2-d network model simulations of miscible two-phase flow displacements in porous media: effects of...

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Physica A 367 (2006) 7–24

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2-D network model simulations of miscible two-phaseflow displacements in porous media: Effects

of heterogeneity and viscosity

Kristen Stevensona,b,1, Martin Ferera,c,�, Grant S. Bromhala, Jared Gumpc,2,Joseph Wildera, Duane H. Smitha,c

aNational Energy Technology Laboratory, US Department of Energy, Morgantown, WV 26507-0880, USAbDepartment of Statistics, West Virginia University, Morgantown, WV 26506, USAcDepartment of Physics, West Virginia University, Morgantown, WV 26506, USA

Received 12 September 2005; received in revised form 2 December 2005

Available online 18 January 2006

Abstract

There are long-standing uncertainties regarding the relative significance of the role of porous medium heterogeneities vs.

the role of fluid properties in determining the efficiencies of various strategies for fluid injection into porous media. In this

paper, we study both the role of heterogeneities and of viscosity ratio in determining the characteristics of miscible, two-

phase flow in two-dimensional (2-D) porous media. Not surprisingly, we find that both are significant in determining the

flow characteristics. For a variety of statistical distributions of pore-throat radii, we find that the coefficient of variation

(the ratio of the standard deviation of the radii to their mean) is a reliable predictor of the injected fluid saturation as well

as the width of the interfacial region. Consistent with earlier results, we find that viscosity ratio causes a crossover from

fractal viscous fingering to standard compact flow at a characteristic crossover time which varies inversely with viscosity

ratio. The studies in this paper show that the power law relating characteristic time to viscosity ratio does not depend upon

the distribution of pore-throat radii or upon the connectivity (coordination number) of the medium each of which affects

the porosity; this suggests that the power law may be entirely independent of the structure of the porous medium. This

power law relationship leads to a robust dependence of the flow properties upon a particular ratio of the saturation to a

given power of the viscosity ratio. This dependence is reminiscent of the empirical ‘‘quarter power mixing rule’’ in three

dimensions. As such, this work provides a physical understanding of the origin and limitations of this empirical mixing

rule.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Miscible drainage; Heterogeneity; Viscous fingering; Pore-level modeling

e front matter r 2006 Elsevier B.V. All rights reserved.

ysa.2005.12.009

ing author. Department of Physics, West Virginia University, Morgantown, WV 26506, USA. Tel.: +1 304 2826160;

5732.

ess: [email protected] (M. Ferer).

ddress: Dana-Farber Cancer Institute, 44 Binney St., Boston, MA 02115, USA.

Address: Indian Head Division, Naval Surface Warfare Center, Research and Technology Department, 101 Strauss

Head, MD 20640, USA.

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ARTICLE IN PRESSK. Stevenson et al. / Physica A 367 (2006) 7–248

1. Introduction

1.1. Background

Miscible two-phase flow in a disordered porous medium has received considerable attention since the 1950sbecause of its great practical importance in many fields within science and engineering, including hydrology,petroleum and chemical engineering, and physics. Applications include enhanced oil recovery, environmentalremediation processes, carbon dioxide sequestration, determination of pore size distribution of catalysts forreactors, etc. The microstructure of a porous medium affects many macroscopic transport properties of themedium, such as permeability and porosity. Because of the geometrical complexity of the medium’smicrostructure and the complexity of the role of the fluid properties in the microstructure, it is difficult toformulate macroscopic models which accurately include effects arising from microscopic properties. Upscalingof this microscale behavior to macroscopic transport is a defining problem in accurately modeling two-phaseand multi-phase flow scenarios. Characterization of the medium’s microstructure and an understanding of thephysics of flow at the pore level is a first step in determining the upscaling to the macroscopic transport of fluidthrough a porous medium.

Two features cause the macroscopic miscible transport to be less efficient than desired. Low-efficiencyfingering arises both from the injection of a less viscous fluid as well as from a significantly heterogeneousporous medium structure. The flow exhibits very low efficiency, fractal viscous fingering if an inviscid fluid isinjected so that the viscosity ratio, M, is zero, where

M ¼ mi=md (1)

and where mi is the viscosity of the injected fluid and md is that of the displaced fluid. This efficiency is furtherreduced by the amount of heterogeneity of the porous medium. A homogeneous medium is defined as amedium where smaller and smaller subdivisions of the medium have equivalent macroscopic parameter values,such as permeability or porosity. When these properties fluctuate over a range of length scales, the porousmedium is heterogeneous. This characterization is scale-dependent because a medium may be homogeneouson one scale but heterogeneous on a smaller scale.

