two-phase multicomponent diffusion and convection for ... · two-phase multicomponent diffusion and...

Two-Phase Multicomponent Diffusion andConvection for Reservoir Initialization

Hadi Nasrabadi, Imperial College London*, and Kassem Ghorayeb** and Abbas Firoozabadi, ReservoirEngineering Research Inst. (RERI)

SummaryWe present formulation and numerical solution of two-phase mul-ticomponent diffusion and natural convection in porous media.Thermal diffusion, pressure diffusion, and molecular diffusion areincluded in the diffusion expression from thermodynamics of ir-reversible processes.

The formulation and the numerical solution are used to performinitialization in a 2D cross section. We use both homogeneous andlayered media without and with anisotropy in our calculations.Numerical examples for a binary mixture of C1/C3 and a multi-component reservoir fluid are presented. Results show a strongeffect of natural convection in species distribution. Results alsoshow that there are at least two main rotating cells at steady state:one in the gas cap, and one in the oil column.

IntroductionProper initialization is an important aspect of reliable reservoirsimulations. The use of the Gibbs segregation condition generallycannot provide reliable initialization in hydrocarbon reservoirs.This is caused, in part, by the effect of thermal diffusion (causedby the geothermal temperature gradient), which cannot be ne-glected in some cases; thermal diffusion might be the main phe-nomenon affecting compositional variation in hydrocarbon reser-voirs, especially for near-critical gas/condensate reservoirs (Ghor-ayeb et al. 2003).

Generally, temperature increases with increasing burial depthbecause heat flows from the Earth’s interior toward the surface.The temperature profile, or geothermal gradient, is related to thethermal conductivity of a body of rock and the heat flux.

Thermal conductivity is not necessarily uniform because it de-pends on the mineralogical composition of the rock, the porosity,and the presence of water or gas. Therefore, differences in thermalconductivity between adjacent lithologies can result in a horizontaltemperature gradient. Horizontal temperature gradients in someoffshore fields can be observed because of a constant water tem-perature (approximately 4°C) in different depths in the seabed floor.

The horizontal temperature gradient causes natural convectionthat might have a significant effect on species distribution(Firoozabadi 1999). The combined effects of diffusion (pressure,thermal, and molecular) and natural convection on compositionalvariation in multicomponent mixtures in porous media have beeninvestigated for single-phase systems (Riley and Firoozabadi1998; Ghorayeb and Firoozabadi 2000a). The results from thesereferences show the importance of natural convection, which, insome cases, overrides diffusion and results in a uniform compo-sition. Natural convection also can result in increased horizontalcompositional variation, an effect similar to that in a thermo-gravitational column (Ghorayeb and Firoozabadi 2001; Nasrabadiet al. 2006).

The combined effect of convection and diffusion on speciesseparation has been the subject of many experimental studies.

Separation in a thermogravitational column with both effects hasbeen measured widely (Schott 1973; Costeseque 1982; ElMataaoui 1986). The thermogravitational column consists of twoisothermal vertical plates with different temperatures separated bya narrow space. The space can be either without a porous mediumor filled with a porous medium. The thermal diffusion, in a binarymixture, causes one component to segregate to the hot plate andthe other to the cold plate. Because of the density gradient causedby temperature and concentration gradients, convection flow oc-curs and creates a concentration difference between the top andbottom of the column. Analytical and numerical models have beenpresented to analyze the experimental results (Lorenz and Emery1959; Jamet et al. 1992; Nasrabadi et al. 2006). The experimentaland theoretical studies show that the composition difference be-tween the top and bottom of the column increases with permeabil-ity until an optimum permeability is reached. Then, the composi-tion difference declines as permeability increases. The process ina thermogravitational column shows the significance of the con-vection from a horizontal temperature gradient.

Hydrocarbon reservoirs in general are not in an equilibriumstate mainly because of the geothermal temperature gradient. Tem-perature gradient produces entropy, and, therefore, the criterion ofthe equilibrium cannot be invoked to initialize composition, fluid-phase distribution, and pressure. If there is a gas/oil contact(GOC), one may assume equilibrium only at the interface betweenthe gas and oil phases. The fluids in the gas cap and the oil leg arenot in equilibrium.

Pederson and Lindeloff (2003) have included thermal diffusionin a 1D vertical model for a single phase at steady state to accountfor thermal diffusion in compositional variation in hydrocarbonreservoirs. They express the influence of temperature gradient interms of partial molar enthalpies of fluid components and comparethe results of their model and the isothermal model with measuredcompositional variation from two fields. Their work shows anincrease of segregation from thermal diffusion.

Høier and Whitson (2001) provide a quantitative comparisonbetween various 1D convection-free (without convection) modelsat steady state with thermal diffusion. They also present two fieldcase histories: one in which an isothermal model can describe thecompositional gradients, and another in which the isothermalmodel does not agree with measured data.

Ghorayeb et al. (2003) provide a model for convection-freecompositional variation accounting for pressure, thermal, and mo-lecular diffusion in a two-phase state. However, to the best of ourknowledge, the combined effect of convection and diffusion intwo-phase hydrocarbon reservoirs has not yet been studied.

