improved mathematical model for mass transfer in fractured...

$: Improved mathematical model for mass transfer in fractured …scientiairanica.sharif.edu/article_3690_9c849e740e0c2ad50f11b5db… · Sharif University of Technology Scientia Iranica$
Scientia Iranica C (2015) 22(3), 955{966

Sharif University of TechnologyScientia Iranica

Transactions C: Chemistry and Chemical Engineeringwww.scientiairanica.com

Improved mathematical model for mass transfer infractured reservoirs during the gas injection process

M. Sakia, M. Masihia;� and S.R. Shadizadehb

a. Department of Chemical & Petroleum Engineering, Sharif University of Technology, Tehran, P.O. Box 11155-9465, Iran.b. Petroleum University of Technology, Ahvaz, P.O. Box 71183-61991, Iran.

Received 24 July 2013; received in revised form 14 July 2014; accepted 5 January 2015

KEYWORDSDi�usion;Injection rate;Di�usion ux;Convection ux;Simulation;Recovery rate.

Abstract. In fractured reservoirs with relatively low matrix permeability, i.e. smallmatrix block size with respect to capillary threshold height, di�usion becomes an importantrecovery mechanism. In this work, we have attempted to model the mass transfer betweenthe matrix and fracture by considering a fractured porous media as a single matrix blockwith an adjacent fracture. An appropriate model should be applicable in the case of thematrix being saturated with both saturated and undersaturated oils. The proposed modelpresents such versatility. The model is a modi�cation for the formulation of Jamili etal. [Jamili, A., Whillhite, G.P. and Green, D.W., Modeling Gas-Phase Mass TransferBetween Fracture and Matrix in Naturally Fractured Reservoir, SPE132622 (2011)], whichsu�ered from several drawbacks such as the use of the classical Fick's law, and of Hua andWhitson [Hua Hu, Whitson, C.H. and Yuanchang, Q.i., A Study of Recovery Mechanismin a Nitrogen Di�usion Experiment, SPE21893 (1991)] to calculate di�usion mass transfercoe�cients between the fracture and matrix, and the use of Darcy's law to model convectionmass transfer between the fracture and matrix. In this work, these drawbacks weresurveyed and amended. Subsequently, the improved model has been validated througha gas injection experiment. Following model validation, the e�ects on the recovery rateof matrix permeability, initial gas saturation and injection rate are investigated. Thenumerical analysis showed that the e�ect of gas injection rate on the recovery rate isconsiderable.c 2015 Sharif University of Technology. All rights reserved.

1. Introduction

It is common to develop the gas invaded zone withinfractured reservoirs by injecting gas and maintainingreservoir pressure. Recovery in such reservoirs willbe the result of a complex interplay of several mech-anisms, such as viscous ow, extraction by moleculardi�usion, gravity drainage, oil swelling, and capillaryforces, where the main mechanism is generally gravity

*. Corresponding author. Tel.: +98 21 66166425E-mail addresses: [email protected] (M.Saki); [email protected] (M. Masihi); [email protected](S.R. Shadizadeh)

drainage. However, in many cases, such as reservoirswith low permeability matrix, small matrix block sizeand high capillary pressure, gravity drainage maybe very low or ine�ective. Low permeability alsoresults in ine�cient viscous displacement. Di�usionis, therefore, the main recovery mechanism in thesecases. Di�usion in fractured reservoirs, unlike inconventional reservoirs, can signi�cantly a�ect thee�ciency of gas injection in oil reservoirs and recyclingin gas/condensate reservoirs. During gas injection infractured reservoirs, the injected gas is inclined to ow through the fractures and may, therefore, push orvaporize the oil in the fractures before any signi�cantgas penetration into the surrounding matrix. Physical

956 M. Saki et al./Scientia Iranica, Transactions C: Chemistry and ... 22 (2015) 955{966

di�usion, similar to gravity, results in a change in thepath of the injected gas species from the fractures tothe matrix, giving rise to a late breakthrough.

The gas injected into the fractures di�uses intothe resident uid inside the matrix and vaporizes thelight oil fractions. The vaporized oil is then transportedinto the fracture by means of convection and di�usion.The process of di�usion/stripping has been describedby Saidi [1]. Theoretically, the recovery of light andintermediate fractions of oil is total (providing su�cientquantities of gas are injected).

The parameters that in uence di�usion includethe nature of the injected gas, the composition of the oilin place, the presence of water, the fracture intensity,the rate of gas injection, and the geometry of the matrixblocks. Among those, the composition of the oil inplace and the nature of the injected gas are the mostimportant.

Various authors have discussed the e�ect of dif-fusion on oil recovery in fractured reservoirs [2-13].Hua and Whitson [5] assumed laminar, incompressible,and steady state ow of injected gas in the fractures,while ignoring the convection mass transfer betweenthe fracture and the matrix. They also assumed thatinside the fracture, gas stream velocity and physicalproperties are constant and una�ected by di�usion.Finally, they proposed an equation for calculatingmass transfer coe�cients between the gas owing inthe fracture and the resident uid inside the matrixblock.