The ‘‘quarter power mixing rule’’ as used ‘‘extensively in refinery calculations’’[1] is an empirical upscalingof the viscosity ratio effects in miscible floods [2,3]. In the earliest report in the open literature, Koval statedthat the quarter power of a mixture of fluids B and C had been represented as the volume weighted average ofthe components

1

ðmBþCÞ1=4¼

SB

ðmBÞ1=4þ

SC

ðmCÞ1=4

, (2)

where S is the volume fraction of each fluid. Therefore, for a two fluid system such as those considered in thispaper, the effective viscosity of the ‘‘mixture’’ is

miþd ¼ md

,Si

ðMÞ1=4þ 1� Si

!4

, (3)

where M is the viscosity ratio from Eq. (1). Therefore, for small viscosity ratios, the effective viscosity of the‘‘mixture’’ is approximately a function of Si=ðMÞ

1=4.For miscible two-phase flow displacements, fingering of the invading fluid into the defending fluid also

increases with increasing heterogeneity of the porous medium, thereby decreasing the saturation of injectedfluid at breakthrough. Koval’s [1] well-known K-factor method was used to show the effect of heterogeneityon this breakthrough saturation (or percent recovery at breakthrough), and results of the method werecompared with experimental values for miscible displacements. For a range of viscosity ratios, Simon andKelsey [4,5] also used a number of distributions of the tube radii to demonstrate the effect of heterogeneity onbreakthrough saturation for displacements in a network model; they compared their results with results fromcore flood experiments. These simulations showed that as their network heterogeneity factor (H ¼ Rmax=Rmin

where R is the tube radius) increased, breakthrough saturation decreased, due to the increased disorder of the

ARTICLE IN PRESSK. Stevenson et al. / Physica A 367 (2006) 7–24 9

network. Their heterogeneity factor, H, is a measure of the variance of the pore size distribution. Anotherstudy of miscible displacement and disordered network geometry was performed by Siddiqui and Sahimi [6].This study also varied a parameter of pore size distribution, similar to H, denoted as lambda (l), such that theradii of their tubes were distributed uniformly in the interval (1� l, 1þ l). As lambda was increased from 0 to1, the viscous fingering patterns became increasingly random in the limit M ! 0. Other studies have focusedon both the effect of pore size distribution and the effect of length scales by stochastically generating models ofporous media and determining correlation lengths [7,8].

Network models attempt to capture the complex structure of a porous medium by using pore or throat sizedistributions and by spatial correlation of throat or pore sizes. A pore-level network model is used in thisstudy to investigate the combined effect of pore structure (using a number of different distributions of throatareas and by changing the mean and variance of the distribution) and fluid properties (using the viscosityratio, M ¼ mi=md) on displacement efficiency of miscible two-phase flow.

1.2. Model description

Many types of network models have been used to study flow through porous media [6,9–16]. Thetopology and geometry of a porous medium can be represented using different configurations of a networkmodel with various assumptions about the pore structure [12,17,18]. A network consists of pore bodies(sites or nodes) connected by a set of bonds (small tubes or throats). The simple network model used to obtainsimulation results in this work consists of tubes with randomly chosen cross-sectional areas connected topore bodies of fixed size and utilizes a two-dimensional (2-D) diamond lattice structure with a unit length, ‘(see Fig. 1). The formulation of the model utilizes only dimensionless parameters; however, propertiesobtained from the simulations can be scaled to a physical situation by choosing this unit length. Thecross-sectional areas of the tubes connecting the pore bodies have been chosen at random from a number ofdistributions (discussed briefly in the next subsection and detailed in Appendix A), thereby incorporatingsome heterogeneity into the model porous medium. The pore bodies each have a spherical volume of ‘3

given by

V ¼ ‘3 ¼4pr3

3; (4)

the cylindrical pore throats have cross-sectional areas between 0 and ‘2, which are randomly chosen using thedistributions to be discussed. Improved statistics from the simulations are obtained by averaging data, over anumber of realizations (different configurations of throat areas chosen from the same distributions, i.e.changing only the random number seed) for the same input parameters and distributions. Once the throat

Fig. 1. Pore and throat structure of the network illustrating the diamond lattice structure.

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Fig. 2. Details of the pore and throat structure of the network.

K. Stevenson et al. / Physica A 367 (2006) 7–2410

areas are chosen, the conductance of each throat, gthroat, is calculated using Poiseuille’s law:

gthroat ¼‘3

8pmd

� �A2

throat

‘4ðxf þ ð1� xf ÞMÞ

" #, (5)

where ‘ is the unit length, Athroat is the area of the throat, mI is the viscosity of the invading fluid, xf is thefraction of the throat filled with defending fluid and M is the viscosity ratio, Eq. (1).

