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THREE-PHASE RELATIVE PERMEABILITY FROM MIXED-WET TRIANGULAR AND STAR-SHAPED PORES J.O. Helland a,∗ , M.I.J. van Dijke b , K.S. Sorbie b , S.M. Skjæveland a a

b

Department of Petroleum Engineering, University of Stavanger, N-4036 Stavanger, Norway

Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh EH14 4AS, UK

Abstract We calculate three-phase relative permability from a capillary bundle model where the pores assume regular star-shaped cross-sections. The angular pore shape allows for several possible cross-sectional fluid configurations, including layers. Accurate three-phase entry pressures have been employed for bulk phase invasion, layer phase invasion and for cases where more than one fluid phase enters the pore simultaneously. This latter displacement type is considered for the first time for gas invasion in angular mixed-wet pores. We also propose a new method to calculate contact angles in three-phase flow accounting for hysteresis. It is demonstrated that, unlike several previous approaches, the method maintains consistency of the capillary entry pressures for both cylindrical and star-shaped pores. Furthermore, the finite element method is used to calculate conductances for different cross-sectional fluid configurations. For single-phase flow, we find that the commonly assumed proportionality between dimensionless conductance and pore shape factor breaks down for star-shaped cross-sections as the number of corners is increased. Relative permeability curves are simulated for gas invasion after waterflood, using star-shaped pores with three corners. The half angle of the corners is allowed to vary from pore to pore. In some cases, the oil saturation So may increase due to simultaneous invasion of bulk gas with accompanying oil layers into water-filled pores. A consequence of this is that along a displacement path the oil relative permeability kro shows hysteresis. For small oil saturations, the dependency kro ∝ So2 is smoothly reduced to a weaker So dependency if half-angle variation is introduced from pore to pore. For oil-wet conditions, we demonstrate that a change max , may change the pore occupancies and in maximum pressure after primary drainage, Pow saturation dependencies of the relative permeabilities. Key words: Three-phase, Mixed wettability, Contact-angle hysteresis, Entry pressure, Consistency, Angular pore, Fluid layers, Conductance, Relative permeability

INTRODUCTION

Three-phase relative permeabilities are required as functions of the saturations to solve the equations for three-phase flow in reservoir simulation. These relationships are normally formulated as correlations with adjustable parameters. In capillarydriven three-phase flow, two of the saturations may vary independently, depending on the combination of the three capillary pressures. As a consequence, an infinite number of displacement paths are possible in the three-phase saturation space. This makes it impractical to perform time-consuming measurements of three-phase relative permeabilities for a wide range of conditions. Most of the three-phase relative permeability correlations available in the literature have therefore been suggested based on knowledge of the relative permeabilities in equivalent two-phase systems (e.g., Stone, 1970, 1973; Baker, 1988; Blunt, 2000). However, both experimental and numerical work have shown that this practice may not be valid (e.g., Blunt, 2000; van Dijke et al., 2001b). Moreover, micromodel studies of three-phase flow have revealed that the fluid distribution and the displacement mechanisms at the pore scale may be more complex than for two phases (Øren and Pinczewski, 1995; Keller et al., 1997). In particular, the form of threephase oil relative permeability in the low oil saturation region, and the residual oil itself, are strongly dependent on the existence and conductances of oil layers in the crevices of the pore space. Drainage of oil through connected layers is believed to be the important mechanism that leads to low residual oil saturation in threephase gravity drainage experiments (Blunt et al., 1995). This work highlights the importance of such layers and adds some new results to the underlying physics. To study oil layers and three-phase relative permeability, we use a capillary bundle model where the pore cross-sections are assumed to be star-shaped (van Dijke and Sorbie, 2006c). Such a simple model does not incorporate the interconnectivity which characterizes real porous rocks. Hence, phase entrapment does not occur and residual saturations are absent. However, the angular pore geometry allows for other important physical processes, such as the development of mixed wettability at the pore scale (Kovscek et al., 1993; Hui and Blunt, 2000), and the layer-drainage mechanism along the corners (e.g., Zhou et al., 1997; Hui and Blunt, 2000). Moreover, the regular star-shape introduced by van Dijke and Sorbie (2006c) makes it possible to vary the angles of all corners in a pore simultaneously. Thus, in a pore with narrow corners a large fraction of the cross-sectional area may be occupied by fluid layers. Several three-phase fluid configurations may occur in mixed-wet angular pore crosssections (e.g., Piri and Blunt, 2005), and capillary displacements may either occur as piston-like displacement of the fluids occupied in the bulk or as piston-like dis∗ Corresponding author. Email address: [email protected] (J.O. Helland).