Jamili et al. [13] simulated experiments numberM5 of Morel et al. [14] and number M25 of LeRomancer et al. [15]. They treated the fracture as aboundary condition for the matrix and discretized thecompositional material balance equations that governthe ow of uid in porous matrix by the �nite di�erencemethod. The di�usion terms are modeled by Fick'slaw. The mass transfer coe�cients between fractureand matrix were calculated by Hua and Whitson [5].Convection between matrix and fracture was de�nedin the model based on Darcy's law. There, theresults of the simulations matched the experimentaldata well.

In this work, we model the mass transfer betweenthe matrix and fracture by considering fractured porousmedia as a single matrix block with an adjacentfracture. The proposed model is a modi�cation of thatof Jamili et al. [13].

Accurate prediction of di�usion coe�cients, in-cluding the o�-diagonal elements, is a key issue. Weuse the multicomponent-di�usion-coe�cient model ofGhorayeb and Firoozabadi [16] to calculate the fulldi�usion matrix as a function of temperature, pressure,and composition. In this work, we use the modelby Leahy Dios and Firoozabadi [17], in place ofHayduk and Minhas [18], in the model of Ghorayeb

and Firoozabadi [16] to predict the in�nite dilutioncoe�cients.

Viscosity and interfacial tension are calculatedfrom Lohrenz et al. [19] and Parachor methods [20],respectively.

First, the governing equations, with regard tomatrix and fracture uid ow, are presented sepa-rately, and afterwards, the matrix/fracture interactionis applied by using appropriate equations to model themass transfer through the matrix/fracture interface (orboundary).

We assumed that there is no oil initially in thefracture and the fracture is fully saturated with gas.When a fracture saturated in gas is in the vicinityof a matrix saturated with undersaturated oil, thegas-gas and oil-oil di�usions will not start becauseof phase discontinuity. The gas oil mass transfer is,therefore, crucial to model di�usion properly. Thecommonly used approach, which is based on the Filmtheory, assumes thermodynamic equilibrium at the gas-oil interface and the continuity of component uxesacross the interface. An appropriate model should beapplicable in the case of the matrix being saturatedwith both saturated and undersaturated oils. Themodel suggested by Hoteit [21] presents such a ver-satility and is used in this paper.

The proposed model is validated by a gas injectionexperiment (experiment no. M5 of Morel et al. [14]).Following model validation, the e�ects on recoveryrate of matrix permeability, initial gas saturation andinjection rate are investigated.

2. Mathematical model

In this work, a fractured porous media is considered asa single matrix block with an adjacent fracture. Thisapproach is a �ne-scale representation of a naturallyfractured reservoir, since it allows one to study the uid ow between the fracture and the matrix block.

2.1. The defects of the model of Jamili et al.[13] and modi�cations

There are several drawbacks within the model of Jamiliet al. [13]:

(a) Using Classical Fick's law: The classical Fick's lawis the most utilized model in reservoir engineeringliterature and in commercial and academic reser-voir simulators [2,22-25]. Current practice is to usethis model in the context of e�ective di�usivity,where di�usion in a multicomponent mixture isassumed to behave as pseudo binary [3]. Thismodel is elegant and simple from a computationalpoint of view and may provide reasonable resultsfor many applications. However, it may not honorthe equi-molar condition which states that the to-tal di�usion ux must be zero. In some cases, this

M. Saki et al./Scientia Iranica, Transactions C: Chemistry and ... 22 (2015) 955{966 957

model may fail to provide an even qualitativelycorrect description of di�usion behavior [26]. Thefailing of the classical Fick's law is a result of ne-glecting the dragging e�ect that is described by theo�-diagonal elements in the di�usion coe�cientmatrix [27]. Krishna and Standart [28] arguedthat the classical Fick's law could be only validfor cases with special conditions, such as ideal bi-nary mixtures and ideal multicomponent mixtureshaving di�usion coe�cients that can be regardedas equal. Nevertheless, these conditions, whichspecify the validity range for the classical Fick'slaw, are su�cient but may not be necessary. Thebottom line is that it is di�cult to anticipate whenFick's law does and does not work. Petroleum uids, even at low pressures, are not ideal becauseof the diversity in size of molecules. A realisticmodel of di�usion should include both diagonaland o�-diagonal entries in the matrix of di�usioncoe�cients. To overcome this issue, we use thegeneralized Fick's law.

(b) Using Hua and Whitson [5] to calculate di�usionmass transfer coe�cients between fracture andmatrix: There are several assumptions in the workof Hua and Whitson [5]: (1) Gas stream velocityand physical properties are constant along thefracture and una�ected by occurring mass transferbetween the matrix and fracture; (2) Incompress-ible and steady state ow along the fracture; (3)The di�usion terms are modeled by classic Fick'slaw; and (4) The convection mass transfer betweenthe matrix and fracture is ignored.

The Hua and Whitson relationship to cal-culate mass transfer coe�cients is inadequate,because the applied assumptions are not alwayssatis�ed. Therefore, we must do our best to modeldi�usion mass transfer by applying more reliableassumptions.

(c) Using Darcy's law to model convection mass trans-fer between fracture and matrix: We know thatmatrix block geometry is an e�cient parameteron mass transfer between matrix and fracture,therefore, Darcy's law is ine�cient for modelingconvection mass transfer. A transfer function thatregards the e�ect of matrix block geometry onmass transfer by shape factor is de�nitely betterthan Darcy's law. The shape factor is related tothe geometry and size of the unfractured matrixblocks and can be used as a tuning parameteragainst well pressure tests or experimental data.

