Elliptic Regions and Stable Solutions for Three-Phase Flow in

Transport in Porous Media 48: 249–269, 2002.c© 2002 Kluwer Academic Publishers. Printed in the Netherlands. 249

Elliptic Regions and Stable Solutions forThree-Phase Flow in Porous Media

MATTHEW D. JACKSON and MARTIN J. BLUNTCentre for Petroleum Studies, T. H. Huxley School, Imperial College London, SW7 2BP, U.K.e-mail: [email protected]

(Received: 21 September 2000)

Abstract. In the limit of zero capillary pressure, solutions to the equations governing three-phaseflow, obtained using common empirical relative permeability models, exhibit complex wavespeedsfor certain saturation values (elliptic regions) that result in unstable and non-unique solutions. Weanalyze a simple but physically realizable pore-scale model: a bundle of cylindrical capillary tubes,to investigate whether the presence of these elliptic regions is an artifact of using unphysical relativepermeabilities. Without gravity, the model does not yield elliptic regions unless the most non-wettingphase is the most viscous and the most wetting phase is the least viscous. With gravity, the modelyields elliptic regions for any combination of viscosities, and these regions occupy a significantfraction of the saturation space. We then present converged, stable numerical solutions for one-dimensional flow, which include capillary pressure. These demonstrate that, even when capillaryforces are small relative to viscous forces, they have a significant effect on solutions which crossor enter the elliptic region. We conclude that elliptic regions can occur for a physically realizablemodel of a porous medium, and that capillary pressure should be included explicitly in three-phasenumerical simulators to obtain stable, physically meaningful solutions which reproduce the correctsequence of saturation changes.

Key words: elliptic region, three-phase flow, stable solutions, capillary pressure, pore-scale.

1. Introduction

The simultaneous flow of three immiscible phases in porous media occurs duringa variety of oil recovery processes, and also during the migration of non-aqueousphase liquids in the unsaturated zone. The flow is described using a multiphaseversion of Darcys law (e.g. Dullien, 1992)

vp = −Kkrp

µp(∇Pp − ρpg) (1)

where the subscript p labels the phase, v is the Darcy velocity (volume of phasep flowing per unit area per unit time), K is the absolute permeability (when onlyone phase is flowing), P is the pressure, ρ is the density, µ is the viscosity, g is theacceleration due to gravity, and kr is the relative permeability, which is required todescribe the flow of each phase in the presence of other phases.

250 MATTHEW D. JACKSON AND MARTIN J. BLUNT

The experimental measurement of three-phase relative permeabilities is dif-ficult, and typically limited to a small saturation range; moreover, three-phaserelative permeabilities usually depend strongly on saturation path and fluid prop-erties, such as interfacial tension (e.g. Oak et al., 1990; Dullien, 1992; Fenwickand Blunt, 1998a, b). In the oil industry, standard practice is to predict three-phaserelative permeabilities using empirical models extrapolated from two-phase mea-surements (e.g. Stone, 1970, 1973; Baker, 1988; Jerauld, 1996, 1997). Whilst thisapproach is convenient, predictions of the oil relative permeability from differentempirical models in the low oil saturation range of interest often vary by ordersof magnitude (e.g. Baker, 1988; Blunt, 1999). Furthermore, and perhaps of morefundamental concern, these empirical models can yield non-physical solutions tothe governing three-phase flow equations, even though they are used ubiquitouslyin three-phase numerical simulators (Transgenstein, 1989; Guzman and Fayers,1997a).

Most empirical relative permeability models have at least one relativepermeability which is a function of two phase saturations (Baker, 1988). Typic-ally, in a water-wet oil reservoir, the water and gas relative permeabilities will befunctions of their own saturations, while the oil relative permeability will be afunction of both the oil and water saturations (Baker, 1988). For all empiricalmodels of this type, in the limit of zero capillary pressure, there are values ofthe phase viscosities, densities, and saturations, for which the governing equa-tions are elliptic in form rather than hyperbolic (Trangenstein, 1989). Regions ofsaturation space which yield ellipticity are termed elliptic regions (Bell et al.,1986). For saturations within the elliptic region, solutions to the governing equa-tions have complex wavespeeds and are unconditionally unstable (Trangenstein,1989).

Clearly, solutions which have complex wavespeeds are non-physical, yet theyseem an inevitable consequence of most empirical models of three-phase relativepermeability. However, no study has addressed the issue of whether the presence ofelliptic regions is physically plausible. Empirical three-phase relative permeabilitymodels are extrapolated from two-phase data, and do not necessarily representexperimentally measured three-phase permeabilities. When three-phase displace-ment experiments are performed, the relative permeabilities are determined wherethere are smooth changes in saturation with distance and time, which correspondmathematically to solutions with real wavespeeds (Grader and O’Meara, 1988;Sahni et al., 1996). Sharp changes in saturation may span elliptic regions, butexperimental uncertainties in the measurements make the unambiguous identifica-tion of an elliptic region almost impossible. From a theoretical standpoint, severalauthors have constructed three-phase models that use physically-based rules atthe pore scale to predict relative permeability (Heiba et al., 1984; Fenwick andBlunt, 1998a, b; Mani and Mohanty, 1998). Fenwick and Blunt (1998b) developeda procedure to obtain relative permeabilities that gave a sequence of saturationchanges consistent with the observed displacement sequence at the micro-scale. By

THREE-PHASE FLOW IN POROUS MEDIA 251

construction, the regions of saturation space probed were those which correspondto real wavespeeds.