2

placements of the fluids in layers. Three-phase capillary entry pressures for these displacements have recently been derived for uniform (van Dijke and Sorbie, 2003; van Dijke et al., 2004) and mixed-wet (Helland and Skjæveland, 2006a; van Dijke et al., 2006) conditions. These accurate three-phase entry conditions have not previously been taken into account in the calculation of three-phase relative permeability. The possibility of co-existence of more than one fluid phase in the pore crosssection, requires reliable expressions for the conductances of the layer-, cornerand bulk-phase configuration in a pore. Several expressions have been suggested in the literature (Øren et al., 1998; Patzek and Kristensen, 2001; Zhou et al., 1997; Hui and Blunt, 2000; Valvatne and Blunt, 2004; Al-Futaisi and Patzek, 2002), and we have performed numerical calculations to check the accuracy of expressions for corner- and bulk-phase conductances. We also present novel results for singlephase calculations for the conductance in star-shaped pore cross-sections. The paper is organized as follows: First we introduce definitions and concepts used in later sections. Then we describe the main features of the pore model, including fluid configurations, entry pressures and contact angles. In the next section we propose a new method to account for contact-angle hysteresis in three-phase flow. The results from our study on fluid conductances are presented thereafter. Finally, we present simulation results of three-phase relative permeability using our pore model with the above features implemented.

PRELIMINARIES Capillary pressure is defined as Pij = Pi − P j , where Pi and P j are the pressures in the lighter and denser phases, respectively. In a three-phase fluid system (e.g., gas, oil and water) the capillary pressures satisfy Pgw = Pgo + Pow .

(1)

The wetting preference of a solid surface in contact with two fluids is typically characterized by the contact angle. Assuming phase j is wetting relative to phase i , then cos θij ≥ 0, where the contact angle θij is measured through the denser phase j . In a three-phase fluid system, where gas-oil, oil-water and gas-water interfaces may be in contact with the solid surface, Young’s equation yields three force balances of the three distinct fluid-fluid-solid contact lines. Elimination of the fluid-solid interfacial tensions results in the Bartell-Osterhof equation (Zhou and Blunt, 1997; van Dijke and Sorbie, 2002): σgw cos θgw = σow cos θow + σgo cos θgo . 3

(2)

This relationship between the contact angles and the interfacial tensions has also been derived from analysis of true three-phase systems where gas-oil-water contact lines exist (van Dijke and Sorbie, 2006b). van Dijke and Sorbie (2002) formulated linear relationships of cos θgo and cos θgw as functions of cos θow which are consistent with Eq. (2): cos θgo =

1 (Cso cos θow + Cso + 2σgo ), 2σgo

(3)

and

1 ((Cso + 2σow ) cos θow + Cso + 2σgo ), (4) 2σgw where the oil spreading coefficient C so = σgw −σgo −σow is nonpositive and reflects the interfacial tensions measured at thermodynamic equilibrium. Thus assuming that the underlying wettability is known in terms of the oil-water contact angles, calculations of θgo and θgw are possible by Eqs. (3), (4). cos θgw =

Eq. (2) ensures that the capillary entry pressures in cylindrical tubes, which are given by Young-Laplace’s equation, Pcij = 2σij cos θij /R, ij = go, ow, gw, are consistent (van Dijke et al., 2001b). This implies that for a cylindrical pore there exist three distinct regions in the capillary pressure space (e.g., in the (Pow , Pgo ) space) where each region is separated by two of the entry pressures and corresponds to a unique occupancy of water, oil and gas in the cylinder. When wettability is uniform, such that the contact angles do not change with the direction of the process, these regions remain unchanged and the uniqueness of the capillary pressure vs. pore occupancy relation does not depend on the displacement history. Under the conditions of uniform contact angles, van Dijke et al. (2001a,b) have analyzed the different possible pore occupancies and the saturation-dependencies of three-phase capillary pressure and relative permeability from a bundle of cylindrical pores. In angular pores, more than one fluid phase may occupy the cross-section: The wetting phase may reside in the corners after the non-wetting phase has occupied the bulk portion of the pore. In a three-phase situation, the intermediate-wet phase may also occupy the pore as layers between the wetting corner phase and the nonwetting bulk phase. Obviously, this will increase the number of possible pore occupancies. Capillary entry pressures for two-phase piston-like displacements in angular pores can be calculated by the MS–P method (Mayer and Stowe, 1965; Princen, 1969a,b, 1970). This method is based on an energy balance equation which equates the virtual work with the associated change of surface free energy for a small displacement of the interface in the direction along the tube. The energy balance equation relates the entry radius of curvature to the cross-sectional area exposed to change of fluid occupancy, the bounding cross-sectional fluid-solid and fluid-fluid lengths, and the contact angle. It has been demonstrated that the MS–P method can be applied on irregular pore shapes (Mason and Morrow, 1991; Øren et al., 1998; Lago and Araujo, 2001; Lindquist, 2006) and mixed-wet conditions (Ma et al., 1996; Øren et al., 1998; Helland and Skjæveland, 2006b; van Dijke and 4

Sorbie, 2006c). However, the analysis on regular n-cornered cross-sections of uniform wettability is largely simplified as all corners have the same half-angle α and hence the same fluid configuration. There are two scenarios that must be be considered separately depending on the contact angle. As an example, consider invasion of phase i into a uniformly wetted tube initially filled with the denser phase j . If θij
3, the results for these bulk-phase configurations may agree better with conductances calculated for 6-cornered star-shapes, since in each corner the AM produces 2 “new” corners at the fluid-fluid-solid contact lines. The conductance data 19