2.2. Governing equations2.2.1. Fluid ow through matrixThe governing equations for the three-phase(gas/oil/water) compositional ow within the porousmatrix are obtained by the species-balance equations,

overall material balance, Darcy's law, generalizedFick's law, the thermodynamic equilibrium betweenthe phases and the constraint equations.

Species-balance equations. Material balance equa-tions govern the transport of each component in oil andgas by convection and di�usion mechanisms:

r:�xc�o

kkro�o

(rPo � orD)

+ yc�gkkrg�g

(rPg � grD)�

+r:�'�oso

nc�1Xk=1

Dockr:xk

+ '�gsgnc�1Xk=1

Dgckr:yk

�+ qD;fm;c + qC;fm;c

=@@t

(�(�osoxc + �gsgyc)) ;

c = 1; 2; :::; nc � 1: (1)

The �rst derivative term represents the convectionmechanism in the oil and gas phases, which is for-mulated by Darcy's law. The second term representsthe di�usion mechanism. The di�usion uxes aremodeled by the generalized Fick's law. qD;fm;c is thedi�usion rate of c between the matrix and fracture thatoccurs at the fracture/matrix boundary. qC;fm;c is theconvection rate of c between the matrix and fracturethat occurs at the fracture/matrix boundary. In thiswork, the convection term, qC;fm;c, is de�ned by themultiphase transfer function as:

qC;mf;c = �sfVbm��oxc

kxkro�o

�m

[Pf � Pom]

+��gyc

kxkrg�g

�[Pf � Pgm]

�; (2)

where �sf is the matrix shape factor that is calculatedas [29]:

� =1Vbm

nXj=1

Ajlj: (3)

Overall material balance equation. SummingEq. (1) for all the species results in an overall materialbalance equation of the form:

r:��okkro�o

(rPo� orD)+�gkkrg�g

(rPg � grD)�


+ncXc=1

qD;fm;c+ncXc=1

qC;fm;c=@@t

(�(�oso + �gsg)) :(4)

Water material balance. It is assumed that hy-drocarbon and water phases are insoluble, therefore,water transportation occurred only via the convectionmechanism through the porous matrix:

r:��wkkrw�w

(rPw � wrD)�

=@@t

(�wsw): (5)

Chemical equilibrium. The hydrocarbon phasesare considered to be in chemical equilibrium throughthe porous matrix. Chemical equilibrium is stated byequality between the fugacities of each species in bothoil and gas phases:

fc;o = fc;g c = 1; 2; :::; nc: (6)

Capillary pressure. The relationships between thegas, oil and water pressures are governed by thecapillary pressures:

Pcog = Pg � Po; (7)

Pcow = Po � Pw: (8)

Constraint equations. The sum of mole fractionswithin the hydrocarbon phases is equal to one. Thesum of the saturations of gas, oil and water is alsoequal to one:

ncXc=1

xc = 1; (9)

ncXc=1

yc = 1; (10)

so + sg + sw = 1: (11)

The multiphase compositional ow through porousmedia is governed by Eqs. (1) to (11). This systemof equations consists of the (2nc + 6) equation andthe (2nc+6) unknown (Po; Pg; Pw; so; sg; sw; xc; yc; c =1; 2; :::; nc).

2.2.2. Fluid ow through fractureWe assumed that there is no oil initially in the fractureand that the fracture is fully saturated with gas. Theinjected gas di�uses into the porous matrix, throughgas and liquid phases. This causes oil to be vaporizedand then transported by convection and di�usion tothe gas owing in the fracture.

The concentration gradient is usually used tocalculate the di�usion mass transfer at the ma-trix/fracture boundary. The owing gas inside the

fracture also a�ects the mass transfer between thefracture and matrix. Therefore, for mass transfercalculations, not only the concentration gradient butalso the e�ect of owing gas inside the fracturemust be considered. Using laminar ow theory, thespecies mass balance inside the fracture is as fol-lows:

r: (yc�gv)�r: �g

nc�1Xk=1

Dgckr:yk

!� qC;mf;c

�qD;fm;c =@@t

(yc�g); c = 1; 2; :::; nc � 1:(12)

Summing Eq. (12) for all species results in an overallMaterial balance equation:

r:(�gv)�ncXc=1

qC;mf;c �ncXc=1

qD;mf;c =@@t

(�g): (13)

2.2.3. Initial conditionsIt is assumed that the system is initially under gravityand chemical equilibriums and that the uid is dis-tributed uniformly through the matrix; therefore, thereare convection and di�usion ow through the porousmatrix.

According to Darcy's law and gravity equilibriumassumption:

(rPp � prD) = 0 p = w; o; g: (14)

For a horizontal plane rD = 0, therefore, @Pp@x and

@Pp@y are equal to zero. This states that the pressure

is constant in a horizontal plane at time zero. For avertical plane rD = 1, therefore, @Pp

@z = p suggestingthat the vertical pressure distribution is given by thecolumn weight. Accordingly, if pressure at a referenceheight is given, then pressure at any point in the modelcan be determined.