The pressure field is calculated by assuming incompressibility of the fluids and using volume conservationlaws. By applying the general equation for flow rate to the net flow out of a pore body through the fourthroats, the pressure field can be calculated. Based on the structure of the network and fluid flow shown inFig. 2, one obtains the following equations for the flow rates through each throat:

qi�1; j�1 ¼ gi�1; j�1ðPi; j � Pi�2; j�2Þ, (6a)

qiþ1; jþ1 ¼ giþ1; jþ1ðPi; j � Piþ2; jþ2Þ, (6b)

qi�1; jþ1 ¼ gi�1; jþ1ðPi; j � Pi�2; jþ2Þ, (6c)

qiþ1; j�1 ¼ giþ1; j�1ðPi; j � Piþ2; j�2Þ. (6d)

Requiring that the net volume flow out of a pore must be zero, the following general equation is obtainedfor the pressure in the pore bodies:

Pi;j ¼

Pþ1nx¼�1

Pþ1ny¼�1

giþnx; jþnyPiþ2nx; jþ2nyPþ1

nx¼�1

Pþ1ny¼�1

giþnx; jþny

. (7)

Using Eq. (7), the program iteratively solves the pressure field until there is a small residual difference betweenpressures in subsequent iterations, which is below a tolerance level chosen to minimize error withoutunnecessarily increasing computer run time. After the pressure field has been solved for one iteration, theinvading fluid can advance through a time interval, Dt, which is also chosen to minimize overall simulationtime without unduly sacrificing accuracy [15]. A constant macroscopic flow rate is maintained by appropriatechanges in the pressure drop at each time step. Our use of Poiseuille’s law precludes the inclusion of non-Darcy fluid flow, e.g. inertial effects and turbulence, as is the case for most pore-level modeling.


1.3. Introductory summary of modeling results

In Section 2, the cross-sectional areas of the pore throats in the model are randomly chosen from a numberof specified distributions with a range of means and standard deviations, as shown in Fig. 3: (i) the uniformdistribution, (ii) Beta distributions, (iii) the triangle distribution and (iv) the exponential distribution. All thesedistributions are described in detail in Appendix A. This allows for variations in the heterogeneity of themodel porous media and enables us to study the effect of heterogeneity upon saturation and fingering, both ofwhich affect the efficiency of oil recovery and of other applications.

In an earlier series of papers, we had observed a crossover from fractal viscous fingering to standardcompact/linear flow at a characteristic time, which varies inversely with the viscosity ratio, i.e. the smaller theviscosity ratio, the larger the characteristic time and the longer the crossover to standard behavior takes[19–21]. In these papers on 2-D models, we showed that the same crossover occurred for the average positionof the injected fluid (related to the interfacial position), the interfacial width, as well as the saturation andfractional flow profiles [19–21]. Furthermore, well-past-crossover (i.e. in the compact flow regime), thecrossover leads to definite predictions for the viscosity ratio dependence of the linear interfacial advance, andthe interfacial width, as well as the saturation and fractional flow profiles. Notably, the fractional flow profilewas predicted to be a function of S/M0.068; this prediction was consistent with well-past-crossover results fromour modeling [21]. This is suggestive of a 0.068 power mixing rule for 2-D miscible systems. Indeed, our earlyresults for small three-dimensional (3-D) systems produced a power in the range 0:16! 0:21 much closer tothe empirically observed value of one fourth [22]. However, all of these earlier results were for a uniformdistribution of randomly chosen throat areas on a square lattice. It is essential to determine if this power(0.068) depends upon the structure of the porous medium before we argue that we have determined the originof the empirical mixing rule. To this end, in Section 3, we present results exhibiting the crossover not only for alog-normal distribution of throat radii on a square lattice, but also for a uniform distribution on honeycomblattice with threefold coordination (connectivity). If these significant changes in the distribution of throat radiiand coordination have no observable effect upon the power law dependence, this would strongly support thehypothesis that this power law is as universal as the empirical ‘‘quarter power’’ seems to be in 3-D. Other than

Fig. 3. Parameter space for the mean and standard deviation of the distributions of cross-sectional throat areas used in the simulations.


just providing a 2-D ‘‘mixing rule’’, this work also provides a physical understanding of the ‘‘mixing rule:’’ (i)it is a result of the fractal to compact crossover, (ii) the ‘‘mixing rule’’ does not really require actual mixing ofthe two fluids because there is no real mixing in our model and (iii) the ‘‘mixing rule’’ rule is only really validwell-past-crossover and not in the fractal fingering regime.

2. Effects of heterogeneity

The following sections discuss the effect of using distributions with different variances and means for therandomly chosen cross-sectional throat areas. These different distributions of throat areas represent differentlevels of heterogeneity of the porous medium. For example, a sharply peaked distribution with only one valueof the throat areas (zero variance) represents a porous medium which is homogeneous at all scales. However,as the variance of a distribution increases, the level of heterogeneity of the porous medium also increases.Clearly, one expects the heterogeneity to affect the flow as evidenced by the saturation at breakthrough, theeffective fractal dimension and the interfacial width. We will use the standard deviation (s, the square root ofthe variance) and mean /AS to characterize the heterogeneity of the network. The modeling in this sectionsimulated miscible floods to breakthrough in 70� 70 networks where the throat areas were chosen randomlyfrom the distributions discussed in detail in Appendix A.