0.04

0.04

gb/A2b vs. Gb gb/(ApAb) vs. Gp

0.03

3/5Gp vs. Gp 0.5222Gb vs. Gb c

g /A2

0.02

c

gb/A2b, gb/(ApAb)

0.03

0.02 Computed data Valvatne corr. 3/5Gc

0.01

0.01

Hui & Blunt corr. Zhou et al. corr. 0 0

0.02

0.04 G ,G b

0.06

0 0

0.08

0.01

0.02

0.04

0.05

0.06

0.07

0.04

0.05

0.06

0.07

G

c

(a)

(b)

0 −0.1

0.03

p

0.04 Scaled dimensionless conductance 13th order polynomial fit

Computed data Patzek correlation 0.03

−0.2

gc/A2c

−0.3 −0.4 −0.5

0.02

0.01

−0.6 −0.7 0

0.01

0.02

0.03

0.04

0.05

0.06

0

0.07

Gc

0

0.01

0.02

0.03

Gc

(c)

(d)

Figure 8. (a) Calculated bulk-phase conductances compared with straight-line approximations. (b) Calculated corner-phase conductances compared with correlations. (c) Dimensionless corner-phase conductances scaled by Eq. (B.6). (d) Corner-phase conductances compared with Eq. (B.7).

are also organized in Fig. 8(a) according to a correlation employed by Valvatne and Blunt (2004), gb = C A b A p G p (22) with C = 3/5, which gives a mean relative error of 26.70%. The equation yields a good concentration of most of the calculated data, although conspicuous devia(k) tions occur for relatively large values of α and bij . Most of these deviations occur when the AMs are concave, for which the configurations are unlikely to be stable. Therefore, we use Eq. (22) with C = 3/5 as the expression for the bulk-phase conductance in the calculation of relative permeability. The calculated corner conductances gc are presented in Fig. 8(b). A comparison with Eq. (21) using C = 3/5 results in a mean relative error of 15.54%. The data are also compared with previously suggested correlations which are listed in Appendix B. The mean relative errors of the correlations by Zhou et al. (1997) and Hui and Blunt (2000) (Eqs. (B.1)-(B.5)) are 15.07% and 23.74%, respectively. The total error when both expressions are considered as one correlation is 17.80%. Valvatne and Blunt (2004) suggested to use Eq. (21) with C = 0.364 + 0.28G c /G 0 , where 20

(1)

G 0 is the shape factor for the corner assuming that the AM at position b ij is flat, i.e., the capillary pressure is zero. This corner shape factor is given by G0 =

sin α cos α . 4(1 + sin α)2

(23)

As shown in Fig. 8(b) this correlation gives an excellent match with the calculated data. The mean relative error is 3.69%. In Fig. 8(c) we follow the method by Patzek and Kristensen (2001) and scale the dimensionless conductances by Eq. (B.6). Patzek and Kristensen (2001) fitted a 2nd order polynomial to the universal curve they obtained, which resulted in a good match with the finite element calculations. However, in their work, calculations for shape factors in the range G c < 0.015 were lacking. As shown in Fig. 8(c) the universal curve exhibits a pronounced turn in this region of G c . As a result, a 2nd order polynomial failed to match our data satisfactorily. To obtain an equally good match with our data as reported by Patzek and Kristensen (2001), a polynomial of a much higher order is required. In the figure we have used a 13th order polynomial which seems to best describe Eq. (B.6) in the range of G c considered here. Fig. 8(d) shows the re-scaled conductances using Eq. (B.7). The mean relative error of this correlation using the 13th order polynomial is 4.10%. Thus, for the corner phase we employ the expression suggested by Valvatne and Blunt (2004) in our model, since it has proven to be more reliable than other suggested correlations in the literature. We have not performed conductance calculations on fluid layers. In our model we use the correlation proposed by Al-Futaisi and Patzek (2002), which has been shown to be superior to other correlations for layer conductances suggested in the literature (Zhou et al., 1997; Hui and Blunt, 2000). In a general fluid configuration with an arbitrary number of AMs, this correlation for the conductance g L of a layer in a corner can be written as follows for a layer phase m separated by jm AM k on the corner side and im AM l on the bulk side: (l)

g L = (bim )4 exp(a + λ f (z)),

(24)

where a = −7.9998, λ = 1.7474, f (z) = 1.3062z + 4.9465, and

(25)

z =0.0797 ln(α) + 0.5540 ln(G L ) + 0.7698 ln (k)

− 0.1494θ L − 0.2679

bjm

(l)

bim

AL

(l)

(bim )2

(26)

,

where A L is the area of the layer, and G L is the layer shape factor defined in Ap21

o

Rmin

g

w

w

Rw

Rmax

o

g

Rw

Rmin

w

Rmax

(b)

(a) w

Rmin

o

w

Rw

g

Rmax

(c) Figure 9. (a)–(b) Bulk-phase pore occupancies during gas invasion for strongly oil-wet max is (a) large and (b) small. (c) Pore occupancies for weakly oil-wet conditions when Pow max is small. The small pores which remain water-filled at the end of conditions when Pow primary drainage are not indicated. The location of the oil-water boundary where gas started to invade is denoted by pore size Rw .

pendix A. The contact angle θ L is given by  (l) (l) θim if Iim = 1, θL = (l) (l) π − θim if Iim = −1.