This paper is organized as follows: first, the mathematicalmodel describing the convection/diffusion flow of two-phase mul-ticomponent fluids is presented. Second, we briefly describe thenumerical approach used to solve this problem. Finally, we presentresults for a binary mixture of C1 and C3 and a 10-componentreservoir fluid.

Mathematical FormulationSpecies Conservation Equation. The mass conservation for com-ponent i in a two-phase, nc-component fluid can be written as (seedetails in Appendix A):

�

�t��czi� = Ui, i = 1, . . . , nc, . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

* Now with Texas A&M U.** Now with Schlumberger.

Copyright © 2006 Society of Petroleum Engineers

This paper (SPE 66365) was first presented at the 2001 SPE Reservoir Simulation Sym-posium, Houston, 11–14 February, and revised for publication. Original manuscript receivedfor review 6 October 2004. Revised manuscript received 18 April 2006. Paper peer ap-proved 31 July 2006.

530 October 2006 SPE Reservoir Evaluation & Engineering

where

Ui = � � �j=1

2 �cjkkrjxji

�j��p + �jgz�� − � ��

j=1

2

�SjJji�� + qi,

i = 1, . . . , nc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

In the above equations, � and c are porosity and total molar den-sity; p, g, k, Sj, cj, qi, krj, xji, Jji, �j, and �j are (respectively)pressure, acceleration due to gravity, permeability, saturation ofphase j, molar density of phase j, molar production rate of com-ponent i per unit volume, relative permeability of phase j, compo-sition of component i in phase j, diffusion flux of component i inphase j, viscosity of phase j, and mass density of phase j. zi is theoverall composition of component i, czi�∑2

j�1Sjcjxji.

Diffusion Flux. The expression for diffusion mass flux of com-ponent i, i�1, . . . , nc–1, in phase j, j�1, 2 is given by:

Jj = −cj�DjM � �x + Dj

T�T + DjP�P�, . . . . . . . . . . . . . . . . . . . . . (3)

where Jj≡ [Jji], DjM≡ [Djik

M ], DjT≡ [Dji

T], DjP≡ [Dji

P], and �x≡ [�xji].Djik

M , DjiT, and Dji

P are the molecular-diffusion coefficients, the ther-mal-diffusion coefficients, and the pressure-diffusion coefficients,respectively. Details of the diffusion coefficients are provided inGhorayeb and Firoozabadi (2000b) and Firoozabadi et al. (2000).The above diffusion coefficients are, in general, different for gasand liquid phases.

To include the effect of porous media on molecular diffusioncoefficients, we use (Brigham et al. 1961; Perkins and Johnston 1963)

D�

D=

1

F�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

where D�, D, and F are the molecular-diffusion coefficient inporous media, the molecular-diffusion coefficient in open space,and the formation resistivity factor. The term 1/F� usually variesbetween 0.15 and 0.7, depending on the lithology of the porousmedia (Pirson 1958; Brigham et al. 1961). For the effect of porousmedia on thermal- and pressure-diffusion coefficients, to the bestof our knowledge, there is not much work in the literature. How-ever, the porous media clearly would have an effect on the ther-mal- and pressure-diffusion coefficients. In this work, we also useEq. 4 for thermal- and pressure-diffusion coefficients. This as-sumption can be justified using the fact that thermal and pressurediffusion are molecular fluxes caused by temperature and pressuregradient, respectively.

Pressure Equation. Using the volume-balance method by Acset al. (1985), we write the pressure equation (see Appendix B):

d�

dp

�p

�t− �CT

�p

�t− �

i=1

nc

VTiUi = 0, i = 1, . . . , nc, . . . . . . . . . (5)

where

VTi = ��Vf

�Ni�

p,Nj, j�i,j=1,...,nc

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

and

CT = −1

Vf��Vf

�p �T,Nj , j=1,...,nc

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

The terms VTi and CT represent two-phase partial molar volumeand two-phase isothermal compressibility, respectively.

The above formulation applies to the general case of a 3Ddomain. In this work, we focus on a 2D vertical rectangular porousmedium of width b and height h (see Fig. 1).

Temperature Profile. We assign a temperature field in the do-main and do not include the energy equation in our calculationsbecause it would unduly complicate the problem. Temperaturemeasurements in a petroleum reservoir with modern tools aremuch easier than composition measurements. Vertical temperature

gradients are often linear based on extensive measurements in theliterature. Horizontal temperature gradients may or may not belinear. We have examined several measured horizontal tempera-tures from fields in different parts of the world. The data indicatelinear temperature gradients in some fields. Note that natural con-vection is from the horizontal temperature gradient. Without thehorizontal temperature gradient, there is no convection.

Similar to Riley and Firoozabadi (1998), we assume that thereservoir is bounded by rock that has constant temperature gradi-ents in the horizontal and vertical directions. We also assume thatthe conductive flow of heat in the reservoir is much greater thanthe convective heat flow. Using these assumptions, the solution ofthe energy equation will have roughly the same temperature gra-dients as the bounding rock (Riley and Firoozabadi 1998). There-fore, for temperature we write T�Txx+Tzz+a, where Tx and Tz arethe temperature gradients in the horizontal and vertical directions,respectively. If we set the temperature at x�x0 and z�z0 equal toT0, then T�Tx(x−x0)+Tz(z−z0)+T0.