Are elliptic regions an inevitable consequence of three-phase flow, or are they anon-physical artifact of extrapolating three-phase relative permeability functionsbeyond the saturation range investigated experimentally? Any physically-basedmodel may never have elliptic regions, since the relative permeabilities depend notonly on the saturations, but the whole saturation path. On the other hand, it couldbe that elliptic regions can always be found, but are not a problem since stableand unique solutions are obtained if capillary pressure is included in the governingequations.

This paper addresses the issue of whether or not elliptic regions are observedin a physically-based model of three-phase flow. The model used is a bundle ofcylindrical capillary tubes of different size. Whilst this is a simplistic representationof a porous medium, it is, in principle, a physically realizable system which obeysa multiphase Darcy’s law (Equation 1). We demonstrate that the model does yieldelliptic regions which proves that elliptic regions are physically possible, and arenot a mathematical artifact of an unphysical extrapolation of relative permeabilityfunctions beyond the range measured experimentally. It is reasonable to hypothes-ize that a more sophisticated model, or indeed real porous media, will also yieldelliptic regions.

We then couple the pore-scale model with a macroscopic three-phase flow simu-lator, extending the approach of Heiba et al. (1986) for two-phase flow. We demon-strate that if capillary pressure is included in the governing equations, stable andunique solutions may be obtained. If the governing equations include a diffusiveterm, such as capillary pressure, or numerical dispersion in a numerical solution,they become parabolic. In this case, high frequency instabilities within the ellipticregion are damped out. However, numerical solutions for systems with ellipticregions have so far been obtained using numerical dispersion as the diffusive force,rather than capillary pressure (e.g. Bell et al., 1985). Solutions for problems withelliptic regions are not necessarily unique, and the true solution may be governedby the nature of the diffusive forces even as these become infinitesimal (Holden,1987; Azevedo and Marchesin, 1995). We demonstrate that properly convergedsolutions which include capillary pressure can be significantly different from solu-tions obtained using numerical dispersion as the diffusive force. This suggeststhat large-scale numerical simulations of three-phase flow should include capillarypressure if they are to yield stable, physically meaningful solutions which properlycapture the displacement sequence.

2. Analysis of Elliptic Regions and Relative Permeability Models

Consider incompressible, immiscible three-phase flow in one dimension. Conser-vation of volume and use of the multiphase Darcy’s law, Equation (1), leads to thefollowing conservation equations for phases 1 and 3


∂S1

∂tD+ ∂f1

∂xD= 0, (2)

∂S3

∂tD+ ∂f3

∂xD= 0, (3)

where S is the saturation, and xD and tD are dimensionless distance and time,respectively defined by

xD = x

L, (4)

where x is distance and L is the length of the system, and

tD =t∫

0

q

φLdt, (5)

where q = v1 + v2 + v3 is the total volumetric flow rate of all three phases, andφ is the porosity. The fractional flow of phases 1 and 3 is given by (Dullien, 1992;Guzman and Fayers, 1997b)

f 1 = λ1

λt

{1 + λ2K

qL

(∂Pc21

∂xD+�ρ21gxL

)+ λ3K

qL

(∂Pc31

∂xD+�ρ31gxL

)},

(6)

f 3 = λ3

λt

{1 − λ2K

qL

(∂Pc32

∂xD+�ρ32gxL

)− λ1K

qL

(∂Pc31

∂xD+�ρ31gxL

)},

(7)

where the mobility of each phase λp = krp/µp, the total mobility λt = λ1 +λ2 + λ3, gx is the component of gravity along the flow direction, and the densitycontrast �ρpq = ρq − ρp. The capillary pressure is given by Pcpq = Pp − Pq, andthe saturation of phase 2 is found from S2 = 1 − S1 − S3.

We begin by considering solutions for a Riemann problem, with

S1 = SL1 , S3 = SL3 , xD � 0

at tD = 0

S1 = SR1 , S3 = SR3 , xD > 0

(8)

in the limit of negligible capillary pressure. These solutions are functions of adimensionless wavespeed vD = xD/tD, which consist of shocks (sharp changesin saturation with vD), rarefactions (smooth changes in saturation with vD), andconstant states (constant values of saturation with vD) (e.g. Bell et al., 1986; Fayers,1989; Transgenstein, 1989; Guzman and Fayers, 1997a, b). The wavespeed is givenby (e.g. Hicks and Grader, 1996; Guzman and Fayers, 1997a)

vD = f11 + f33

2± 1

2

√(f11 − f33)

2 + 4f13f31 (9)


using the notation fpq = ∂fp/∂Sq. The wavespeeds are the eigenvalues of theeigenvectors

dS3

dS1= f31

vD − f33. (10)

Equation (9) gives real wavespeeds vD when

(f11 − f33)2 + 4f13f31 � 0. (11)

If, for all values of saturation, the left-hand side of Equation (11) is positive, givingtwo distinct wavespeeds, then the system is strictly hyperbolic. In this case analyticsolutions can be obtained using extensions of Buckley–Leverett theory for two-phase flow (e.g. Bell et al., 1986; Fayers, 1989; Transgenstein, 1989; Guzmanand Fayers, 1997a, b). If Equation (11) is always obeyed, but there are values ofthe saturation for which the two wavespeeds are the same, termed umbilic points,then the system is non-strictly hyperbolic. In this case the solutions may be verysensitive to initial condition and include transitional shocks (Isaacson et al., 1990;Hurley and Plohr, 1995; Marchesin and Plohr, 1999). If there are saturation valuesfor which Equation (11) is not obeyed, then the wavespeeds are complex, and thegoverning equations are elliptic in form.