(27)

If the same fluid phase is located on the corner and bulk side of the layer, i.e., if (k) (k+1) i = j , then l = k + 1 and Iim = −Iim .

RELATIVE PERMEABILITY

In the numerical examples of relative permeability calculations which we present here, the pore cross-sections are modelled as 3-cornered star-shapes (n = 3). A capillary bundle of 500 pores uniformly distributed between R min = 1μm and Rmax = 50μm is used. Furthermore, we model the interfacial tensions σ gw = 0.030 N/m, σgo = 0.012 N/m and σow = 0.020 N/m which also were used in a previous secmax , tion. We simulate primary drainage to a predetermined capillary pressure Pow followed by a water flood to an initial water saturation Swi where gas is injected. The contact angle in primary drainage is always set equal to zero, θ ow,pd = 0◦ . We model capillary bundles with three different combinations of corner half angles and method of contact angle calculation: (i) The advancing oil-water contact angle, θowa , is used to calculate the contact angles θgor , θgwr . This approach should be valid if hysteresis in subsequent displacement processes is absent, i.e., if θ owa = θowr . Furthermore, a uniform half angle α is used in all pores. ∗ is calculated by the method describe previously, from (ii) The contact angle θow which θgor , θgwr are obtained. A uniform half angle is used in all pores. (iii) The contact angles are calculated as in case (i). The half angles are assumed 22

S

1

g

0.2

0.8

0.8

0.6

0.6

0.6

krg

0.4

Swi = 0.1

0.4

0.4

S = 0.9 wi

0.8

So

0.2

0.2 0.2

0.4

0.6

Sw

0.8

0 0

0.2

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0.6

0.8

1

S

g

(a)

(b) 200 180 160

ow

θ* (°)

140 120 100 80 60 40 20 0 0

10

20

30

40

50

R (μm)

(c) Figure 10. Simulated results for case (ii) using θowa = 180◦ and α = π/9. (a) Saturation ∗ paths. (b) Gas relative permeability krg as a function of Sg . (c) Calculated contact angle θow ∗ as a function of R. The contact angle θow also increases with Swi.

to vary linearly between α1 (Rmax ) and α2 (Rmin ) as α=

(R − Rmin )(α1 − α2 ) + α2 . Rmax − Rmin

(28)

This corresponds to a continuous change in shape factor G p as a function of pore size R by Eq. (A.4). max = We first consider the above numerical experiments with θ owa = 180◦ and Pow 20 kPa. This represents strongly oil-wet conditions for which the bulk-phase pore occupancies in the capillary bundle are expected to follow the scenario shown in Fig. 9(a). Helland and Skjæveland (2006a,c) have demonstrated that the choice of max is important when θ ◦ max Pow owa > 90 in mixed-wet pores. For small Pow , nonmonotonic invasion order of the pore sizes may occur during the water flood (Helland and Skjæveland, 2006b). This changes the pore occupancies occurring during the gas invasion, and examples are given in Fig. 9(b) and (c). Furthermore, van Dijke et al. max is required in order for several lay(2006) demonstrated that a large value of Pow ers to exist in the corners. In our numerical experiments, α = π/9 in cases (i) and

23

0

10

10

−2

Oil layers displaced N→I

−4

10

10

Layer drainage regime2 k ~S ro

o

kro

k

ro

10

−6

10

10

−8

10

10

−5

10

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−3

−2

−1

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10

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0

0

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−8 −5

10

10

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10

S

−3

10

−2

10

−1

10

0

S

o

o

(a) Case (i) 10

k

ro

10

10

10

10

(b) Case (ii)

0

−2

−4

−6

−8 −5

10

−4

10

−3

−2

10

10

−1

10

0

10

S

o

(c) Case (iii) Figure 11. Oil relative permeability as a function of oil saturation. (a) Case (i) with α = π/9 ∗ calculated. (c) Case (iii) with α varying and θowa = 180◦ . (b) Case (ii) with α = π/9 and θow ◦ and θowa = 180 .