Flow patterns at breakthrough of the injected fluid with their corresponding effective fractal dimension areshown in Fig. 4 for a range of viscosity ratios, M ¼ 0:0001–1 for the first realization of each set of simulations.In this figure, increased heterogeneity of the porous medium is due to increasing the standard deviation of thedistribution of cross-sectional throat areas, while keeping the mean throat area fixed at hAi ¼ 0:5. As thestandard deviation of the distribution is increased, the breakthrough saturation of the injected fluid and theeffective fractal dimension both decrease for a given value of viscosity ratio. This is similar to the trend shownby Simon and Kelsey [4]; as their heterogeneity factor (a measure of the variance of their distribution)increased, their values of breakthrough saturation decreased. This effect is seen in the infiltration patterns in

Fig. 4. Visualizations and fractal dimensions for the first realization of each simulation with increasing standard deviation, a constant

mean hAi ¼ 0:5 and a range of viscosity ratios. The effective fractal dimension shown results from standard box counting on the

breakthrough flow patterns.


Fig. 4. As expected for each distribution, the breakthrough saturation increased with viscosity ratio, due toreductions in the viscous fingering [5,11,23]. For the smallest viscosity ratios, the fingering is dendritic (lessrandom with flow along straighter paths) for the smaller variances but then becomes more and more randomwith increasing variance. This effect was observed a number of years ago [10]; Fig. 4 shows how the increasedrandomness persists in larger viscosity ratios.

Fig. 5 shows the effect of changing the mean value of the throat cross-section as well. For the smaller mean,the patterns tend to be more random. This can be understood, by realizing that the larger throats dominate theflow and that the distributions with a smaller mean have a broader distribution of large throats than do thedistributions with a larger mean value, see Fig. A1 in Appendix A. Therefore, the randomness or heterogeneityincreases with increased variance but decreases with increased mean. This suggests that the coefficient ofvariation defined as

CV ¼ s=hAi (8)

might provide a measure of the heterogeneity.Fig. 6 shows that the breakthrough saturation decreases as the coefficient of variation increases, for all the

viscosity ratios. For the larger viscosity ratios, where fingering is less important, the dependence seemsessentially linear.

However, for the smaller viscosity ratios, there is less variation in the dendritic (small CV) regime than thereis in the more random (larger CV) regime. Fig. 7 shows the breakthrough saturation vs. interfacial width forall values of viscosity ratio and heterogeneity. The interfacial width was determined using the methoddescribed in earlier references, where we showed that an expression involving first and second moments of theinjected fluid (i.e. /xS and /x2S) would reproduce the width of a standard flow pattern, i.e. one having asaturation that was constant behind the interface and decreasing smoothly through the interface to zero

Fig. 5. Breakthrough flow patterns and fractal dimensions for different means and standard deviations. The mean is varied while keeping

the standard deviation constant by reversing the parameters in Beta(a;b), as defined in Appendix A. A comparison is also made between

the distributions with the same mean and different standard deviations, for example the Beta(3; 6) vs. the Beta(1; 2).

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0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Sb M=0.0001Sb M=0.001Sb M=0.01Sb M=0.1Sb M=1

SB

CV

Fig. 6. Saturation at breakthrough vs. coefficient of variation. The solid lines show the results of fitting a quadratic polynomial to the

data.

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1

W M=0.0001W M=0.001W M=0.01W M=0.1W M=1

<W

>

SB

Fig. 7. Interfacial width at breakthrough vs. the breakthrough value of the injected fluid’s saturation for all viscosity ratios and

heterogeneities.


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6

8

10

30

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

W M=0.0001W M=0.001W M=0.01W M=0.1W M=1

WB

CV

Fig. 8. For compact flow, M40:01, the interfacial width increases with the coefficient of variation. For more fractal flow, Mo0:01, theinterfacial width is effectively constant with negligible variation between the dendritic growth at small values of CV and the random fractal

growth at larger values of CV.

K. Stevenson et al. / Physica A 367 (2006) 7–24 15

[16,20]. Not surprisingly, there is a consistent decrease of interfacial width with increasing values ofbreakthrough saturation.

Given the correlation between the value of saturation at breakthrough and the interfacial width atbreakthrough, one would expect a consistent dependence of the interfacial width upon the coefficient ofvariation. Fig. 8 shows this dependence. As expected, the interfacial width decreases with increasing viscosityratio. In the compact flow regime, i.e. the largest viscosity ratios, the interfacial width increases with increasingheterogeneity. However, for the three smallest viscosity ratios, the interfacial width is essentially constant,indicating that there is little difference in the interfacial width between the dendritic fingering patterns (e.g.M ¼ 0:0001 and small CV in Figs. 4 and 5) and the more random fingering for the larger heterogeneities inFigs. 4 and 5.