According to the generalized Fick's law and chem-ical equilibrium assumption:

nc�1Xk=1

Dockr:xk = 0; c = 1; 2; :::; nc � 1; (15)

nc�1Xk=1

Dgckr:yk = 0; c = 1; 2; :::; nc � 1: (16)

This results in no composition gradient within thematrix; hence rxc;ryc = 0 (c = 1; 2; :::; nc). If uidcomposition and PT (pressure and temperature) con-ditions are known, one can determine the compositionand saturation of hydrocarbon phases through ashcalculations.


2.2.4. Boundary conditionsIn this work, a fractured porous media is consideredas a single matrix block with an adjacent fracture.Many researchers seal some sides of the matrix [5,9,13-15]. This is to say that there are fractures contactingsome sides (hence, a mass transfer between the matrixand fracture), while other sides are in no contact withany fracture (no mass transfer between the matrix andfracture). So, there are two kinds of boundary for aporous matrix block, namely, ow boundaries and no- ow boundaries, also known as sealed boundaries.

Sealed boundaries. The di�usion and convection uxes for any species and any phases (oil, gas andwater) are zero.

Therefore, the convection ux at the boundary is:

�pkkrp�p

(rPp � prD) = 0; p = w; o; g; (17)

and the di�usion ux is:

nc�1Xk=1

Dockr:xk = 0; c = 1; 2; :::; nc � 1; (18)

nc�1Xk=1

Dgckr:yk = 0; c = 1; 2; :::; nc � 1: (19)

Flowing boundaries. At these boundaries, massexchange occurs between the matrix and fracture. It isassumed that there is no oil initially in the fractureand that the fracture is fully saturated with gas.When a fracture saturated in gas is in the vicinityof a matrix saturated with undersaturated oil, thegas-gas and oil-oil di�usions will not start becauseof phase discontinuity. The gas-oil mass transfer is,therefore, crucial to model di�usion properly. Thecommonly used approach, which is based on the Filmtheory, assumes thermodynamic equilibrium at the gas-oil interface and continuity of component uxes acrossthe interface.

Hoteit and Firoozabadi [10] suggested using thecross-phase equilibrium concept at the matrix-fractureinterface in discrete fracture models. This conceptassumes that the fracture gas is in thermodynamicequilibrium with the matrix oil within a thin regionadjacent to the fracture. With this approach, a thin,two-phase region is introduced between the fractureblocks and the matrix blocks (Figure 1). Therefore,gas-in-gas and oil-in-oil di�usions between the matrixand the fracture blocks can occur via the two-phaseregion. The introduced region should be thin enoughto reduce the gridding e�ect. It was found that atreservoir scale with the fracture aperture around 1 mm,for example, the two-phase gridblock thickness should

Figure 1. Cross ow equilibrium concept: Gas-gas andoil-oil di�usions occur across two-phase region [21].

be in the range of 10 cm [10]. This approach providesreasonable accuracy and is simple to apply in single-porosity and discrete-fracture models. However, it isnot clear how to apply this approach in the context ofdual-porosity models [21].

In this work, the approach proposed by Hoteit [21]is used to model mass transfer at the matrix/fractureinterface (or boundary).

In an isothermal system, it is assumed that thereis a thin transition region at the gas-oil contact wherethe two uids totally mix and, consequently, are inchemical equilibrium. If no reaction occurs at theinterface, the continuity of component molar uxesacross the interface holds. Consider the situation wherea fracture gridblock saturated with gas is adjacent to amatrix gridblock saturated with oil. A sketch of the twoblocks is shown in Figure 2. We note that the gas-oilinterface coincides with the matrix-fracture interface,i.e., the gridblock boundary.

An appropriate model should be applicable in thecase of the matrix being saturated with both saturatedand undersaturated oils. The proposed model presentssuch versatility.

According to Hoteit [21], the following is consid-ered for the governing equations of mass transfer at the owing boundary:

Figure 2. Mass transfer at the gas-oil interface: Gas-gasand oil-oil di�usion occur across a thin �lm at theinterface [21].


1. The di�usion and convection molar uxes de�nedfrom both sides of the boundary are continuous,therefore:(a) Continuity of di�usion uxes across the bound-

ary:(Jgc + Joc )m = Jgcf ; c = 1; 2; :::; nc � 1:

(20)

By applying generalized Fick's law to Eq. (20),we have: '�gsg

nc�1Xk=1

Dgck@yk@x

!m

+

'�oso

nc�1Xk=1

Dock@xk@x

!m

= � �g

nc�1Xk=1

Dgck@yk@x

!f;x=0

c = 1; 2; :::; nc � 1; (21)

where:�@yk@x

�f

=yk;mf � ykf

Fa2

; (22)�@xk@x

�m

=xkm � xk;mf

�x2

; (23)�@yk@x

�f

=yk;mf � ykf

Fa2

; (24)

(b) Continuity of convection uxes across theboundary:��

�okxkro�o

@Po@x

�+��gkxkrg�g

�@Pg@x

�m

= �sfVbm��okxkro�o

�m

[Pf � Pom]

+��gkxkrg�g

�m

[Pf � Pgm]�; (25)

where:@Po@x

=Pom � Pboundary

�x2

; (26)

@Pg@x

=Pgm � Pboundary

�x2

: (27)

2. The hydrocarbon phases are under chemical equi-librium, therefore:fc;o (P; T; xc;mf ; c = 1; 2; :::; nc � 1)

= fc;g (P; T; yc;mf ; c = 1; 2; :::; nc � 1)

c = 1; 2; :::; nc: (28)

This system of equations consists of 2nc equa-tions and 2nc unknowns (Pf ; V F; yc;mf ; xc;mf ; c =1; 2; :::; nc � 1).