(ii), and in case (iii), α1 = π/6 and α2 = π/18. In all these strongly oil-wet cases it turns out that the pore occupancies shown in Fig. 9(a) occur, even though the contact angles and the half angle vary with pore size in case (ii) and (iii), respectively. Before gas is injected, configuration E exists in the larger pores while configuration C exists in the smaller pores filled with oil. Gas invasion into the water-filled pores results in the displacement sequence from configuraton E to N to I. In the larger oil-filled pores the displacement from configuration C to N to I occurs, and in the smaller oil-filled pores, the displacement C to I occurs. Therefore, the saturation-dependencies of capillary pressure and relative permeability are similar in all the three cases. Only the gas relative permeability, krg , and the oil-water capillary pressure Pow are functions of two saturations. Fig. 10(a) and (b) show the saturation paths and the gas relative permeability calculated for case (ii). The corresponding results for case (i) and (iii) are very similar. In ∗ which were calculated from the values Fig. 10(c) we have plotted the 9 sets of θow of Pow at the 9 different initial water saturations Swi where gas was injected. The ∗ increases with S . The good agreement with case (i) where contactlevel of θow wi angle hysteresis is neglected is a satisfactory result since it indicates that as long as 24

12000

P

cgo

10000

(N → I)

(C → I, E → I)

P

(N → I)

cgo

min{P

cgo

Pgo (Pa)

8000

6000

(N → I)}

6000

4000

4000

2000

2000

C 0 0

(C → N, E → N)

P

cgo

max{Pcgo(N → I)}

8000

P

cgo

Pcgo (C → I, E → I)

10000

Pgo (Pa)

12000

Pcgo (C → N, E → N)

C 10

20

30

40

E

0 0

50

R (μm)

10

20

30

40

E

50

R (μm)

(a) Case (ii)

(b) Case (iii)

Figure 12. Gas-oil capillary entry pressure plotted as a function of pore size for gas invasion when Swi = 0.5. The vertical line separates pore sizes occupied by configuration C and E ∗ calculated. (b) Case (iii) with α after the water flood. (a) Case (ii) with α = π/9 and θow ◦ varying and θowa = 180 .

the contact angles are chosen consistently, the results will be similar even though the resulting sets of contact angles differ. This extends the predictive capability of the model. The oil relative permeabilities are plotted in Fig. 11. Clearly, they are functions of only the oil saturation. However, different trends occur for small k ro . This range is important as drainage of oil by connected layers may be responsible for very small residual oil saturations in real rock samples (e.g., Blunt et al., 1995; Blunt, 2000). In this oil-layer regime, k ro ∝ So2 , which is a consequence of the proportionality between the layer-phase conductances and the layer area squared, g L ∝ A2L (e.g., Hui and Blunt, 2000). In case (i) the layer drainage regime is terminated by the displacement N to I which removes all layers at the same entry pressure in all pores. Notice that for the corresponding range of So , the oil relperm is no longer proportional to the oil saturation squared. Subsequently, displacements proceed directly from C to I until So = 0. This produces distinct regions with different trends. In case (ii), the separation between the different regions is smoothed out since con∗ , and consequently θ tact angle θow gor and θgwr , varies with R in the oil-filled pores, but also in this case k ro is clearly not proportional to So2 . This is demonstrated in Fig. 12(a) where the entry pressures for the gas invasion starting from Swi = 0.5 is plotted. The displacement N to I occurs first in pores which were oil-filled initially. The entry pressure for this event increases smoothly to its maximum value which occurs for the oil layers that exist in the initally water-filled pores where ∗ = θ θow owa . Therefore, this maximum value is equal to the entry pressure for the same displacement in case (i). At this value, all remaining oil layers are displaced simultaneously. This causes the oil relative permeability to become horizontal at the end of the region of oil-layer displacements. As mentioned earlier, the contact ∗ also vary with S , which in turn causes a variation of the entry presangles θow wi sures. This effect is demonstrated by a slight spread of k ro curves in the oil-layer 25

S

0.9

g

0.8

0.2

0.8

0.7 0.6

0.6 kro

0.4

0.5 0.4

0.6

0.4

0.8

0.3 0.2

0.2

0.1

S

o

0.2

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S

0 0

w

0.8

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1

So

(a)

(b) 1 0.9 0.8 0.7

k

rw

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

S

0.8

1

w

(c) Figure 13. Simulation results for case (iii) with α1 = π/30, α2 = π/6, θowa = 120◦ max and Pow = 3.0 kPa. (a) Saturation paths. (b) Oil relative permeability. (c) Water relative permeability.

displacement region in Fig. 11(b). Fig. 11(c) shows the oil relative permeability for case (iii). In this case k ro is proportional to the oil saturation squared only in a region around S o = 0.01. For lower oil saturations the relative permeabilities deflect horizontally as layers are displaced. The entry pressures are plotted in Fig. 12(b), which shows that layers exist for a larger range of Pgo in the pores with narrower corners. In this case the regime of oil-layer displacements, determined by the region between the entry pressure for displacement N to I and the corresponding minimum value, is larger. Although the results are not shown here, we mention that we also have performed simulations where α is a linearly decreasing function of R. The same trend in the k ro curves were observed. Finally, we consider case (iii) with α1 = π/30, α2 = π/6. Thus α is a decreasing function of R, and the largest pores have narrowest corners. The advancing oil-water contact angle is set as θowa = 120◦ which represents weakly oil-wet conditions, i.e., θgwr < 90◦ . Furthermore, we model two cases with different reversal max = 3.0 kPa, for points after primary drainage. In the first example we use Pow 26