3. Viscosity ratio effects: viscous fingering to compact crossover

A number of years ago, we had studied miscible floods for viscosity ratios less than unity where the throatareas were chosen from the uniform distribution. The study was motivated by the discovery that for very smallviscosity ratios the flows were described by diffusion limited aggregation (DLA), which produces self-similarfractals where the interface advances faster than linearly with time because of the fingering [10,24]. Indeed, fornet flow in the x direction, the relation between the average position, /xS, of the injected low-viscosity fluidand the amount of that fluid, m (proportional to time, t, for constant injection rate) is given by the fractaldimension, Df ,

hxi ¼ cðm=W Þ1=ðDf�1Þ ¼ ct1þe, (9)

where c is an undetermined constant, W is the width of the porous medium along which fluid is injected and1=ðDf � 1Þ ¼ 1þ e � 1:4 for DLA in 2-D. Summarizing our earlier results for viscosity ratios greater thanzero, we found that the initial flows were DLA fractals, but that the flow behavior would crossover to


compact/linear behavior at a characteristic time inversely related to the viscosity ratio [19–22]. That is, thelarger the viscosity ratio, the sooner the flows would become compact.

In an attempt to determine the viscosity ratio dependence of this characteristic time, we found that thefollowing leading power law dependence of the characteristic time,

t /M�0:17, (10)

represented the longer-time data quite well. However, to best represent the viscosity ratio dependence of all thedata, we found that a correction to this leading dependence was needed to represent the intermediate-timedata. Effectively, this correction provides a viscosity ratio-dependent shift of the time origin that isproportional to the characteristic time in Eq. (10), so that the data for the average position (from Fig. 9) canbe represented as a function

hxi ¼ Ct1:4wðvÞ (11)

of the variable

v ¼ ftþ b=M�0:17g=M�0:17. (12)

As shown in Fig. 10, use of the value b ¼ 8, in these expressions accurately represents the viscosity ratiodependence of the data in Fig. 9.

Furthermore, we showed that this dependence of average position upon time and viscosity ratio given inEqs. (11) and (12) as shown in Fig. 10 leads to definite predictions for the saturation and fractional flowprofiles for times well past crossover when the flows were compact [21]. For these compact flows, we showed

x/t (3)

x/t (10)

x/t (30)

x/t (100)

x/t (300)

x/t (10,000)

1/M

2.5

2

1.5

1

0.5

x / t

0 10 20 30 40 50 60 70 80t

Fig. 9. Average position of the injected fluid divided by the time will be constant for compact flow, but will grow as te � t0:4 for DLA

fractal flow. All of the flows, initially, follow the fractal dependence but then break away approaching a constant at a characteristic time,

which decreases with increasing viscosity ratio.

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100.1

0.3

0.6

1

x/t

1.4

v = {t+8/M-0.17}/M-0.17

x/t**1.4 (3)

x/t**1.4 (10)

x/t**1.4 (30)

x/t**1.4 (100)

x/t**1.4 (300)

x/t**1.4 (10,000)

1/M

Fig. 10. The data from Fig. 9, are shown plotted vs. the variable in Eq. (12). The data show the functional form of wðvÞ in Eq. (11).


that the fractional flow should depend on one variable incorporating both saturation an viscosity ratiothrough the characteristic time, specifically

F ¼ IðteSÞ. (13)

Fig. 11 shows data for the fractional flow from times which are well past crossover plotted vs. the scaledvariable in Eq. (13). The predicted behavior does collapse the data to one curve, indicating that the factorte �M0:068 correctly incorporates the viscosity ratio dependence.

As mentioned earlier, this is reminiscent of the empirical 3-D ‘‘quarter power mixing rule’’. However, to beas robust and widely applicable as the 3-D ‘‘quarter power mixing rule’’, it is essential that this behavior doesnot depend upon the structure of the porous medium. To date, we have only shown that it applies to a squarelattice (coordination number four) porous medium with a uniform distribution of throat areas. Todemonstrate the robustness of this 2-D ‘‘0.068 power mixing rule’’, we have performed simulations on (i) asquare lattice porous medium with a log-normal distribution of throat radii and (ii) a honeycomb lattice(coordination number three) porous medium with a uniform distribution of throat radii. As we will show, thepower law in Eq. (10) is the same (i) for the two very different distributions of throat radii and (ii) for the twodifferent coordination numbers, thereby supporting our assertion that the power law (Eq. (10)) for thecharacteristic time is independent of the distribution of throat radii and of the coordination number ofthe porous medium. This supports our assertion that our 2-D ‘‘0.068 power mixing rule’’ is as robust as theempirical 3-D ‘‘quarter power mixing rule’’.


3.1. Viscous fingering to compact crossover: log-normal distribution

For the uniform distribution of the cross-sectional areas of the throats, the probability of finding aparticular throat radius increases linearly with the value of the radius, r; this distribution is shown inFig. 12(a). Fig. 12(b) shows a log-normal distribution of throat radii.

