fraction
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)
b) Argon mole fraction
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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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)
a) Methane mole fraction
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)
b) Argon mole fraction
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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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.