Boundary/Initial Conditions. Appropriate boundary and initialconditions are required to complete the two-phase multicomponentformulation. As mentioned earlier, the cross section is assumed tobe bounded by an impervious rock so that the normal total massflux for all the components vanishes at the boundaries:

�j=1

2

�Sj xjivj + SjJji� � n = 0, i = 1, . . . , nc, x = 0, b, and z = 0, h.

. . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

In Eq. 8, n is the unit normal vector. The objective of our work,as we will explain later, is the steady-state solution of the un-steady-state problem. The steady-state solution should be indepen-dent of initial conditions as long as we place the same number ofmoles of each species in the domain. We can use various initialconditions in the above formulation. One can start, for instance,with a constant overall composition (in either a single-phase or atwo-phase domain) and constant pressure in the entire 2D crosssection. One also can use a specified composition and pressuredistribution based on pressure and composition at a reference pointusing the convection-free model presented by Ghorayeb et al.(2003); a summary of the model is provided in Appendix C. Theexamples presented in this work use both approaches for settingthe initial conditions. In our experience, these two types of initialconditions show successful convergence to the steady-state solu-tion. The steady-state solution is independent of the initial condi-tion as long as the total amount of species in the domain is the same.

Model Assumptions and Limitations. This model is limited tothe conditions when there is no filling or leakage; we assume aclosed boundary. We also assume that the reservoir is in a steady-state condition.

Numerical ApproachThere are a number of approaches that can be used in the numeri-cal solution of the above equations. The two basic approaches are

Fig. 1—Geometry.

531October 2006 SPE Reservoir Evaluation & Engineering

the coupled approach and the decoupled approach. In this work,we apply the decoupled approach based on the volume-balancemethod by Acs et al. (1985). Kendall et al. (1983) and Watts(1986) present a similar scheme, adding a sequential implicit step.

In the volume-balance method, phase-equilibrium equations areseparated from pressure and species-conservation equations (Eqs.5 and 1, respectively). In other words, we first calculate the pres-sure and overall composition in the domain and then performphase-equilibrium calculations unlike the coupled approach [e.g.,Young and Stephenson (1983)] that solves all the equations simul-taneously. As a result, we can implement stability analysis in theformulation in a simple way. In the problem that we are solving inthis paper, it is important to avoid any disturbance in convergenceto steady-state solution. Any disturbance from the phase-behaviorcalculations will cause an excessive increase in CPU time. Whenthe phase-behavior computations are decoupled from pressure andspecies-conservation calculations, we have an easier time findingthe source of possible problems in code testing and developmentand can ensure efficient and robust phase-behavior calculations.

The species-conservation equation and the pressure equation indifference form can be written as (see Appendix D):

N min+1 − N mi

n = Vrm�tU min,n+1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

�dV pmn

dp+ V fm

n C Tmn ��pm

n+1 − pmn �

= V fmn − V pm

n + Vrm�t�i=1

nc

V Tmin U mi

n,n+1, . . . . . . . . . . . . . . . . . . . . (10)

where

U min,n+1 = � � �

j=1

2 �cjnkkrj

n xjin

�jn ��pn+1 + �j

ngz��− � � �

j=1

2

�SjnJji

n� − qi, . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

V Tmin = ��V f

n

�Ni�

p,Nj, j� i,j=1,...,nc

, . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

and

C Tmn = −

1

V fmn ��V fm

n

�p �T,Nj , j=1,...,nc

. . . . . . . . . . . . . . . . . . . . . . . . . (13)

The term Vfmn −Vpm

n on the right side of Eq. 10 represents thevolume error carried over from the last timestep and is used tocorrect the error in the pressure equation in the current timestep(Acs et al. 1985). The terms CTm

n and VTmin are calculated with the

Peng-Robinson equation of state (PR-EOS) (Peng and Robinson1976). In this work, there is no source or sink term; therefore, qi in

Eq. 11 is set equal to zero. The diffusion term �� j�1

2

�SjnJji

n� is

evaluated explicitly.In Eq. 11, for the values of cj

n, krjn , xji

n , and �jn at the interface

of the cells, we use single-point upstream weighting. For saturationand diffusion coefficients in the diffusion term (Eq. 11), harmonicaveraging is found more suitable and would increase the stabilityof the system.

In our scheme, first, pressure is calculated implicitly by solvingEq. 10. Then, using Eq. 9, the number of moles of each componentis updated, and the overall composition of each component be-comes readily available. Subsequently, stability analysis is per-formed using the accelerated successive substitution method sug-gested by Michelsen (1982). If the stability analysis indicates thatthe mixture is in a two-phase state, then flash calculations providesaturation and phase compositions of the mixture using a combi-nation of the accelerated successive substitution iterative (ASSI)method and the Newton method (Michelsen and Mollerup 2004).In this work the above method for stability and flash worked well.However, in some complicated problems, one may need to follow

the approach by Hoteit and Firoozabadi (2006) for a robust sta-bility testing. Phase recognition in single-phase systems is an im-portant part of the simulation; it is performed by calculating thecritical temperature of the mixture (Hoteit et al. 2006).