Having performed simulations for this log-normal distribution, we determined the average position as afunction of time for a wide range of viscosity ratios using the same procedures as were used in generating Fig.9 for the uniform areal distribution. We were able to collapse these data using the same power law exponent inEq. (10), by changing only the value of the constant b in the time-origin shift in Eq. (12) from 8 to 16. Againthis time-origin shift does not affect the long time collapse, only the shorter time collapse. The results areshown in Fig. 13.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5

Uni

form

Dis

trib

utio

n fo

r A

reas

Throat Radius

Probability

IntegratedProbability

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5

Log

Nor

mal

Dis

trib

utio

n w

=0.

5

Throat Radius

Probability

Integrated

Probability

(a) (b)

Fig. 12. Probability of finding a particular value of the throat radius as well as the integrated or cumulative probability. (a) A uniform

areal distribution; (b) a log normal distribution with width 0.5 and most probable value of radius, r ¼ 0:2525.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

M=1/3M=1/10M=1/30M=1/100

F(x

,t,M

)

��S(x,t,M )

Fig. 11. Data for fractional flow and saturation profiles from times well-past-crossover so that the flows exhibiting compact behavior are

plotted to test the dependence predicted in Eq. (13).

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1

0.10.11

x/t

1.4

[t+16/(M-0.17)]/M-0.17

M

1/10

1/30

1/100

1/300

1/10K

Fig. 13. Shows the collapse of the average position data from simulations of the square lattice porous media with a log normal distribution

of throat radii.


The collapse of the data is just as good as that in Fig. 10, showing that the power law for characteristic timeis the same for these two very different distributions of throat radii. Of course, we do not intend to say that thedistributions do not affect the flow behavior: the value of the constant b in the time-origin shift is verydifferent; at short times the value of c ¼ hxi=t1:4 is larger; and the crossover occurs for a larger value of thescaled time variable.

3.2. Viscous fingering to compact crossover: honeycomb lattice

For the same viscosity ratios considered above, we have performed simulations for this honeycomb latticeof pore bodies using the uniform distribution of the cross-sectional areas of the throats (Fig. 14). Using thesame power law for the characteristic time, Eq. (10), we were able to include all of the viscosity ratio and timedependence (i.e. collapse the data) into the one variable in Eq. (12) by simply changing the value of theconstant b in the time-origin shift from 8 to 10. These results are shown in Fig. 15.

Again, the collapse of the data is just as good as that in Figs. 10 and 13, showing that the power law forcharacteristic time is the same for the two very different distributions of throat radii. Again, this does not saythat the distributions do not affect the flow behavior. As before, the same constants (e.g. c and b) are changed,but the power law dependence of the characteristic time upon viscosity ratio is unchanged, thereby preservingthe ‘‘0.068 power mixing rule’’.

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0

0.05

0.1

0.15

0.2

0.25

0.3

1 10

M=1/10000M=1/1000M=1/300M=1/100M=1/30M=1/10

<x

>/t

1.4

(t+10/M-0.17)/M-0.17

Fig. 15. Collapse of the data for the honeycomb lattice using the same power law dependence of the characteristic time, changing only the

value of b in the time-origin shift in Eq. (12).

Fig. 14. A section of the honeycomb lattice used for the simulations in this subsection. Note that this lattice has coordination number

three, where each pore body is connected to three neighboring pore bodies; unlike the square lattice, where each pore body is connected to

four neighboring pore bodies.


4. Conclusions

For the random systems considered, the effects of heterogeneity upon flow characteristics seem to beadequately described by the coefficient of variation, CV in Eq. (8). As the standard deviation (a measure ofmedium heterogeneity) of the distributions of cross-sectional throat areas increased, the effective fractaldimension decreased and more extensive fingering decreased the breakthrough saturation of the injected fluid.Conversely, as the mean of the distribution increased, this breakthrough saturation and fractal dimension


increased. Both effects are satisfactorily incorporated in the coefficient of variation as seen in Figs. 6 and 8,which show that the saturation and interfacial width are relatively smooth functions of CV. It should be notedthat increasing the mean or the standard deviation increases the porosity. However, the porosity is not simplyrelated to the coefficient of variation since this increases with the standard deviation but decreases with themean. We have not considered the effect of correlated heterogeneities.

In Section 3, we have demonstrated the robustness of the DLA-to-compact crossover. Specifically, we haveshown that the same 0.17 power of viscosity ratio describes the characteristic time for two very different pore-throat distributions and for two different connectivities, all of which have different porosities. These resultssupport the ‘‘universality’’ of the exponent of the power law, Eq. (10), i.e. its independence of the structure ofthe porous medium.

Of course other aspects of the flow are affected by the porous medium structure: (i) the constant, c, inEq. (11) multiplying the t1.4 in the DLA behavior, (ii) the constant, b, in Eq. (12) in the time-origin shift and(iii) the constant determining for what value of the variable v in Eq. (12) the crossover occurs.