S

1

g

0.9

0.2

0.8

0.8

0.7 0.6

0.6 krg

0.4 0.6

Swi = 0.1

0.5 0.4

0.4

S = 0.85 wi

0.3

0.8

0.2

0.2

0.1

S

o

0.2

0.4

0.6

S

0 0

w

0.8

0.2

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0.6

0.8

1

Sg

(a)

(b) 1 0.9 0.8 0.7

k

rw

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

S

0.8

1

w

(c) Figure 14. Simulation results for case (iii) with α1 = π/30, α2 = π/6, θowa = 120◦ max and Pow = 20 kPa. (a) Saturation paths. (b) Gas relative permeability. (c) Water relative permeability.

which the results are shown in Fig. 13. After waterflooding, the fluid configurations are arranged as A-D-C-D-E in the order of increasing pore size in most of the simulations. When gas is injected, displacements occur first from configurations C, D and E to N in the large pores. Finally, the layer displacement N to I occurs. In the smaller pores displacements occur directly from configuration C and D to I. This agrees with the pore occupancies sketched in Fig. 9(c). As demonstrated by the saturation paths in Fig. 13(a), the displacements from D and E to N introduce a significant amount of oil in the form of layers into pores which were filled with bulk water initially. This is also the case for the saturation path starting from Swi = 1. An increase of oil saturation up to about 10% occurs during gas invasion. Since these very thick layers are located between water in the corners and gas in the bulk portion, the expected So -dependency on kro is violated in the region where the oil saturation increases. However, when the oil layers become thinner, gas starts to invade oil-filled pores, So decreases and kro becomes a function of only So , as expected. This is demonstrated in Fig. 13(b). Since the water phase mainly occupies the medium-sized pores, it is now the water relative permeability which depends on two saturations. The gas relative permeability depends only on the gas saturation. 27

540

Pcgo(C → N)

520

P

(D → N), Pmax = 3.0 kPa

P

(E → N), Pmax = 3.0 kPa

P

(E → N),

ow P max ow

4.4

4.6

cgo

ow

cgo

500

Pgo (Pa)

cgo

= 20 kPa

480 460 440 420 400 3.2

3.4

3.6

3.8

4

4.2

R (m)

4.8

5 −5

x 10

Figure 15. Favourable capillary entry pressures for the numerical experiments presented in Figs. 13, 14 plotted as a function of pore size when Swi = 0.5. The entry pressure for the displacement C to N is given by a two-phase expression calculated from the MS-P method and is therefore equal in both cases.

This is in agreement with Fig. 9(c) and the results by van Dijke et al. (2001b). Fig. 14 shows the results when these numerical experiments are modelled using a max = 20 kPa. Thus, in this case water is pushed further larger drainage pressure, Pow into the corners before the water flood begins. This also reduces the length of the water-wet surface in the corners of the pores. The main difference from the previous example is that configuration E forms in all the large pores during the water flood. However, in this case the increase of oil saturation during the gas invasion is less significant, as indicated by the saturation paths in Fig. 14(a). The reason for this is that gas starts to invade pores filled by water and oil simultaneously. Thus the increase of oil saturation by oil layer formation is neutralized by the invasion of gas into bulk oil. To clarify this further, we have compared the entry pressures for the favourable displacements in the gas invasions starting at S wi = 0.5 for max values in Fig. 15. The reason for the deviation is the behaviour of the both Pow entry pressure for displacement E to N which increases with pore size even though θgwr < 90◦ , while the entry pressure for displacement D to N, which occurred max = 3.0 kPa, decreases with pore size. As a consequence, the simulation for Pow max = 20 kPa agrees with the pore occupancies illustrated in Fig. 9(b). This for Pow implies that the gas relative permeability becomes a function of two saturations, as shown in Fig. 14(b), while k rw is a function of only Sw , as shown in Fig. 14(c). Furthermore, kro is a function of only So . These examples illustrate that only by max , different trends of relative permeability can occur in a capillary changing Pow bundle of angular pore shapes.

SUMMARY AND CONCLUSIONS

In this paper we have calculated three-phase relative permeability from a capillary bundle model where the tubes have star-shaped pore cross-sections. Accurate three28