Equation (11) is always true if the relative permeability of each phase is afunction only of its own saturation (a Corey-type model) (Transgenstein, 1989).However, there may be one or more isolated umbilic points (Guzman and Fayers,1997a) which can result in solutions that depend on the nature of diffusive forceseven when they become vanishingly small (Marchesin and Plohr, 1999). All otherempirical relative permeability models, in which one or more of the relative per-meabilities are functions of two saturations, yield elliptic regions (Transgenstein,1989). Strategies to manipulate the relative permeabilities to avoid elliptic regionshave been proposed, but they would require relative permeabilities that varied withdensity and viscosity without any physical justification (Holden, 1990).

Analytical and numerical solutions for the Riemann problem in three-phaseflow have been constructed for a variety of cases (Bell et al., 1986; Holden, 1987;Schaeffer and Shearer, 1987; Isaacson et al., 1989, 1990; Transgenstein, 1989;Holden, 1990; Falls and Schulte, 1992a, b; Medeiros, 1992; Azvedo and Marchesin,1995; Hurley and Plohr, 1995; Hicks and Grader, 1996; Guzman and Fayers, 1997a,b; Marchesin and Plohr, 1999). All have been obtained in the limit of negligiblecapillary pressure. They suggest that, for saturation states which lie within anelliptic region, solutions to Equations (8) and (9) are unconditionally unstable(Trangenstein, 1989). States on either side of an elliptic region may be separated bya stable shock (Bell et al., 1986; Trangenstein, 1989; Guzman and Fayers, 1997b).If one state lies within an elliptic region, the solution shocks from that state to oneoutside the elliptic region (Trangenstein, 1989). If both left and right states lie inthe elliptic region, the solution shocks outside the elliptic region from both states(Trangenstein, 1989). Note that elliptic regions obtained using empirical models


typically occupy only around 1% of the total saturation space. This proportionvaries with fluid viscosities and densities with no obvious trend (e.g. Fayers, 1989;Guzman and Fayers, 1997a; Hicks and Grader, 1996).

3. Pore-Scale Model

We will investigate elliptic regions using relative permeabilities obtained from apore-scale model, rather than using empirical relative permeabilities. The essen-tial features of our model are (i) that it is, in principle, physically realizable; (ii)that it obeys the multiphase Darcy’s law (Equation 1) with three-phase relativepermeabilities that are unique functions of saturation, and (iii) that the relativepermeabilities of the most wetting and most non-wetting phase are functions oftheir own saturation, whilst the relative permeability of the intermediate-wet phaseis a function of two saturations. The model is a simple bundle of cylindrical poresof different size, which has been studied by many authors (e.g. Dullien, 1992; Huiand Blunt, 2000), and is derived, in particular, from the work of van Dijke et al.(2000a, b). Within each bundle, the tubes are parallel to the flow direction. Thebundles are aligned in series along the flow direction, and at the junction of twobundles there is pressure communication between the tubes. We assume that onlyone phase may occupy each tube at a time; thus each bundle has some fraction ofthe tubes filled with each phase. However, the phase occupancy of the tubes canvary with distance, since different bundles will have different phase occupancies.

The conductance of each tube C is defined by

Q = C

µ(∇P − ρgx) , (12)

where Q is the volume of fluid flowing per unit time and l is the tube length. Wefind C from Poiseuille’s law:

C = πr4

8. (13)

We define N(r)dr as the number of tubes of radius between r and r+dr. The volumeof each tube (assuming they are of uniform length) is proportional to r2, while theconductance, from Equation (13), is proportional to r4. Within each bundle, weassume that the capillary pressures alone control the displacement sequence, whichis applicable at low flow rates where the viscous pressure drop across a bundle issmall in comparison to the capillary pressures. Then the smallest tubes are filledwith the wetting phase, the largest tubes are filled with the non-wetting phase,whilst tubes of intermediate size are filled with the intermediate-wet phase (vanDijke et al., 2000b).

We will define phase 1 as the most wetting phase, phase 2 as intermediate-wetand phase 3 as most non-wetting. Then phase 1 occupies the smallest tubes, andthe radius of the largest tube occupied by phase 1 is denoted r1. Phase 3 occupies


the largest tubes, and the radius of the smallest tube occupied by phase 3 is denotedr3. Phase 2 occupies tubes with radii between r1 and r3. The phase saturations andrelative permeabilities are then given by (van Dijke et al., 2000b)

S1 =

r1∫0r2N(r) dr

∞∫0r2N(r) dr

, S3 =

∞∫r3

r2N(r) dr

∞∫0r2N(r) dr

, S2 = 1 − S1 − S3, (14)

kr1 =

r1∫0r4N(r) dr

∞∫0r4N(r) dr

, kr3 =

∞∫r3

r4N(r) dr

∞∫0r4N(r) dr

, kr2 = 1 − kr1 − kr3, (15)