Still, this work supports our assertion that for any given, 2-D porous medium, the power law in Eq. (10)accounts for the viscosity ratio dependence of the characteristic time, and that the factor te �M�0:068 inEq. (13) accounts for the viscosity ratio dependence of the fractional flow. Therefore, we have determined the‘‘0.068 power mixing rule’’, a 2-D version of the 3-D ‘‘quarter power mixing rule’’ in that the well-past-crossover flow properties depend on the saturation, S, and viscosity ratio through the combination S/M0.068 .Furthermore, we have provided a physical understanding of the ‘‘mixing rule:’’ (i) it is a result of the fractal tocompact crossover (ii) the ‘‘mixing rule’’ does not really require actual mixing of the two fluids because there isno real mixing in our model and (ii) the ‘‘mixing rule’’ is only really valid well-past-crossover and not in thefractal fingering regime. Our early preliminary work on 3-D miscible flow suggested a value of the power in te

closer to 0.25 than the 2-D value of 0.068. We intend to perform simulations on larger 3-D systems to see howclose a more reliable modeling result is to the empirical value of 0.25.

Acknowledgements

K. Stevenson gratefully acknowledges the support of the US Department of Energy through the NETLUniversity Partnership Program. M. Ferer and J. Gump gratefully acknowledge the support of the USDepartment of Energy, Office of Fossil Energy. M. Ferer also acknowledges helpful discussions with M. Piri.

Appendix A

For a medium to be considered homogeneous, samples of a given size, taken from the medium, should havestatistically indistinguishable pore structures, so that the distribution of pore sizes for these samples shouldhave a small variance (as defined in this work). In contrast, samples of a given size, which are heterogeneous,show statistically different pore morphologies; the pore size distribution obtained from sampling in this casewould have a larger variance compared with that of a homogeneous medium. Obviously, a medium may behomogeneous on a larger scale but heterogeneous on a smaller scale, so that determining whether a medium ishomogeneous or heterogeneous is largely scale dependent. Macroscopic porous media normally encounteredin application are usually considered to be heterogeneous, due to the spatial variability of the pore structureand variation of permeability or porosity on multiple length scales. The distributions used in Section 2 of thepaper are defined in the following sections.

A.1. Uniform distribution

Let X denote a random variable (like cross-sectional area) that has a uniform distribution in the interval a tob, denoted X~Uniformða; bÞ. Then the probability of finding a particular value of X is given by

f ðxÞ ¼1

b� a; apxpb (A.1)


with mean and variance:

hxi ¼aþ b

2, (A.2)

s2 ¼ðb� aÞ2

12. (A.3)

A.2. Beta distribution

Let X denote a random variable that has a beta distribution, denoted X~ Betaða;bÞ. Then the probability offinding a particular value of X is given by [25]

f ðxÞ ¼Gðaþ bÞGðaÞGðbÞ

xa�1ð1� xÞb�1; 0oxo1, (A.4)

where

GðaÞ ¼ ða� 1Þ! (A.5)

for integers a and similarly for b. The mean and variance of a beta distribution are given as follows:

hxi ¼a

aþ b, (A.6)

s2 ¼ab

ðaþ bþ 1Þðaþ bÞ2. (A.7)

It should be noted that a beta distribution with parameter a ¼ 1, b ¼ 1 has a uniform distribution, f ðxÞ ¼ 1,on the interval (0, 1). Additionally, reversing the parameters for a beta distribution gives different means, butdoes not change the variance. For example, a Beta(1,2) has a mean of 0.333 and a Beta(2,1) distribution has amean of 0.667; however, the variance is the same in both cases, 0.0556.

A.3. Exponential distribution

Let X denote a random variable that has an exponential distribution, denoted X~ Exp(l). Then theprobability of finding a particular value of X is given by

f ðxÞ ¼ le�lx; 0oxo1 (A.8)

with parameter l with the following form [25]: with mean and variance:

hxi ¼1

l, (A.9)

s2 ¼1

l2. (A.10)

A.4. Triangle distribution

Let X denote a random variable that has a triangle distribution, denoted X~ Triangle(a, b). Then X follows atriangle distribution with parameters a and b (denoting the interval) with the following form [26]:

f ðxÞ ¼2ðx� aÞ

ðb� aÞðc� aÞif apxpc, (A.11)

ARTICLE IN PRESS

0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

Beta(2,18)Beta(3,6)

Beta(6,6)Triangle

Beta(6,3)Beta(18,2)

f(x)

x

Fig. A1. Probabilities of finding a particular value of x in the range 0–1 for a number of distributions discussed in the Appendix and used

in Section 2 of the paper.