3. Numerical scheme

The di�erential equations governing compositionalmultiphase ow in porous media are presented in theprevious section. Some of these equations are nonlin-ear. The numerical technique replaces all derivativesby the �nite di�erence approximations resulting in a setof nonlinear algebraic equations. Then, the resultantequations are linearized and solved by the iterativeNewton-Raphson method.

The Young and Stephenson [30] numericalmethod is used as the numerical scheme in this work,which is an IMPESC (implicit pressure/explicit sat-uration composition) type numerical model. Youngand Stephenson [30] de�ned W and F as �wsw and�oso + �gsg, respectively. According to Young andStephenson [30], W;F; Pmo; Pf ; V; zmc; ymc; yfc; c =1; 2; :::; nc � 1 are considered primary variables, andsom; sgm; swm; xcm; c = 1; 2; :::; nc � 1 as secondaryvariables. Transmissibilities and di�usion terms in the ow equations are evaluated explicitly. The values ofthe primary and secondary variables in each time stepare used as the initial guess for next time step.

In this work, matrix and fracture gridding areperformed �rst. The gridding is carried out in such away that no gridblock is found partially in the matrixand fracture.

As mentioned earlier, the matrix/fracture interac-tion is accounted for through matrix/fracture interfacecalculations. Since uid ow through the matrix andfracture are not independent, their governing equationsmust be solved simultaneously. Therefore, the Jacobianmatrix structure is slightly di�erent to that of Youngand Stephenson [30].

The gridblock types can be recognized, withrespect to the number of unknowns:

a) 2nc + 6 unknowns (Po; Pg; Pw; so; sg; sw; xc; yc; c =1; 2; :::; nc) for each matrix gridblock containing twohydrocarbon phases;

b) nc+4 unknowns (Pp; Pw; sp; sw; xc; c = 1; 2; :::; nc; p= oil or gas) for each matrix gridblock containinga single hydrocarbon phase;

c) nc + 1 unknowns (Pf ; yc; c = 1; 2; :::; nc) for eachfracture gridblock containing the gas phase.

To ascertain convergence and stability, the equa-tions and unknowns must be arranged within theJacobian matrix, in proper order. The fracture owequations were added to the Young and Stephenson [30]structure in di�erent ways. It was observed that forthe same initial guess, the precision and convergenceof the solutions are nearly the same, suggesting thatthe fracture ow equations can be added to the Youngand Stephenson [30] structure arbitrarily. In thismethod, the Newton/Raphson scheme is used to solve


for primary variables in each gridblock. The secondaryvariables are then evaluated using primary variables.

4. Model validation

The proposed model is validated by a gas injectionexperiment (experiment no. M5 of Morel et al. [14]).The experiment is designed in one dimension. It isperformed to model the mass exchange between the owing gas in the fracture and the uid saturatingthe horizontal matrix block. Species recovery and gassaturation along the matrix block are measured. Localsaturation through the porous media is measured bythe gamma-ray attenuation method [14]. Jamili etal. [13] conducted this experiment, assuming that sincenitrogen injection is a three component system, the ab-sorption coe�cient, densities and molar compositionsof hydrocarbon phases change continually. This resultin local saturation is impossible to calculate exactlywithout having local uid properties. To override theproblem, they calculated saturations for two extremes:

a) Matrix uid is a binary mixture of methane andpentane;

b) Matrix uid is a binary mixture of nitrogen andpentane.

Obviously, the real process is an intermediatestate between (a) and (b).

Table 1 shows the input data used for simulation.The relative permeability and capillary pressure dataare presented in Table 2.

Binary interaction coe�cients are set to zero.The Peng-Robinson equation of state is used for thephase behavior description. The pressure at thematrix/fracture boundary is assumed to be constantduring simulation and is set equal to the injection pres-sure of 1479 psia. The capillary pressure (Table 2) is

Table 2. Relative permeabilities and capillary pressure[14].

Sg Kro Krg Pcog (psi)

0 1 0 2.228650.1 0.9 0.0002 2.35480.2 0.586 0.004 2.480950.3 0.316 0.02 2.60710.4 0.153 0.045 2.712230.5 0.063 0.1 2.838380.55 0.037 0.15 2.933060.6 0.02 0.21 3.02760.65 0.0096 0.3 3.153750.7 0.0039 0.5 3.27890.8 0 0.9 3.99475

Figure 3. Experiment no. M5 of Morel et al. [14].

reported at reference interfacial tension (2.9 dynes/cm)and must be modi�ed with respect to local interfacialtension:

Pc = P refc

� ��ref

�: (29)

The system is a one-dimensional two phase uid systemsaturated with three components. Figure 3 shows theschematic of the system. All sides of the core are

Table 1. Models input for simulation, experiment no. M5 of Morel et al. [14].