The robustness and efficiency of phase-equilibrium calcula-tions has a direct and strong effect on the performance of ourmodel. Because reaching steady state is the main objective in ourcalculations, the robustness of phase-equilibrium calculations isextremely important; for example, a single failure in a cell in atimestep in phase-equilibrium calculations can delay reaching thesteady state and increase CPU time enormously, and it even cancause divergence in the solution. The phase-equilibrium calcula-tions take a significant part of the total CPU time. Therefore,increasing the efficiency of these calculations can increase theoverall efficiency significantly.

We sometimes have encountered a problem in phase-split cal-culations. When the amount of phase (either gas or liquid) is verysmall, the Newton method diverges frequently to unphysical val-ues. We have solved this problem by returning to the ASSImethod. From our experience, the ASSI method can overcome thedeficiency of the Newton method. Also, for increasing the effi-ciency of the flash calculations, we use the ASSI method only inthe first timestep. For subsequent timesteps, for cells in the two-phase region, we use the K-values of the previous timestep as aninitial guess for the Newton method and perform flash calculationswithout stability-analysis testing. It is known that the Newtonmethod converges quadratically from a good initial guess. Ourexperience shows that K-values from the previous timestep oftenprovide a good initial guess for the Newton method. When theNewton method does not converge to a correct solution, we usestability analysis to check if the mixture is in a two-phase or asingle-phase state. In rare cases in which the stability analysis showsthat the mixture is in a two-phase state, a flash calculation is per-formed with the ASSI method. Using this approach drastically re-duces the computational cost because it not only saves us from doingstability analysis for all the cells at every timestep, but it also de-creases significantly the number of iterations in flash calculations.

The phase recognition is performed only in the first timestep. Ifthe mixture at a mesh point is in a single-phase state at timestep(n–1) and remains single-phase at timestep n, we assume that nophase change has occurred during this timestep. When the mixtureis in a single-phase state at timestep n while it was in a two-phasestate at timestep (n–1) with high gas (liquid) saturation, the mix-ture is gas (liquid) at timestep n. This procedure has been testedextensively (in our examples) by comparing the results with thoseobtained with flash and phase-recognition calculations for all themesh points at each timestep in the entire domain.

Numerical ResultsThe model presented above has been verified with experimentaldata of species separation in a thermogravitational column (Nas-rabadi et al. 2006). In this work, we have selected two sets of ex-amples. In the first set, a binary mixture is considered. In the secondset, a 10-component reservoir fluid is studied in a large reservoir.In the reservoir scale problem, we allow for both homogeneousand layered media with and without anisotropy in permeability.

In these examples, the Peng-Robinson equation of state is usedto calculate the density and phase behavior, and the viscosity iscalculated with the correlation of Lohrenz et al. (1964). We as-sume the term 1/F� to be equal to 0.2 and multiply the molecular-,thermal-, and pressure-diffusion coefficients by this value. The pur-pose of the calculations is to investigate the combined effect of dif-fusion and convection on compositional variation and GOC location.

Binary Example. Let us consider a 2D vertical rectangular porousmedium of width 1000 m and height 200 m saturated with a binarymixture of C1/C3. The vertical and horizontal temperature gradi-ents are −3 K/100 m and 1.5 K/km, respectively. The verticaltemperature gradient of −3 K/100 m is of the order observed inmany reservoirs. The horizontal temperature gradient of 1.5 K/kmis also in line with measured data in offshore Brazilian fields andunpublished data from many reservoirs. We have performed cal-


culations for two permeabilities: k�0.1 and 10 md. The porosityis set to 20% .The gridding considered in this example is 41×41.Various sensitivity studies demonstrated that this level of griddingis adequate for this problem. We use linear relative permeability inthis example. We also tested nonlinear relative permeability(krj�Sj

3) and found that the steady-state results are not affected,although it took longer to reach steady state.

The first initial condition for this system is selected to be theconvection-free state, which is obtained using the model describedin Ghorayeb et al. (2003). The convection-free solution is obtainedon the basis of the pressure and composition of a reference point(located at the center of the cavity: x�500 m, z�100 m). At thereference point, the composition, temperature, and pressure are setat 20% C1/80% C3, T�346 K, and p�57.5 bar (slightly above thebubblepoint pressure, see Fig. 2), respectively. The reason for thechoice of the pressure close to the bubblepoint pressure is to allowfor the GOC to be near z�100 m and x�500 m (the middle ofthe cavity). The predicted molecular-, thermal-, and pressure-dif-fusion coefficients (in open space) at the reference point areDM�5.19E–9 m2/sec, DT–6.35E–13 m2/sec.K, and DP�7.97E–18 m2/sec.Pa, respectively. These values are obtained using themodel suggested by Ghorayeb and Firoozabadi (2000b). Note thatin a binary mixture, one component will be used as reference.Here, we use C3 as the reference. A negative thermal-diffusioncoefficient for methane implies that methane will segregate towardthe hot side, but because of the gravity effect (that is, pressurediffusion), there is more methane segregation to the top.