∂kr2

∂S1= −∂kr1

∂S1= −r2

1 ,∂kr2

∂S3= −∂kr3

∂S3= −r2

3 . (16)

Note that the relative permeabilities of phase 1 and 3 are functions of their ownsaturation: kr1 = kr1(S1). The relative permeability of phase 2 is a function of twosaturations: kr2 = kr2(S1, S3). The capillary pressures are given by

Pc12 = 2σ12 cos θ12

r1, Pc23 = 2σ23 cos θ23

r3, Pc13 = Pc12 + Pc23, (17)

where σpq is the interfacial tension between phases p and q, θpq is the contact anglebetween phases p and q (measured through phase q), and σ31 cos θ31 = σ32 cos θ32+σ21 cos θ21 (Hui and Blunt, 2000).

4. Analytic Solutions Using the Pore-Scale Model

Our pore-scale model allows one of the relative permeabilities to be a functionof two saturations. Trangenstein (1989) demonstrated that empirical relative per-meability models of this type will yield elliptic regions. We will now investigatewhether this is the case for our physically-realizable pore-scale model. We willshow that, without gravity, the model does not yield elliptic regions unless the mostnon-wetting phase is the most viscous, and the wetting phase is the least viscous.With gravity, we will demonstrate the existence of elliptic regions with an example.

First we consider the case without gravity. To test if Equation (11) is obeyed,we need the following derivatives:

f11 = λ11

λ2t

(λt − A1λ1) , (18)

f33 = λ33

λ2t

(λt − A3λ3) , (19)


f13 = −A3λ1λ33

λ2t

, (20)

f31 = −A1λ3λ11

λ2t

, (21)

where λij = ∂λi/∂Sj , A1 = (µ2 − µ1)

/µ2, and A3 = (µ2 − µ3)

/µ2. Equation

(11) becomes

(λ11 (λt − A1λ1)− λ33 (λt − A3λ3))2 � − 4A1A3λ11λ1λ33λ3. (22)

From Equation (15), it is clear that, for our pore-scale model, λ1, λ11, λ3, and λ33

are always non-negative. Consequently, Equation (22) is obeyed if both A1 and A3

are positive or negative (A1A3 > 0). Thus our model does not give elliptic regionsif we ignore gravity, and the intermediate-wet phase is either the most viscous orthe least viscous fluid.

The proof when A1A3 < 0 is more involved. We re-write the first term inEquation (22) using Equations (14)–(16)

λ33 (λt − A3λ3)− λ11 (λt − A1λ1)

= 1

µ1µ2µ3

[r2

3 (µ1 + kr1 (µ2 − µ1))− r21 (µ3 + kr3 (µ2 − µ3))

](23)

By construction r3 � r1. Since 1 � kr � 0, both terms in the square bracket on theright-hand side of Equation (23) are positive. Imagine that we have a given valueof r3. As we increase r1 from zero, the first term on the right-hand side of Equation(23) is larger than the second. The right-hand side of Equation (23) is positive. Nowconsider the largest possible value of r1: r1 = r3. In this limit kr2 = 0 and kr3 = 1−kr1 (Equation 15). Then Equation (23) becomes

λ33 (λt − A3λ3)− λ11 (λt − A1λ1) = r21

µ1µ3[kr1 (A1 + A3)− A1] . (24)

If A1 > 0 and A3 < 0, the right-hand side of Equation (24) is negative. Thismeans that there is a value of r1 for which the right-hand side of Equation (23)is zero. In Equation (22) this will lead to an elliptic region, since the inequalityis no longer obeyed. Thus elliptic regions are possible for A1 > 0 and A3 < 0,or µ3 > µ2 > µ1. This is physically unlikely, since the most non-wetting fluid(usually gas) is required to be the most viscous.

If A1 < 0 and A3 > 0, then µ1 > µ2 > µ3, the right-hand side of Equation (24)is positive and the right-hand side of Equation (23) is positive for all values of r1.The magnitude of Equation (23) is smallest when r1 = r3, and it is in this limit thatelliptic regions may be present. Putting r1 = r3 in Equation (22) yields

[kr1 (A1 − A3)− A1]2 � − 4A1A3kr1 (1 − kr1) . (25)


This can be re-arranged as

A21 + k2

r1 (A1 + A3)2 − 2A2

1kr1 � 0 (26)

It is easy to show that this inequality is always obeyed; that is, there are no ellipticregions in this case. Thus, without gravity the model does not give elliptic regionsunless µ3 > µ2 > µ1.

In the absence of an elliptic region, there should be no regions of saturationspace which cannot be accessed by a smooth change in saturation (a rarefaction).For our pore-scale model, and a given set of fluid viscosities, we can plot selectedrarefaction paths for both fast and slow wavespeeds using Equations (9) and (10)(e.g. Falls and Shulte, 1992; Hicks and Grader, 1996; Guzman and Fayers, 1997b).We first choose a start point. We then calculate the wavespeed using Equation (9),and the corresponding change in saturation using Equation (10). We repeat thisprocess and track the movement of each rarefaction path through saturation space.Relative permeabilities at each point are again given by our pore-scale model, usingEquations (14) and (15).