Rock material Paris basin chalkCore length (m) 0.357Core cross section (m2) 0:032� 0:032Core porosity 0.40Core permeability (md) 2Water saturation (%) 0Oil residual saturation (%) 0.2Pressure (psi) 1479Temperature (�C) 38.5Initial gas saturation (%) 0.25Mole fraction C1 0.524Mole Fraction C5 0.476N2 ow rate in the fracture (cm3/hr) 4 until 14.4 days, then 16


closed except the left, where a stream of nitrogen ispassed at a constant rate. Red arrows depict thepossible directions of ow. The uid ows through theporous matrix in the x direction only. Injected gas ows inside the fracture in the z direction, and massexchange between the matrix and fracture occurs inthe x direction. Thus, ow inside the fracture is two-dimensional.

According to Figure 3, the boundary conditionsfor uid ow through the fracture are as follows:

yc = yc;inj at z = 0

yc = yc;mf at x = 0

@yc@x

= 0 at x = Fa (c = 1; 2; :::; nc); (30)

where yc;mf is the mole fraction of component c insidethe gas phase at the matrix/fracture boundary; yc;injis the mole fraction of component c inside the gas phaseat the injection point; and Fa is fracture width at thex direction.

It was assumed that the linear velocity inEqs. (12)-(13) is constant (equal to injection linearvelocity) and una�ected by mass transfer between thematrix and fracture.

In this work, the system is �rst simulated by theproposed model, the results of which are compared tothe experimental data reported by Jamili [13]. Follow-ing model validation, the e�ects of matrix permeability,initial gas saturation and injection rate on recovery rateare investigated.

The core was simulated with 20 grids in the xdirection. The fracture was simulated with 5 grids inthe z-direction. All runs were performed on a 2.5 GHZ,Core 2 Due PC. Runtime was less than 10 minutes forall runs.

Figures 4 and 5 show the methane and pentanerecovery, respectively. As can be seen, the simulationresults well match the experimental data, indicatingthe very good performance of the proposed model in

Figure 4. Variation of the methane molar recoverythrough simulation and experimental results; theexperiment no. M5 of Morel et al. [14].

Figure 5. Variation of the pentane molar recoverythrough simulation and experimental results; theexperiment no. M5 of Morel et al. [14].

Figure 6. Variation of the gas saturation distributionalong the core at t = 8 days through simulation andexperimental results; experiment no. M5 of Morel et al.[14].

predicting the production behavior of methane andpentane.

Figure 6 shows gas saturation distribution alongthe core after 8 days. As expected, the saturationdistribution is placed between two extremes. This alsoindicates that the proposed model predicts saturationdistribution successfully.

Values of the di�usion coe�cients in liquid andgas phases at a corresponding composition for thematrix gridblock adjacent to the fracture at 99.77 atmand 311.65 k are presented in Tables 3 and 4.

5. Numerical analysis and discussion

Following model validation, the e�ects of matrix per-meability, initial gas saturation and injection rate onrecovery rate are investigated.

5.1. The e�ect of matrix permeability onrecovery rate

0.1, 0.5 and 2 md are selected to investigate thepermeability e�ect on recovery rate.

This permeability range of magnitude selected fordi�usion remains a main production mechanism.

Figure 7 shows the permeability e�ect on the


Table 3. Molecular di�usion coe�cients (cm2/sec) in gas phase, 99.77 atm and 311.65 k.

Components C1 C5 N2

C1 2:154� 10�3 6:35� 10�4 �1:12� 10�5

C5 �1:22� 10�5 1:10� 10�3 �1:35� 10�5

N2 �1:39� 10�3 �8:60� 10�4 7:58� 10�4

Composition (mol%) 62.79 4.58 32.63

Table 4. Molecular di�usion coe�cients (cm2/sec) in liquid phase, 99.77 atm and 311.65 k.

Components C1 C5 N2

C1 1:23� 10�3 5:94� 10�4 �3:43� 10�5

C5 1:92� 10�5 1:06� 10�3 �1:53� 10�7

N2 5:12� 10�5 4:51� 10�4 1:06� 10�3

Composition (mol%) 28.26 64.32 7.42

Figure 7. Variation of the methane molar recovery ratethrough simulation results (permeability e�ect),experiment no. M5 of Morel et al. [14].

Figure 8. Variation of the pentane molar recovery ratethrough simulation results (permeability e�ect),experiment no. M5 of Morel et al. [14].

methane recovery rate. The methane recovery ratedecreases negligibly with a decrease in matrix per-meability. This negligible decrement is due to theconvection role in methane production. Since methaneis produced by di�usion mainly [13] and there isno signi�cant relationship between permeability anddi�usion, the change in methane recovery due to changein matrix permeability is not considerable.

Figure 8 shows the permeability e�ect on thepentane recovery rate. Since the main mechanism for

Figure 9. Variation of the methane molar recovery ratethrough simulation results (gas injection rate e�ect);experiment no. M5 of Morel et al. [14].

pentane production is convection [13] and the e�ect ofpermeability on convection is direct (Darcy's law) withan increase in permeability, the pentane recovery rateincreases considerably.

5.2. The e�ect of injection rate on recoveryrate

To investigate the e�ect of gas injection rate on re-covery, simulations were performed with two injectionrates of 4 and 7 cc/hr.