Fig. 3a presents the contour plot of methane composition at theinitial condition using the convection-free model. The calculationsare first performed for k�0.1 md. The results for k�0.1 md atsteady state are used to perform the calculation for k�10 md. Thesteady state is considered to be reached when the relative differ-ence of pressure and compositions in any point of the cavity doesnot change from one timestep to another within a given relativetolerance (5.0E–11 for pressure and 1.0E–11 for composition).

The contour plot of C1 composition at different times fork�0.1 md using the convection-free model is shown in Fig. 3b.Note that the GOC becomes horizontal toward steady state after

Fig. 2—Pressure-composition diagram for the C1/C3 mixture atT=346 K.

Fig. 3—Plot of C1 composition at different times; convection-free initialization, k=0.1 md, binary mixture, h=200 m, b=1000 m, withthermal diffusion.


natural convection is included. Fig. 4 shows the contour plot of C1

composition at steady state with and without thermal diffusion.One can see that thermal diffusion decreases the variation in com-position (in the liquid column) and results in a more uniformmixture. Fig. 5 shows the velocity profiles at the same times as inFig. 3b. The evolution of the convection cells shown in Fig. 5reveals the development of two main rotating cells (one in the gasregion and one in the oil region) at steady state.

We also assumed a uniform pressure and composition in thedomain as the initial condition. The pressure and composition ischosen such that there is nearly the same total number of moles ofeach component in the entire domain, as in the previous test. Thisleads to p�57.1 bar and 23.8% C1/76.2% C3 at t�0. Here, theentire domain is initially in a two-phase state.

Figs. 6 and 7 show the plot of C1 composition and velocityprofiles at different times for k�0.1 md using a uniform initial

condition, respectively. These two figures show a clear differencebetween the results using the uniform initial condition and theconvection-free initial condition at early times, but close to steadystate, the difference becomes less; at steady state, the results aresimilar, showing that the steady-state solution of the problem ispath-independent. This difference can be seen for the velocityprofiles at 3.0×104 years. In the latter, there is a nearly largetwo-phase region in the middle of the cavity that does not exist inthe former because phase segregation takes longer in uniform ini-tialization. The contour plots of C1 composition at steady state(k�0.1 md) with and without thermal diffusion are the same asthose in Fig. 4; once again, we demonstrate that different initialconditions do not affect the steady-state results.

Next, we study the effect of permeability on segregation. Fig. 8shows the contour plot of C1 composition at different times fork�10 md using the results of k�0.1 md at steady state as theinitial condition. It can be seen that, in the oil column, an increasein permeability increases the uniformity of the mixture. Fig. 9compares steady-state results for k�10 md with and without ther-mal diffusion. It shows a significant effect of thermal diffusion onspecies segregation.

Fig. 10 presents a velocity profile revealing a difference be-tween k�0.1 md and 10 md. Here, there are two main rotatingcells in the gas cap, while for k�0.1 md, there is only one mainrotating cell. However, in the oil column, similar to k�10 md,there is one main rotating cell.

Multicomponent Example. In the past, Ghorayeb and Firooza-badi (2000a) have studied multicomponent convection and diffu-sion in single-phase systems for a near-critical gas/condensate res-ervoir reported by Lee and Chaverra (1998). We use the multi-component fluid and change pressure so that the mixture is in atwo-phase state. Here, we apply a uniform initial condition (i.e.,uniform pressure and composition) in the entire domain. Table 1lists relevant reservoir data as well as the initial uniform pressure,temperature at the reference point, and gridding used in this ex-ample. Table 2 presents the fluid overall composition and criticalproperties, acentric factors, and binary interaction coefficients be-tween methane and other components used for the PR-EOS. Bi-nary interaction coefficients between CO2 and the other compo-nents are 0.150; the remainder of the coefficients are set to zero.The composition of equilibrium phases at the reference point atinitial time (T�422 K and p�150 bar) is listed in Table 3. Pre-dicted molecular-, thermal-, and pressure-diffusion coefficients foreach phase (in open space) at the reference point at initial time(T�422 K and p�150 bar) are presented in Tables 4 through 6.Here, with the multicomponent mixture, we have chosen exampleswith an isotropic homogeneous domain and an anisotropic layereddomain. We use linear relative permeability for the homogeneous

Fig. 4—Effect of thermal diffusion on compositional variation atsteady state; convection-free initialization, k=0.1 md, binarymixture, h=200 m, b=1000 m, with and without thermal diffusion.

Fig. 5—Velocity streamlines at different times; convection-free initialization, k=0.1 md, binary mixture, h=200 m, b=1000 m, withthermal diffusion.


domain and nonlinear relative permeability (krj�Sj3) for the lay-

ered domain. Although the nonlinear relative permeability does notaffect steady-state results, we use it for the layered domain toexamine the performance of our model at unsteady state.

Homogeneous Reservoir. Fig. 11 shows the contour plots ofC1 composition at different times for k�0.1 md. One observes thatthe methane composition varies in the gas cap from nearly 76% atthe top left corner to nearly 73% at the bottom right corner. Thecompositional variation in the oil column is not pronounced.