Figure 1. Ternary diagram showing rarefaction curves for our bundle of capillary tubes modelwithout gravity; model and fluid properties are described in the text. The fast rarefaction wavesare shown in plot (a), and the slow waves in plot (b).

Figure 1 shows selected fast and slow rarefaction curves, in the absence ofgravity, for our pore-scale model with

N(r) = Nt (1 − γ )

r1−γmax − r

1−γmin

r1−γ , rmax � r � rmin

(27)= 0, r > rmax, r < rmin,

where Nt is the total number of tubes in each bundle, γ = −1.8, rmax = 100and rmin is arbitrarily small. The phase viscosities are µ1 = 1 × 10−3 Pas (1cp),µ2 = 7.5 × 10−4 Pas (0.75cp), and µ3 = 2 × 10−4 Pas (0.2cp), which yieldsA1 = −1/3 and A3 = 11/15. These viscosities are physically plausible for an oil


reservoir if phase 1 corresponds to water, phase 2 corresponds to oil and phase 3corresponds to gas; note that an elliptic region is not predicted.

Figure 1 confirms that there is no elliptic region; rarefaction curves can be plot-ted to fill the entire saturation space. Rarefactions follow each curve from the rightor initial (R) to the left or final (L) state (Equation 8) as long as the wavespeed in-creases monotonically from the left to the right state (Guzman and Fayers, 1997a).In Figure 1, fast rarefactions correspond to solutions in which the saturation ofphase 3 increases from the right state to the left state, while slow rarefactions cor-respond to solutions in which the saturation of phase 2 increases from the right stateto the left state. In a water-wet oil reservoir, the fast rarefactions would thereforecorrespond to solutions in which the gas saturation increases (for example, duringgas injection), while the slow rarefactions would correspond to solutions in whichthe oil saturation increases (for example, during the formation of an oil bank). Notethat an umbilic point, at which the two wavespeeds coincide, does not appear to bepresent, which implies that in this example, the governing equations are strictlyhyperbolic. However, we have not undertaken a detailed search for an umbilicpoint.

We now consider flow with gravity. Equations (18)–(21) become

f11 = λ11

λ2t

((λt − A1λ1)+ ε

g

1 (λ1 (λt (A1 − 1)− A1λ2)+ λtλ2) ++ εg3λ3 (λt − A1λ1)

), (28)

f33 = λ33

λ2t

((λt − A3λ3)− ε

g

2 (λ3 (λt (A3 − 1)− A3λ2)+ λtλ2)−− εg3λ1 (λt − A3λ3)

), (29)

f13 = −λ33λ1

λ2t

(A3 + ε

g

1 (λt (1 − A3)− A3λ2)− εg

3 (λt − A3λ3)), (30)

f31 = −λ11λ3

λ2t

(A1 − ε

g

2 (λt (1 − A1)+ A1λ2)+ εg

3 (λt − A1λ1)), (31)

where

εg

1 = K�ρ21gx/qµ2, ε

g

2 = K�ρ32gx/qµ2,

εg

3 = K�ρ31gx/qµ2 = ε

g

1 + εg

2 . (32)

The parameters εg represent a dimensionless ratio of gravity to viscous forces.We re-plot Figure 1 but this time include gravity by setting εg1 = 1.5, εg1 = 1.5and εg3 = 3. These values are large (i.e. they assume a significant contributionfrom gravity forces) but are again physically plausible for an oil reservoir. Figure 2shows the result: an elliptic region shaded black occupies approximately 16% of the


Figure 2. Ternary diagram showing elliptic region and rarefaction curves for our bundle ofcapillary tubes model with gravity; model and fluid properties are described in the text. Theelliptic region is denoted by the black dots, which represent sample points at which Equation(11) does not hold; see text for more details. The fast rarefaction waves are shown in plot (a),and the slow waves in plot (b).

saturation space. Also shown are selected fast and slow rarefaction curves outsidethe elliptic region. The region was mapped by sampling the saturation space withapproximately 45,000 points, and testing whether Equation (11) was obeyed ateach. Note that, although not resolved by the sampling scheme, the shape of therarefaction curves indicates that the elliptic region extends to the S1 = 0 axis atboth the top, and bottom left, corners of the ternary diagram. Note also that theelliptic region is considerably larger than those obtained using empirical relativepermeability models, even when their properties are tuned to emphasize the sizeof the region (Trangenstein, 1989; Guzman and Fayers, 1997a). Our model para-meters were chosen arbitrarily, and no attempt was made to maximize or minimizethe size of the elliptic region. Unlike the rarefaction curves shown in Figure 1, thewavespeeds along each curve shown in Figure 2 do not always increase mono-tonically towards a given increase in saturation. If an elliptic region interrupts thecurve, then as the curve approaches the region the wavespeed decreases until it iszero at the boundary of the region. Consequently, rarefactions sharpen to formBuckley–Leverett type shocks (Fayers, 1989) if they move towards the ellipticregion.

We conclude that a physically realizable pore-scale model does indeed yieldelliptic regions, and that those regions can occupy a significant fraction of thesaturation space. For the example shown, the distribution of the elliptic regionthroughout saturation space is such that, within an oil reservoir, common displace-ment processes such as water or gas injection would yield saturation changes whichoften enter, leave or span the region. We obtain similar results for different fluidviscosities and densities.