As seen in Figure 9, the recovery rate of methaneincreases as the injection rate increases. This is dueto the e�ect of gas convection inside the fracture onthe mass exchange between the matrix and fracture.The mass transfer coe�cients between the matrix andfracture increase by increasing the injection rate.

Increasing the injection rate also increases therecovery rate of pentane (Figure 10), although not ase�ectively as is the case for methane.

5.3. The e�ect of initial gas saturation onrecovery rate

The initial gas saturation of the matrix block dependson the initial pressure of the matrix. To investigatethe e�ect of this parameter, the initial pressure is


Figure 10. Variation of the pentane molar recovery ratethrough simulation results (gas injection rate e�ect);experiment no. M5 of Morel et al. [14].

Figure 11. Variation of the methane molar recovery ratethrough simulation results (initial gas saturation e�ect);experiment no. M5 of Morel et al. [14].

changed and the initial gas saturation is calculatedby performing ash calculation. The simulation isdone for initial gas saturations of 0.25, 0.30 and0.34.

Figure 11 shows the e�ect of change in initialgas saturation on methane recovery rate. Increasinginitial gas saturation increases the methane recoveryrate negligibly. Two kinds of di�usion process oc-cur at the matrix/fracture boundary, namely, gas-gasdi�usion and oil-gas di�usion. With an increase ingas saturation, the gas-gas di�usion increases and oil-gas di�usion decreases. As the role of the formeris more signi�cant, the recovery rate of methane in-creases.

Figure 12 shows the e�ect of change in the initialgas saturation on pentane recovery rate. It can beseen that an increase in initial gas saturation has asmall e�ect on decreasing the pentane recovery rate.As mentioned before, the main mechanism of pentanerecovery is oil convection, and this decreases with adecrease in oil saturation (relative permeability e�ect).Thus, there is a decrease in pentane with an increasein the initial gas saturation. This is in accordancewith the results obtained by Morel et al. [14], whichcan be seen as another factor in validation of themodel.

Figure 12. Variation of the pentane molar recovery ratethrough simulation results (initial gas saturation e�ect);experiment no. M5 of Morel et al. [14].

6. Conclusions

In order to obtain a better model for mass exchangebetween the fracture and matrix during gas injection,several modi�cations were proposed to the Jamili et alformulation [13].

The validity of the new formulation was veri�edusing experimental data. Then, e�ects of matrix per-meability, gas injection rate and initial gas saturationon recovery rate were investigated. The followingresults were obtained:

� The e�ect of permeability on the recovery rate ofpentane is signi�cant, since the main productionmechanism of pentane is by convection.

� The e�ect of permeability on the recovery rateof methane is negligible, as the main productionmechanism is di�usion for methane.

� The e�ect of initial gas saturation on the recoveryrate of methane and pentane is negligible.

� The e�ect of gas injection rate on the recovery rateof methane and pentane is considerable. An increasein the injection rate leads to greater recovery ratesfor both methane and pentane. The e�ect is morenoticeable in case of methane, due to di�usion beingits main production mechanism.

Acknowledgment

The authors would like to express their gratitude toS.M.R. Pishvaie and all other colleagues from SharifUniversity of Technology who helped in undertakingthis research.

Nomenclature

D Depth, mDij Di�usion coe�cient, m2.s�1

f Fugacity, PaFa Fracture width, m


kr Relative permeabilitync Number of componentsP Pressure, PaPc Capillary pressure, PaPcog Oil/gas capillary pressure, PaPcow Oil/water capillary pressure, Pas Saturationv Linear velocity, m.s�1

V Bulk volume, m3

VF Vapor fractionx Liquid mole fractiony Gas mole fraction

Greek letters

Speci�c weight, Pa.m�1

� Viscosity, Pa.s� Molar density, mol.m�3

� Interfacial tension, N.m�1

� Porosity

Subscripts

b Bulkc ComponentC ConvectionD Di�usionf Fractureg Gasm Matrix blockmf Matrix/fracture boundaryo Oilw Water

Superscripts

g Gaso Oilref Reference

References

1. Saidi, A.M., Reservoir Engineering of Fractured Reser-voirs, Total Edition Press (1987).

2. Coats, K.H., Implicit Compositional Simulationof Single-Porosity and Dual Porosity Reservoirs,SPE18427 (1989).

3. Da Silva, F.V. and Belery, P., Molecular Di�usion inNaturally Fractured Reservoirs: A Decisive RecoveryMechanism, SPE19672 (1989).

4. Thomas, L.K., Dixon, T.N., Pierson, R.G. and Her-mansen, H. \Eko�sk nitrogen injection", SPE (June1991).

5. Hua Hu, Whitson, C.H. and Yuanchang, Q.i., AStudy of Recovery Mechanism in a Nitrogen Di�usionExperiment, SPE21893 (1991).

6. Fayers, F.J. and Lee, S.T., Cross Flow Mechanismsby Gas Drive in Heterogeneous Reservoirs, SPE 24934(1992).

7. Riazi, M.R., Whitson, C.H. and Da Silva, F. \Modelingof di�usional mass transfer in naturally fracturedreservoirs", JPSE., 10, pp. 239-253 (1994).