Fig. 12 shows composition contour plots of C1 and C7+ fork�0.1 md at steady state with and without thermal diffusion.There is considerable compositional variation in the gas region.However, in the oil column, there is less variation in composition.Also, it can be seen that thermal diffusion has a pronounced effecton compositional variation, especially in the oil column.

Heterogeneous Anisotropic Reservoir. We introduced fivehorizontal layers with different permeabilities in a domain to studythe effect of permeability layering. We also included permeabilityanisotropy of kv /kh�0.1, where kv and kh are vertical permeabilityand horizontal permeability, respectively. Fig. 13 shows the dis-tribution of horizontal permeability in the domain. The horizontal

permeabilities of layers are 0.1, 0.05, 1, 10, and 2 md from bottomto top. The thickness of each layer is the same. We also have madea run with kv/kh�1 (results are not shown for the sake of brevity).There is less compositional variation than in the results with per-meability anisotropy.

Figs. 14 and 15 present the contour plots of C1 and C7+ com-position at different times. Comparing steady-state results fromthese figures and Fig. 12 (the results with thermal diffusion in ahomogeneous reservoir) shows the pronounced effect of perme-ability heterogeneity and anisotropy on compositional variation. Itcan be seen that introducing the layers has affected both the trendand values of species distribution significantly.

Conclusions

A model for two-phase multicomponent diffusion and convectionin porous media is presented. Results for the binary and multicom-ponent mixtures show the importance of natural convection on theGOC in a nonisothermal medium. Because of natural convection,the GOC is almost horizontal in the entire 2D cross section. Naturalconvection clearly has an important effect on the phase distribution.

Fig. 6—Plot of C1 composition at different times; uniform initialization, k=0.1 md, binary mixture, h=200 m, b=1000 m, with thermaldiffusion.

Fig. 7—Velocity streamlines at different times; uniform initialization, k=0.1 md, binary mixture, h=200 m, b=1000 m, with thermaldiffusion.


We believe the model presented in this work will set the stagefor the proper initialization of hydrocarbon reservoirs in which thenonequilibrium effects are important.

Nomenclatureb � reservoir width, mc � total molar density, mol/m3

cji � molar density of component i in phase j, mol/m3

cj � molar density of phase j, mol/m3

CT � two-phase isothermal compressibility, 1/PaDij

M � molecular-diffusion coefficient, m2/sDi

p � pressure-diffusion coefficient, m2/s.Pa

DiT � thermal-diffusion coefficient, m2/s.K

fji � fugacity of component i in phase j, PaF � formation resistivity factorg � acceleration of gravity, m2/sh � reservoir height, mJj � vector of molar diffusive flux of phase jJji � molar diffusive flux of component i in phase jk � permeability, m2

krj � relative permeability of phase jn � normal vector

nc � number of componentsNi � total number of moles of component ip � pressure, Paqi � molar production rate of component i per volume,

mol/s.m3

Q � total molar production rate per volume, mol/s.m3

Sj � saturation of phase jt � time, s

T � temperature, KUi � total molar flux of component i per volume, mol/s.m3

vji � velocity of component i in phase j, m/svj � bulk velocity of phase j, m/sV � volume, m3

x, z � coordinatesxji � mole fraction of component i in phase j, dimensionlesszi � overall composition of component i, dimensionless� � porosity

Fig. 9—Effect of thermal diffusion on compositional variation;k=10 md, binary mixture, h=200 m, b=1000 m, with and withoutthermal diffusion.

Fig. 10—Velocity streamlines at different times; k=10 md, binarymixture, h=200 m, b=1000 m, with thermal diffusion.

Fig. 8—Plot of C1 composition at different times; k=10 md,binary mixture, h=200 m, b=1000 m, with thermal diffusion.


�j � viscosity of phase j, kg/m·s�j � mass density of phase j, kg/m3

Subscripts

0 � reference volume elementf � fluidi � componentj � phase

m � general volume elementr � rock

Superscripts

n � timestep

AcknowledgmentsThe funding for this work was provided by the member companiesof the Reservoir Engineering Research Inst. (RERI). Their supportis appreciated. We also thank H. Hoteit of RERI for his help andcomments throughout the work.

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Appendix A—Derivation ofSpecies-Balance EquationThe mass balance for component i in a two-phase, nc-componentfluid reads:

�

�t��

j=1

2

�Sjcji� + � ��j=1

2

�cjivj + SjJji�� − qi = 0,

i = 1, . . . , nc, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-1)where vj and Jji are the bulk velocity of phase j and the molardiffusive flux (molecular, pressure, and thermal) of component i inphase j.