phase entry pressures have been employed for bulk phase invasion, layer phase invasion and for cases where more than one fluid phase enters the pore simultaneously, either by invasion of a bulk phase surrounded by layers, or by invasion of more than one layer phase. This latter displacement type is considered for the first time for gas invasion in angular mixed-wet pores. Furthermore, we have proposed a new method to account for contact-angle hysteresis in three-phase flow which maintains consistency of the capillary entry pressures. The main idea is to invert the MS–P method and, instead of calculating entry ∗ . This quantity in turn is used to pressure, solve for an oil-water contact angle θow calculate gas-oil and gas-water contact angles to be employed in gas invasion. In ∗ is important as it is used in three-phase disangular pores, the calculated value θow placements where gas invasion into the pore occurs together with a displacement of oil by water or vice versa. Reliable relative permeability calculations require accurate expressions for fluid conductances for the possible fluid configurations that can occur in the pore crosssections. We have calculated conductances using the finite-element method and compared the results with existing expressions available in the literature to find the most accurate expression to use in each case, i.e. bulk and corner configurations. We calculated relative permeability for a 3-cornered star-shape accounting for the above features. The specific conclusions are as follows: (i) The method for the calculation of the contact angles has been demonstrated for cylindrical pores. Contrary to the unique cross-over which we find between the different pore-occupancy regions in the absence of hysteresis, our method instead produces a unique line in the region where the calculated contact angle ∗ varies. Although the method requires further development to account fully θow for angular pores, we have provided preliminary results for star-shaped pores which show consistency. (ii) For single-phase flow, we have demonstrated that the commonly assumed proportionality between dimensionless conductance and pore shape factor, g p / A2p ∝ G p , breaks down as the number of corners in a star-shaped pore is increased. Additionally, our calculations of bulk and corner phase conductances demonstrate that the Valvatne and Blunt (2004) expressions are most accurate among the correlations reported in the literature. (iii) For equal half angle star-shaped pores, we observe a kro ∝ So2 dependency at higher oil saturations that goes over sharply to a weaker k ro vs. So dependency at low oil saturations, corresponding to layer displacement. This transition remains but is made more smooth if we introduce half angle variation from pore to pore. (iv) For certain gas displacements, we predict that oil saturation may increase due to simultaneous invasion of accompanying oil layers. A consequence of this is that along a gas displacement path the oil relative permeability shows hys29

teresis. We are uncertain as to whether such displacements should be allowed to occur since they depend on the availability of the surrounding oil phase. (v) For oil-wet conditions, a change in maximum pressure after primary drainage, max , may change the pore occupancies and saturation dependencies of the Pow relative permeability in gas invasion after waterflooding.

Acknowledgements

Support for Johan Olav Helland was provided by Statoil through the VISTA program. The work at Heriot-Watt University was supported in part by EPSRC under grant number EP/D002435/1.

A

GEOMETRY, NOTATION AND DEFINITIONS

A regular star-shaped cross-section is characterized by the radius of the inscribed circle, R, the half angle of the corner, α, and the number of corners n. Because of symmetry it suffices in our analysis to consider only one half of a corner, indicated 1 of by the shaded area in Fig. A.1. For an n-sided star-shape this part represents 2n the entire pore-cross-section. The area A and the distance d are given by A=

R 2 sin(α + πn ) sin πn , 2 sin α

(A.1)

R sin πn d= . (A.2) sin α The area and perimeter of the entire pore cross-section is given by A p = 2n A and d p = 2nd, respectively. Contrary to convex shapes, such as the triangle or the quadrilateral, thestar-shape remains regular when the half angles of all corners are varied simultaneously. All half-angles in the range 0 < α ≤ αmax ,

where

αmax =

π π − , 2 n

(A.3)

are allowed. The largest half-angle αmax corresponds to the limiting n-sided regular convex cross-section. As indicated in Fig. A.1, the limiting case α = α max corresponds to the case when the shaded area forms a right triangle. For clarity, the inscribed radius R of this limiting shape is changed in Fig. A.1. However, the parameters α and R may be varied independently. Mason and Morrow (1991) introduced the shape factor G p , defined as the crosssectional area divided by the perimeter squared, to analyze capillary behaviour 30

Gsym a

a max W

d

G

A

p n

R Gsym

1 Figure A.1. Illustration of geometric parameters defined on 2n of an n-sided star-shaped cross-section. The dashed lines illustrate the geometry when α = αmax . The domain  and associated boundaries  and sym employed in the single-phase conductance calculations are also indicated.

in strongly water-wet irregular triangular cross-sections. For the symmetric starshapes considered here, this parameter may be expressed as Gp =

sin(α + πn ) sin α Ap = . d 2p 4n sin πn

(A.4)

A larger half-angle α corresponds to a larger shape factor G p . In the following we define the parameters introduced to describe three-phase capillary entry pressures in regular star-shaped pore cross-sections for piston-like displacements between the fluid configurations shown in Fig. 1. A similar notation was employed by Helland and Skjæveland (2006a) to describe fluid configurations in equilateral triangular cross-sections. AMs formed by the same pair of phases in a corner are numbered in order from the corner towards the center of the crosssection. The kth AM between a lighter phase i and a denser phase j is referred to as ij AM k. We introduce an indicator function defined by

Iij(k)

⎧ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0

if ij AM k separates bulk phase i and corner phase j , if ij AM k separates bulk phase j and corner phase i , otherwise,