2ðb� xÞ

ðb� aÞðb� cÞif cpxpb, (A.12)

where apcpb: with mean:

hxi ¼ðaþ bþ cÞ

3. (A.13)

The triangle distribution can also be obtained from a transformation using two random variablesdistributed uniformly. Let X1 and X2 be two independent random variables each having a uniformdistribution. Then, the variable X ¼ ðX 1 þ X 2Þ=2 would have a triangular distribution with a mean of 0.5 onthe interval [0, 1] (Fig. A1).

References

[1] E.J. Koval, A method of predicting the performance of unstable miscible displacement in heterogeneous media, Soc. Pet. Eng. J.—

Trans. AIME 228 (1963) 145–154.

[2] M.R. Todd, W.J. Longstaff, The development, testing, and application of a numerical simulator for predicting miscible flood

performance, JPT (1972) 874–882.

[3] F.J. Fayers, M.J. Blunt, M.A. Christie, Accurate calibration of empirical viscous fingering models, Rev. L’Inst. Fr. Petrol. 46 (1991)

311–325.

[4] R. Simon, F.J. Kelsey, The use of capillary tube networks in reservoir performance studies: I. Equal viscosity miscible displacements,

SPEJ 11 (1971) 99–112.

[5] R. Simon, F.J. Kelsey, The use of capillary tube networks in reservoir performance studies: II. Effect of heterogeneity and viscosity

ratio on miscible displacement efficiency, SPEJ 12 (1972) 345–351.

[6] H. Siddiqui, M. Sahimi, Computer simulations of miscible displacement processes in disordered porous media, Chem. Eng. Sci. 45

(1990) 163–182.

[7] U.G. Araktingi, F.M. Orr, Viscous fingering in heterogeneous porous media, in: 63rd Annual Technical Conference and Exhibition of

Petroleum Engineers, SPE 18095, Houston, TX, 1988.


[8] J.R. Waggoner, J.L. Castillo, L.W. Lake, Simulation of EOR processes in stochastically generated permeable media, in: 11th SPE

Symposium on Reservoir Simulation, Anaheim, CA, 1991.

[9] I. Fatt, The network model of porous media, Trans. AIME 207 (1956) 144–164.

[10] J.-D. Chen, D. Wilkinson, Pore-scale viscous fingering in porous media, Phys. Rev. Lett. 55 (1985) 1892–1895.

[11] R. Lenormand, E. Touboul, C.J. Zarcone, Numerical models and experiments on immiscible displacements in porous media, J. Fluid

Mech. 189 (1988) 165–187.

[12] M. Blunt, P. King, Macroscopic parameters from simulations of pore scale flow, Phys. Rev. A 42 (1990) 4780–4787.

[13] Y.C. Yortsos, B. Xu, D. Salin, Phase diagram of fully developed drainage in porous media, Phys. Rev. Lett. 79 (1997) 4581–4584.

[14] E. Aker, et al., A two dimensional simulator for two phase flow in porous media, Transport Porous Media 32 (1998) 163–186.

[15] M. Ferer, G.S. Bromhal, D.H. Smith, Pore-level modeling of immiscible drainage: validation in the invasion percolation and DLA

limits, Physica A 319 (2003) 11–35.

[16] K. Stevenson, et al., Miscible, vertical network model 2-D simulations of two-phase flow displacements in porous media, Physica A

343 (2004) 317–334.

[17] M.J. Blunt, et al., Detailed physics, predictive capabilities, and macroscopic consequences for pore-network models of multiphase

flow, Adv. Water Res. 25 (2002) 1069–1089.

[18] P.E. +ren, S. Bakke, O.J. Arntzen, Extending predictive capabilities to network models, SPEJ 3 (1998) 324–336.

[19] M. Ferer, et al., Crossover from fractal to compact growth from simulations of two phase flow with finite viscosity ratio in two-

dimensional porous media, Phys. Rev. E 47 (1993) 2713.

[20] M. Ferer, D.H. Smith, The dynamics of growing interfaces from the simulation of unstable flow in random media, Phys. Rev. E 49

(1994) 4114.

[21] M. Ferer, et al., The fractal nature of viscous fingering in two-dimensional pore level models, AIChE J. 49 (1995) 749.

[22] M. Ferer, J. Gump, D.H. Smith, The fractal nature of viscous fingering in three-dimensional pore-level models, Phys. Rev. E 53

(1996) 2502–2508.

[23] G.M. Homsy, Viscous fingering in porous media, Annu. Rev. Fluid Mech. 19 (1987) 271–311.

[24] G. Daccord, J. Nittmann, H.E. Stanley, Radial viscous fingers and diffusion-limited aggregation: fractal dimension and growth sites,

Phys. Rev. Lett. 56 (1986) 336.

[25] R.V. Hogg, A.T. Craig, Introduction to Mathematical Statistics, fifth ed, Prentice Hall, New Jersey, 1995.

[26] M. Evans, N. Hastings, B. Peacock, Triangular Distribution Statistical Distributions, third ed., Wiley, NY, 2000, pp. 187–188.

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