In terms of the mole fraction of component i in phase j, xji, Eq.A-1 can be written as:

�

�t��

j=1

2

�Sjcj xji� + � ��j=1

2

�cj xjivj + SjJji�� − qi = 0,

i = 1, . . . , nc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-2)

The bulk velocity of phase j is given by Darcy’s law:

vj = −kkrj

�j��p + �jgz�, j = 1, 2. . . . . . . . . . . . . . . . . . . . . . . . (A-3)

Note that Eq. A-3 is valid when the capillary pressure is neglected.The total molar density c can be expressed by:

c = �j=1

2

�Sjcj�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-4)

Using Eqs. A-3 and A-4, Eq. A-2 can be written as

�

�t�� czi� = Ui, i = 1, . . . , nc, i = 1, . . . , nc, . . . . . . . . . (A-5)

where

Ui = � � �j=1

2 �cjkkrj xji

�j��p + �jgz�� − � ��

j=1

2

�SjJji�� + qi,

i = 1, . . . , nc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-6)

Appendix B—Derivation of Pressure EquationWe use the concept of volume-balance method by Acs et al.(1985), which states that the total volume of fluid occupying thepore space, Vf , must be equal to the pore-space volume, Vp:

Vf = Vp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-1)Let Vr be the rock volume; then,

Vp = �Vr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-2)From Eqs. B-1 and B-2, the number of moles of component i in thepore space is

Ni = czi�Vr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-3)Eq. B-3 gives

�Ni

�t= Vr

�� czi�

�t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-4)

Fig. 11—Plot of C1 composition at different times; k=0.1 md, multicomponent mixture, h=1.5 km, b=10 km, with thermal diffusion.


From Eq. B-1,

dVf

dt=

dVp

dt, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-5)

but, from Eq. B-2,

dVp

dt= Vr

d�

dp

�p

�t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-6)

Also, we have

dVf

dt=

�Vf

�p

�p

�t+ �

i=1

nc �Vf

�Ni

�Ni

�t. . . . . . . . . . . . . . . . . . . . . . . . . . . (B-7)

Using Eqs. B-5 through B-7, one can derive

�Vf

�p

�p

�t+ �

i=1

nc �Vf

�Ni

�Ni

�t= Vr

d�

dp

�p

�t. . . . . . . . . . . . . . . . . . . . . . (B-8)

Combining Eqs. B-4 and B-8,

�Vf

�p

�p

�t+ Vr �

i=1

nc �Vf

�Ni

�� czi�

�t= Vr

d�

dp

�p

�t. . . . . . . . . . . . . . . (B-9)

By using Eqs. A-5, A-6, and B-2, we can write Eq. B-9 in the form

d�

dp

�p

�t− �CT

�p

�t− �

i=1

nc

VTiUi = 0, i = 1, . . . , nc, . . . . . . (B-10)

where

VTi = ��Vf

�Ni�

p,Nj, j� i, j=1,...,nc

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-11)

and

CT = −1

Vf��Vf

�p �T,Nj, j=1,...,nc

. . . . . . . . . . . . . . . . . . . . . . . . . . . (B-12)

Appendix C—Convection-Free ModelWe restrict the following presentation to phase j and omit thesubscript j for the sake of brevity. The mass diffusion flux basedon the molar average velocity J�(J�1, . . . , J�n−1) in a nonideal fluidmixture of n components reads (Ghorayeb and Firoozabadi 2000b):

J = −c�DM � �x + DT � �T + DP � �P�. . . . . . . . . . . . . . . . . (C-1)

Detailed expressions for the above coefficients are provided inGhorayeb and Firoozabadi (2000b). Eq. C-1 also can be written as

J = −c�D.M.L.W.F.�x +Mnxn

RT2D.M.L.Q�T + D.M.L.V�p�,

. . . . . . . . . . . . . . . . . . . . . . . . . (C-2)

where L≡[Lij] is the matrix of the phenomenological coefficients.The symbols in Eq. C-2 are defined by:

D ≡ �Dij� = � RLii

cMiMnxixnij� i, j = 1, . . . , nc − 1,

F ≡ �Fij� = �� lnfi

�xj�

xj, T, P� i, j = 1, . . . , nc − 1,

M ≡ �Mij� = �Mixi

Liiij� i, j = 1, . . . , nc − 1,

W ≡ �Wij� = �Mjxj + Mnxnij

Mj� i, j = 1, . . . , nc − 1. . . . . (C-3)

In the above expressions, R, fi, and ij denote the universal gasconstant, the fugacity of component i, and the Kronecker delta,respectively; the subscript xj is defined by

�x1, . . . , xj−1, xj+1, . . . , xn−1�. . . . . . . . . . . . . . . . . . . . . . . . . . (C-4)

The column vector Q is given by

Q ≡ �Q*iMi

−Q*nMn� i = 1, . . . , nc. . . . . . . . . . . . . . . . . . . . . . . (C-5)Fig. 13—Distribution of horizontal permeability (kh) in md for the

multicomponent example, h=1.5 km, b=10 km.

Fig. 12—Effect of thermal diffusion on compositional variation at steady state; k=0.1 md, multicomponent mixture, h=1.5 km, b=10km, with and without thermal diffusion.