5. Numerical Solutions Using the Pore-Scale Model

We now explore the consequences of elliptic regions by considering numericalsolutions of the flow equations in one dimension. We discretize our system intoN gridblocks, each of size �xD = 1/N . Using standard upstream weighting, thedifferential Equations (2) and (3) become (Aziz and Settari,1979)

Sn+11,i = Sn1,i −

�tD

�xD

(f n1,m − f n1,n

), (33)

Sn+13,i = Sn3,i −

�tD

�xD

(f n3,m − f n3,n

), (34)

where the subscript i labels the grid block, the superscript n labels the time step,and m = i, n = i − 1 if the solution is locally propagating to the right, or m =i + 1, n = i if the solution is locally propagating to the left. The fractional flows,Equations (6) and (7), are discretized centrally using

f n1,i = λ1(Sn1,i)

λt (Sn1,i, S

n3,i)

1 + εPc1 kr2(S

n1,i , S

n3,i)

2�xD(G1 + PcD21(S

n1,i+1)−

−PcD21(Sn1,i−1))

+ εPc3 kr3(S

n3,i)

2�xD(1 − A3)(G3 + PcD31(S

n1,i+1, S

n3,i+1)−

−PcD31(Sn1,i−1, S

n3,i−1))

,

(35)

f n3,i = λ3(Sn3,i)

λt (Sn1,i, S

n3,i)

1 − εPc3 kr2(S

n1,i , S

n3,i)

2�xD(G2 + PcD32(S

n3,i+1)−

−PcD32(Sn3,i−1

))

− εPc1 kr1(S

n1,i)

2�xD(1 − A1)(G3 + PcD31(S

n1,i+1, S

n3,i+1)−

−PcD31(Sn1,i−1, S

n3,i−1))

,

(36)

where

PcD21 = rmax

r1, PcD32 = rmax

r3, PcD31 = PcD21 + PcD32, (37)

εPc1 = 2Kσ21 cos θ21

qµ2Lrmax, ε

Pc2 = 2Kσ32 cos θ32

qµ2Lrmax,

εPc3 = 2Kσ31 cos θ31

qµ2Lrmax= ε1 + ε2, (38)


G1 = εg

1�xD

εPc1

, G2 = εg

2�xD

εPc2

, G3 = εg

3�xD

εPc3

(39)

The parameters εPc represent a dimensionless ratio of capillary to viscous forcesover the scale of the whole simulation (xD = 1) and on physical grounds shouldbe small for large-scale flow, such as at the scale of an oil reservoir. The ratioεPc

/�xD = NεPc represents a dimensionless magnitude of capillary to viscous

forces at the grid-block scale. For the dominant diffusive force to come from ca-pillary pressure, rather than numerical dispersion (i.e. to obtain physically validsolutions) this must be large; in the absence of capillary pressure, numerical dis-persion yields an effective value for NεPc of ∼ 1. The parameters G represent adimensionless ratio of gravity to capillary forces at the grid-block scale.

We present numerical solutions both with and without capillary pressure, for thepore-scale model and phase properties used to obtain the analytic results shown inFigure 2. Where capillary pressure is included we set εPc

1 = 0.01, εPc2 = 0.01

and εPc3 = 0.02. This ratio is physically plausible for large-scale displacements

within an oil reservoir: a typical oil/water capillary pressure may be of the order104 Pa (0.1 atm), while a typical pressure drop between wells will be less than106 Pa (10 atm). In all cases, solutions were obtained using N = 2,000, 4,000 and8,000 gridblocks; note that in the presence of capillary pressure, the ratio NεPc =20, 40 and 80, respectively. This is sufficiently large for the dominant diffusiveforce to come from capillary pressure. Refining the grid allowed us to check thatsolutions with capillary pressure were properly converged, and also to investigatethe effect of reducing numerical dispersion in solutions without capillary pressure.Solutions with capillary pressure were found to be stable and properly convergedusing �tD = �xD/100. This small timestep is required for stability by our explicitnumerical scheme, which is consequently not numerically efficient, however, weare interested only in the nature of the solutions, rather than in numerical efficiency.Unless otherwise stated, solutions are shown at a dimensionless time of tD = 0.15.

We begin by presenting a numerical solution with initial (R) and final (L) states(Equation 8) given by

SL1 = 0.1, SL3 = 0.8, SR1 = 0.1, SR3 = 0.2. (40)

The solution follows a fast rarefaction which may be obtained analytically, and isshown in Figure 3. We present this solution (i) to demonstrate that our numericalscheme can reproduce the rarefaction, and (ii) to demonstrate that, for solutionswhich do not span or enter the elliptic region, the amount of capillary pressure weinclude in the numerical solutions does not significantly change their nature. Notethat the gas saturation profile predicted from the rarefaction is not monotonic; thisreflects the decrease in wavespeed along the rarefaction curve as it approaches theelliptic region, and suggests that the analytic solution without capillary pressurewould feature a shock. In the absence of capillary pressure, the numeric solu-tion closely reproduces the rarefaction over the saturation range modelled. Adding


Figure 3. Analytic and numerical solutions with and without capillary pressure, for our bundleof capillary tubes model with gravity, at tD = 0.15. Model and fluid properties are describedin the text; the initial (R) and final (L) states in the numerical solutions are given by Equation(40). Both states lie away from the elliptic region. Plot (a) shows the saturation path in theternary diagram; the curved dashed lines denote the boundary of the elliptic region as plottedin Figure 2. The straight dashed line denotes a fast rarefaction obtained analytically; the boldline denotes the numerical solutions with and without capillary pressure. They are indistin-guishable. Plot (b) shows the saturation of phase 3 against dimensionless distance. The dottedline denotes the rarefaction obtained analytically. The bold line denotes the numerical solutionwith capillary pressure; the fine line denotes the numerical solution without capillary pressure.In both cases, N = 2000.

capillary pressure simply smoothes the profile, and does not alter the saturationpath.