(A.5)

where ij = go, ow, gw. Furthermore, the number of AMs between phases i and j present in a corner before the displacement is denoted N ijinit , and the number after the displacement is denoted N ijfin . The cross-sectional bulk area bounded by ij AM (k) k is denoted A ij(k). The bounding solid-fluid and fluid-fluid lengths are denoted L sij (k)

and L fij , respectively. With reference to Fig. A.2, these parameters are defined as 31

bij(k )

j q ij(k ) L(kfij)

b

(k ) ij

rij

b ij(k )

i

Aij(k )

bij(k ) r ij

R

i L(kfij)

j

q ij(k )

Aij(k )

R

L(ksij)

L(ksij) (a) Iij(k) = 1

(b) Iij(k) = −1

Figure A.2. Representation of the cross-sectional parameters of ij AM k.

follows: (k) Aij

⎧ (k) (k) (k) rij bij sin(βij + α) rij2 βij ⎪ ⎪ ⎨A − + 2 2 = (k) (k) 2 β (k) ⎪ r r b sin(β − α) ij ⎪ ij ij ij ij ⎩A − − 2 2 (k)

and

(k)

(k)

if Iij = 1, if

(k) Iij

(A.6)

= −1,

L fij = rij βij ,

(A.7)

(k) = d − bij(k), L sij

(A.8)

where (k) bij

(k)

=

rij sin βij sin α

,

(A.9)

⎧π ⎨ − α − θ (k) if I (k) = 1, ij ij (k) βij = π2 (A.10) (k) (k) ⎩ +α−θ if I = −1. ij ij 2 (k) Eqs. (A.6)–(A.10) are written with a general contact angle θij which may be equal

and

(k) to θijh if the AM is hinging, θ ijr if the AM is receding, θija if the AM is advancing or θij∗ if the contact angle was calculated from the MS-P method to ensure that Eq. (2) is satisfied. The latter case is only considered for oil-water interfaces in this paper.

We also define shape factors for the area in the pore cross-section occupied by bulk fluid, and for the area in one corner occupied by corner and layer phase. By using the above notation, the bulk (b), corner (c) and layer (L) shape factors may be defined as Ai i = b, c, L , (A.11) Gi = 2 , di where di is the perimeter of Ai . The total cross-sectional bulk area is given by Ab = 6 Aij(k), and the bulk shape factor G b is defined based on the area and perimeter of the bulk-phase occupancy as Ab Gb =   . (k) (k) 2 6(L sij + L fij ) 32

(A.12)

The total cross-sectional area of the corner phase in one corner is given by A c = (1) 2(A − Aij ), and the corner shape factor may be expressed as

Ac Gc =   = (1) (1) 2 2(bij + L fij )

sin α 2

 sin(β (1) + α) sin β (1) ij

(1)

ij

sin α  4 sin2 βij(1) 1 + βij(1)

− βij sin α 

 ,

(A.13)

(1)

sin βij

implying that G c does not depend on bij(1) . In a general fluid configuration with an arbitrary number of AMs, the expression for the layer shape factor can be written as follows for a layer phase m separated by jm AM k on the corner side and im AM l on the bulk side: GL =

AL (k) (2(L sjm

−

(l) L sim

(k)

(l)

+ L fjm + L fim ))2 (k)

,

(A.14)

(l)

where the area of the layer is given by A L = 2(Ajm − Aim ). If the same fluid phase is located on the corner and bulk side of the layer, i.e., if i = j , then l = k + 1 and (k) (k+1) Iim = −Iim .

B

CORNER-PHASE CONDUCTANCES

The correlation for corner conductance gc developed by Zhou et al. (1997), which is valid for θij(1) < π/2 − α, can be written as follows when no-slip conditions are assumed along the AMs: (1)

gc = where

A2c (1 − sin α)2 (φ2 cos θij − φ1 )φ32 12 sin2 α(1 − φ3 )2 (φ2 + φ1 )2 π − α − θij(1) , 2 (1) (1) φ2 = cot α cos θij − sin θij , φ1 =

and φ3 =

π 2

,

(B.1)

(B.2) (B.3)



− α tan α.

(B.4)

Hui and Blunt (2000) extended Eq. (B.1) to allow for contact angles in the range (1) θij ≥ π/2 − α. For no-slip conditions along the AMs, this correlation is given by gc =

A2c tan α(1 − sin α)2 φ32 12 sin2 α(1 − φ3 )(1 + φ3 )2 33

.

(B.5)

Patzek and Kristensen (2001) proposed to scale dimensionless conductances as follows:

e1   gc π 1 π e2 − 0.02 sin α − (B.6) g˜ c = ln cos α − − Gc 4π 6 6 A2c where e1 = 7/8 and e2 = 1/2 when no-slip conditions are assumed along the AMs. They demonstrated that Eq. (B.6) yields a universal curve when plotted against G c . The dimensionless conductance was approximated by a polynomial as a function of the corner shape factor, g˜ c = f (G c ). The dimensional conductance is eventually obtained from Eq. (B.6) as gc =

A2c exp

 f (G c ) + 0.02 sin(α − π )  6 1 ( 4π − G c )e1 cose2 (α − π6 )

.

(B.7)

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