Qi* is the net heat of transport of component i given by

Q*i = −�Ui

i+��

j=1

n xj�Uj

j� Vi

�j=1

n

xjVi

i = 1, . . . , nc, . . . . (C-6)

where �Ui is the partial molar internal energy departure of com-ponent i and i��Ui

vap/�Uivis; Vi is the partial molar volume of

component i, �Uivap and �Ui

vis are the energy of vaporization andthe energy of viscous flow of pure component i, respectively. �Ui

and Vi are calculated using the PR-EOS; the value of i used in thiswork is fixed equal to 4 based on a previous study by Shukla andFiroozabadi (1998). At steady state, the diffusion flux vanishes;Eq. C-2 then reduces to

W.F.�x +Mnxn

RT2Q�T + V�p = 0. . . . . . . . . . . . . . . . . . . . . (C-7)

At isothermal conditions, Eq. C-7 reads

W.F.�x + V�p = 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (C-8)

which provides the condition for thermodynamic equilibrium in anisothermal n-component mixture. Eq. C-7 is integrated using aforward first-order finite difference scheme to calculate composi-tion and pressure in the entire domain (Ghorayeb et al. 2003).

Appendix D—Discretization

From Eq. B-1, one writes

V fmn+1 = V pm

n+1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D-1)

where n is the timestep number (n+1 stands for current timestep)and m is the block number. Then, using a first-order Taylor ex-pansion, one may obtain:

Fig. 15—Plot of C7+ composition at different times; multicomponent mixture, layered anisotropic media, h=1.5 km, b=10 km, withthermal diffusion.

Fig. 14—Plot of C1 composition at different times; multicomponent mixture, layered anisotropic media, h=1.5 km, b=10 km, withthermal diffusion.


V fmn +

dV fmn

dt�t = V pm

n +dV pm

n

dt�t. . . . . . . . . . . . . . . . . . . . . . . (D-2)

By combining Eqs. B-6, B-7, and D-2, after discretization, we have

V fmn +

�V fmn

�p�pm

n+1 − pmn � + �

i=1

nc ��V fmn

�Nmi��N mi

n+1 − N min �

= V pmn +

dV pmn

dp�pm

n+1 − pmn �. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D-3)

From Eqs. A-5 and B-4, after discretization, one may obtain

N min+1 − N mi

n = Vrm�tU min,n+1. . . . . . . . . . . . . . . . . . . . . . . . . . . . (D-4)

From Eqs. D-3 and D-4, the pressure equation in difference formcan be obtained as:

�dV pmn

dp+ V fm

n C Tmn ��pm

n+1 − pmn �

= V fmn − V pm

n + Vrm�t�i=1

nc

V Tmin U mi

n,n+1, . . . . . . . . . . . . . . . . . . . (D-5)

where

U min,n+1 = � � �

j=1

2 �cjnkkrj

n xjin

�jn ��pn+1 + �j

ngz��− � � �

j=1

2

�SjnJji

n� − qi, . . . . . . . . . . . . . . . . . . . . . . . . . . (D-6)

V Tmin = ��V f

n

�Ni�

p,Nj, j� i, j=1,...,nc

, . . . . . . . . . . . . . . . . . . . . . . . . . (D-7)

and

C Tmn = −

1

V fmn ��V fm

n

�p �T,Nj, j=1,...,nc

. . . . . . . . . . . . . . . . . . . . . . . . (D-8)

SI Metric Conversion Factorsbar × 1.0* E+05 � Pa

ft × 3.048* E−01 � mft3 × 2.831 685 E−02 � m3

°F (°F−32)/1.8 � °C°F (°F+459.67)/1.8 � K

mile × 1.609 344* E+00 � km

*Conversion factor is exact.

Hadi Nasrabadi is a visiting assistant professor in the PetroleumEngineering Dept. at Texas A&M U. e-mail: [email protected]. His research interests include compositional varia-tion in hydrocarbon reservoirs, phase behavior of petroleumfluids, and reservoir simulation. Nasrabadi holds a BS degree incivil engineering from Sharif U. of Technology, Tehran, Iran, anda PhD degree in petroleum engineering from Imperial College,London. Kassem Ghorayeb is a principal research scientist inthe Schlumberger Dhahran Carbonate Research Center,Saudi Arabia, where he heads the EOR research program. Priorto this assignment with Schlumberger, he was involved in differ-ent development projects, where his main responsibility wasfield management, including surface/subsurface integratedmodeling. Before joining Schlumberger, he worked as a scien-tist at the Reservoir Engineering Research Inst., Palo Alto, Cali-fornia, where he conducted research on numerical simulationand mathematical modeling of compositional variation andfluid flow in homogeneous and fractured hydrocarbon reser-voirs. He is the author or coauthor of more than 20 technicalpapers and serves as a Technical Editor for SPEREE. Ghorayebholds BS, MS, and PhD degrees in fluid mechanics from the U.of Toulouse, France. Abbas Firoozabadi is a senior scientist anddirector at RERI. He also teaches at Yale U. and at ImperialCollege, London. e-mail: [email protected]. His main research ac-tivities center on thermodynamics of hydrocarbon reservoirsand production and on multiphase/multicomponent flow infractured petroleum reservoirs. Firoozabadi holds a BS degreefrom Abadan Inst. of Technology, Abadan, Iran, and MS andPhD degrees from the Illinois Inst. of Technology, Chicago, all ingas engineering. Firoozabadi is the recipient of both the 2002SPE Anthony Lucas Gold Medal and the 2004 SPE John FranklinCarll Award.


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