We now consider a solution where the initial and final states lie on either sideof the elliptic region

SL1 = 0.36, SL3 = 0.50, SR1 = 0.45, SR3 = 0.05. (41)

This solution might represent gas injection into an oil reservoir which containswaterflood residual oil, and is shown in Figure 4. It is clear that the inclusion ofcapillary pressure has a large impact on this solution. Significantly, the saturationpath is changed: in the absence of capillary pressure, the solution initially followsa fast rarefaction, from state R to R1 (cf. Figure 2), before shocking across theelliptic region to state R2. This shock ‘overshoots’ the injection (final) state (L),and is followed by another shock to state R3. This shock again overshoots to stateR4, and is followed by a final shock to the injection state L. Refining the gridreduces the amount of numerical dispersion, and increases the overshoot in eachcase. The solution is unstable and sensitive to the grid resolution: it appears tooscillate around the injection state, from one side of the elliptic region to another.In contrast, with capillary pressure, the solution moves smoothly from the initial(R) state to the injection (L) state. Thus it would appear that, using our pore-scalemodel, the presence of an elliptic region may have a significant effect on solutions


Figure 4. Numerical solutions with and without capillary pressure, for our bundle of capillarytubes model with gravity, at tD = 0.15. Model and fluid properties are described in the text;the initial (R) and final (L) states are given by Equation (41). The states lie on either side of theelliptic region. Plot (a) shows the saturation path in the ternary diagram, while plot (b) showsan enlargement of the diagram around the final (L) state. In both cases, the dotted lines denotethe boundary of the elliptic region. Plot (c) shows the saturation of phase 3 against dimen-sionless distance. In all plots, the bold line denotes the solution with capillary pressure andN = 2000. The fine lines denote solutions without capillary pressure. The solution closest tothe capillary pressure solution has N = 2000, that furthest from the capillary pressure solutionhas N = 8000, while that in between has N = 4000.

to physically plausible displacements within an oil reservoir. In contrast, Bell et al.(1986) concluded that, using Stone I relative permeabilities and neglecting capil-lary pressure, the existence of an elliptic region does not affect typical reservoirsimulation solutions.

We next consider a solution in which the initial state lies outside the ellipticregion, but the final state lies within the elliptic region

SL1 = 0.48, SL3 = 0.40, SR1 = 0.45, SR3 = 0.20. (42)

Again, the inclusion of capillary pressure has a significant effect on the solution,changing the saturation path (Figure 5). With capillary pressure, the solution movessmoothly from the initial (R) to the injection (L) state. Without capillary pressure,the solution is unstable and sensitive to the grid resolution. Figure 6 shows a solu-tion in which the initial state lies within the elliptic region, but the final state liesoutside the elliptic region

SL1 = 0.45, SL3 = 0.50, SR1 = 0.55, SR3 = 0.20. (43)

Solutions without capillary pressure are unstable, causing the saturation of phase 3to become non-physical at early times; consequently, only the solution with capil-lary pressure is shown. Solutions at early times (tD = 0.015) are shown in Figure 7;in the absence of capillary pressure the saturation of phase 3 becomes non-physical


Figure 5. Numerical solutions with and without capillary pressure, for our bundle of capillarytubes model with gravity, at tD = 0.15. Model and fluid properties are described in the text;the initial (R) and final (L) states are given by Equation (42). The initial state lies outside theelliptic region; the final state lies within the elliptic region. Plot (a) shows the saturation pathin the ternary diagram, while plot (b) shows an enlargement of the diagram around the final (L)state. In both cases, the dotted lines denote the boundary of the elliptic region. Plot (c) showsthe saturation of phase 3 against dimensionless distance. In all plots, the bold line denotes thesolution with capillary pressure and N = 2000. The fine lines denote solutions without capil-lary pressure. The solution closest to the capillary pressure solution hasN = 2000, that furthestfrom the capillary pressure solution has N = 8000, while that in between has N = 4000.

Figure 6. Numerical solution with capillary pressure, for our bundle of capillary tubes modelwith gravity, at tD = 0.15 and with N = 2000. Model and fluid properties are described in thetext; the initial (R) and final (L) states are given by Equation (43). The initial state lies withinthe elliptic region; the final state lies outside the elliptic region. Plot (a) shows the saturationpath in the ternary diagram; the dotted lines denote the boundary of the elliptic region. Plot(b) shows the saturation of phase 3 against dimensionless distance.


Figure 7. Numerical solutions with and without capillary pressure at early times (tD = 0.015),corresponding to the solution shown in Figure 6. In both plots, the bold line denotes the solu-tion with capillary pressure and N = 2000. The fine lines denote solutions without capillarypressure; that closest to the capillary pressure solution has N = 2000, while that furthest fromthe capillary pressure solution has N = 4000.

because the solution shocks outside the elliptic region and overshoots the S3 = 0axis.