8. Saidi, A.M., Twenty Years of Gas Injection Historyinto Well-Fractured Haft Kel Field (Iran), SPE35309(1996).

9. Lenormand, R., Le Romancer, J-F.X., Le Gallo, Y. andBourbiaux, B., Modeling the Di�usion Between Matrixand Fissure in a Fissured Reservoir, SPE49007 (1998).

10. Hoteit, H. and Firoozabadi, A. \Numerical modelingof di�usion in fractured media for gas injection andrecycling schemes", SPEJ., 14(2), pp. 323-337 (2009).

11. Alavian, A. and Whitson, C.H., Modeling CO2 In-jection in a Fractured Chalkexperiment, SPE125362(2009).

12. Moortgat, J., Firoozabadi, A. and Morravej, M., ANew Approach to Compositional Modeling of CO2 In-jection in Fractured Media Compared to ExperimentalData, SPE124918 (2009).

13. Jamili, A., Whillhite, G.P. and Green, D.W., Model-ing Gas-Phase Mass Transfer Between Fracture andMatrix in Naturally Fractured Reservoir, SPE132622(2011).

14. Morel, D., Latil, M. and Borbiaux, B., Di�usionE�ect in Gas Flooded Light Oil Fractured Reservoirs,SPE20516 (1990).

15. Le Romancer, J-F.X., De�ves, D.F. and Fernandes,G., Mechanism of Oil Recovery by Gas Di�usion inPresence of Water, SPE27746 (1994).

16. Ghorayeb, K. and Firoozabadi, A. \Molecular, pres-sure, and thermal di�usion in nonideal multicompo-nent mixtures", AIChE J., 46(5), pp. 883-891 (2000).

17. Leahy-Dios, A. and Firoozabadi, A. \Uni�ed modelfor non-ideal multicomponent molecular di�usion co-e�cients", AICHE J., 53(11), pp. 2932-2939 (2007).

18. Hayduk, W. and Minhas, B. \Correlations for predic-tions of molecular di�usions in liquids", Can. J. Chem.Eng., 60, pp. 295-299 (1982).

19. Lohrenz, J., Bray, B.G. and Clark, C.R. \Calculatingviscosities of reservoir uids from their composition",JPT. Trans. AIME., 225, pp. 177-184 (Oct. 1964).

20. Reid, R.C. and Sherwood, T.K., The Properties ofGases and Liquids: Their Estimation and Correlation,3rd Ed., McGraw-Hill (1977).

21. Hoteit, H. \Modeling di�usion and gas-oil mass trans-fer in fractured reservoirs", JPSE., 105, pp. 1-17(2013).

22. Sherivastava, V., Nghiem, L.X., Moore, R. andOkazawa, T. \Modeling physical dispersion in miscibledisplacement-Part 1: Theory and the proposed numer-ical scheme", JCPT., 44(2), pp. 25-33 (2005).


23. Darvish, G.R., Lindeberg, E., Holt, T. and Utne, S.A.,Reservoir-Conditions Laboratory Experiments of CO2

Injection into Fractured Cores, SPE99650 (2006).

24. Kazemi, A. and Jamialahmadi, M., The E�ect of Oiland Gas Molecular Di�usion in Production of Frac-tured Reservoir During Gravity Drainage Mechanismby CO2 Injection, SPE120894 (2009).

25. Alavian, A. and Whitson, C.H., Modeling of CO2

Injection in a Fractured Chalk Experiment, SPE135339(2010).

26. Duncan, J. and Toor, H. \An Experimental study ofthree component gas di�usion", AIChEJ., 8(1), pp.38-41 (1962).

27. Krishna, R. and Wesselingh, J.A. \The Maxwell-Stefanapproach to mass transfer", Chem. Engng. Commun.,59, pp. 33-64 (1991).

28. Krishna, R. and Stangart, G. \Mass and energytransfer in multicomponent system", Chem. Engng.Commun., 3, pp. 201-275 (1979).

29. Zhang, X., Morrow, N.R. and Ma, S. \Experimentalveri�cation of a modi�ed scaling group for spontaneousimbibitions", SPERE, 11, pp. 280-285 (1996).

30. Young, L.C. and Stephenson, R.E. \A generalized com-positional approach for reservoir simulation", SPEJ.,23(4), pp. 727-742 (1983).

Biographies

Mohammad Saki obtained his BS degree in Pro-duction Engineering from the Petroleum University ofTechnology, Iran, and his MS degree in Reservoir Engi-neering from Sharif University of Technology, Tehran,Iran. His research interests include reservoir modelingand simulation, fractured reservoir, phase equilibriumand EOR.

Mohsen Masihi obtained a PhD degree in PetroleumEngineering from Imperial College, London, UK,in 2006. He is presently Associate Professor ofPetroleum Engineering at Sharif University of Tech-nology, Tehran, Iran. His research interests include thepercolation approach in reservoir modeling, fracturedreservoir studies, enhanced oil recovery techniques,reservoir modeling and simulation and reservoir char-acterization.

Seyyed Reza Shadizadeh has a PhD degree inPetroleum Engineering. He is presently Professor ofPetroleum Engineering at the Petroleum Universityof Technology, Tehran, Iran. His research interestsinclude fractured reservoir studies, enhanced oil recov-ery techniques, reservoir modeling and simulation andreservoir characterization.

improved mathematical model for mass transfer in fractured...

Documents