We now consider a case where both the initial and final states lie within theelliptic region

SL1 = 0.48, SL3 = 0.40, SR1 = 0.55, SR3 = 0.20. (44)

Again, solutions without capillary pressure are unstable, causing the saturation ofphase 3 to become non-physical at early times; consequently, only the solution withcapillary pressure is shown in Figure 8. Figure 9 shows solutions at early times;in the absence of capillary pressure the solution again shocks outside the ellipticregion and overshoots the S3 = 0 axis. In contrast, the solution with capillarypressure moves smoothly from initial (R) to injection (L) state regardless of thepresence of the elliptic region.

We finally examine the behaviour of solutions with capillary pressure at latetimes. Figure 10 shows solutions at tD = 1.5 which correspond to those shown attD = 0.15 in Figures 4 and 5; the initial (R) and final (L) conditions are given byEquations 41 and 42, respectively. Both the length- and time-scales of the late-timesolutions are increased by a factor of 10 over those shown in Figures 4 and 5. Theywere obtained using the same value of �xD in order to maintain the balance ofcapillary and viscous forces at the grid-block scale. The late-time solutions shownin Figure 10 were obtained with N = 20, 000 (NεPc = 20); solutions obtainedwith N = 40, 000 (NεPc = 40) demonstrate that they are properly converged.In both cases, it is clear that the saturation change does not simply sharpen withtime; rather, the change becomes oscillatory, although the solutions are stable.Qualitatively, the solutions appear to approach the form of those obtained without


Figure 8. Numerical solution with capillary pressure, for our bundle of capillary tubes modelwith gravity, at tD = 0.15 and with N = 2000. Model and fluid properties are described inthe text; the initial (R) and final (L) states are given by Equation (44). Both states lie withinthe elliptic region. Plot (a) shows the saturation path in the ternary diagram; the dotted linesdenote the boundary of the elliptic region. Plot (b) shows the saturation of phase 3 againstdimensionless distance.

Figure 9. Numerical solutions with and without capillary pressure at early times (tD = 0.015),corresponding to the solution shown in Figure 8. In both plots, the bold line denotes the solu-tion with capillary pressure and N = 2000. The fine lines denote solutions without capillarypressure; that closest to the capillary pressure solution has N = 2000, while that furthest fromthe capillary pressure solution has N = 4000.

capillary pressure and with only small amounts of numerical dispersion (i.e. thoseobtained with N = 8000); see Figures 4 and 5. This is perhaps not surprising:because the length-scale of the late-time solutions is increased by a factor of 10, theeffective ratio of capillary to viscous forces over the scale of the whole simulationis decreased by a factor of 10. Consequently, the contribution of capillary forces issmaller. The results indicate that solutions which encounter the elliptic region areexceptionally sensitive to the balance of diffusive and viscous forces.


Figure 10. Numerical solutions with capillary pressure, at tD = 0.15 (plotted against the loweraxis) and tD = 1.5 (plotted against the upper axis), with N = 2000 and 20000 respectively;�xD is therefore the same for both simulations, though the lengthscale of the late-time solution(tD = 1.5) has increased by a factor of 10 (see text for details). The solution shown in plot (a)has initial (R) and final (L) states given by Equation (41), and corresponds to that shown inFigure 4. The solution shown in plot (b) has initial (R) and final (L) states given by Equation(42), and corresponds to that shown in Figure 5.

6. Conclusions

Elliptic regions introduce mathematical complexities into solutions of the govern-ing three-phase flow equations in the limit of negligible capillary pressure. We havedemonstrated that elliptic regions, occupying a significant fraction of the saturationspace, are present for a physically realizable model of a porous medium. The modelwe used was a bundle of parallel capillary tubes, where the relative permeabilityof the intermediate-wet phase is a function of two independent saturations. Thepresence of elliptic regions for this physically-realizable model demonstrates thatelliptic regions are not an artifact of extrapolating two-phase data into the three-phase region; rather, they appear to be an inevitable consequence of a non-trivialmodel of three-phase flow.

We have also presented numerical solutions for three-phase flow which includea physically consistent treatment of capillary pressure. The ratio of capillary toviscous forces was representative of field-scale displacements within oil reser-voirs. Previous studies have used numerical dispersion to provide diffusive forces(Bell et al., 1986); in our study diffusive forces were provided by the capillarypressure. Solutions which encountered an elliptic region, obtained using numericaldispersion as the diffusive force, were unstable, and qualitatively different to theconverged solutions obtained with capillary pressure. Moreover, they were excep-tionally sensitive to the grid resolution, which in the absence of capillary pressure,governed the balance of viscous and diffusive forces. This suggests that numericalsolutions of three-phase flow, even at large length-scales where capillary forces are


small, must properly account for capillary pressure to be physically meaningfuland reproduce the correct sequence of saturation changes.

Acknowledgements

The authors are grateful to Rink van Dijke for helpful suggestions and comments.The members of the Imperial College consortium on pore-scale modeling (Statoil,Schlumberger, JNOC, BHP, Enterprise Oil, and the Department of Trade and In-dustry) are thanked for their financial support.

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