3 Supplementary and Closure Relations
3.4 Capillary Pressure and Relative Permeability
Cubic Equations of State
For cubic equations of state, Equation 3.14 can be evaluated analytically if we use Equation 3.4 as mixing rule. We get [68]
ln Φκα= bκα bα
(Zα−1)−ln(Zα−Bα?) + A?α
B?p
u2α−4wα
bκα bα −δκα
ln2Zα+Bα?(uα+p
u2α−4wα) 2Zα+Bα?(uα−p
u2α−4wα) , (3.15) with the abbreviations
A?α= aαpα
(RTα)2 , Bα? = bαpα
RTα
, and δκα=
√aκα 2aα
X
λ
xλα(1−¯kακλ) q
aλα.
Ideal Mixtures
Often, we assume a fluid phase to be an ideal mixture of its constituent components. In this case, all fugacity coefficients are independent of the composition of the phases, and we get the relation
fκ=pαxκαΦκα(Tα, pα).
For ideal gases, Φκα is equal to1; for liquid phases, the fugacity coefficient and the fluid pressure are usually combined into a single coefficient
Pακ(Tα, pα) :=pαΦκα(Tα, pκα).
For components that exhibit a low miscibility with the liquid, this coefficient is called HENRY
coefficient; otherwise, the component is the dominant constituent of the liquid, andPακis either called RAOULTcoefficient,vapor pressureorsaturation pressure.
Capillary Pressure and Relative Permeability 37
Sw
pc,nw
1.0 0.75
0.50 0.25
VANGENUCHTEN BROOKS-COREY
(a)
Sw
kr,α
1.0 0.75
0.50 0.25
1.0
0.75
0.50
0.25
kr,wVANGENUCHTEN kr,nVANGENUCHTEN kr,wBROOKS-COREY kr,nBROOKS-COREY
(b)
Figure 3.2: Principle shapes of the BROOKS-COREY and VAN GENUCHTEN functions for two-phase flow in porous media. The parameters of the functions have been converted using the approach proposed by LENHARD, PARKER, and MISHRA[50].
(a)Capillary pressure. (b)Relative Permeability.
GENUCHTEN. These functions will also be the subject of the following subsections and are illustrated in Figure 3.2.
3.4.1 Generic Two-Phase Relations for Relative Permeability
Before we deal with the saturation-dependent two-phase capillary pressure curves, we need to discuss the connection between the capillary pressure and the relative permeability for them. As it turns out that, although in general we can regard the relative permeabilitieskr,α
and the capillary pressurespc,αas independent quantities, they have been shown to be related in the most common cases [39]. If only two fluid phases are involved, and the capillary pressure between those only depends on the fluid saturations, the relative permeabilities are given by the equations [39]
kr,w =SwA
RSw
0
pc,nw( ˆSw)−B
d ˆSw R1
0
pc,nw( ˆSw)−B
d ˆSw
C
and (3.16)
kr,n = (1−Sw)A
R1
Sw
pc,nw( ˆSw)−B
d ˆSw R1
0
pc,nw( ˆSw)−B
d ˆSw
C
(3.17)
with some constants A, B, andC. We note that it is not possible to directly apply these equations to capillary pressure functions without choosing concrete values forA,B, andC.
3.4.2 Relation of BROOKSand COREY
The most widely applied approach for capillary pressure in two-phase flows is the one proposed by BROOKSand COREY[13, 39]. This approach considers two fluid phases, the wetting and the non-wetting phases. There, the fluid for which the contact angle of the fluid-fluid interface with the surface of the solid is smaller than90◦is defined as thewettingfluid, the other fluid is callednon-wetting. The capillary pressure relation proposed by BROOKSand COREYnow assumes that the only quantity on which the difference in pressure between the wetting and the non-wetting phase depends, is the saturation of the wetting phase. Under these preconditions, BROOKSand COREYproposed [13] the relation
pc,nw =pn−pw =peS−
1 λbc
w , (3.18)
wherepeis theentry pressureandλbcis ashape parameter. Both of these parameters are highly specific to the properties of the porous medium as well as to the properties of the considered fluids [39]. For this reason, these two parameters are usually obtained experimentally [39].
In conjunction with the BROOKS-COREYcapillary pressure curve, the BURDINE parameteriza-tionA=B = 2, andC= 1is normally used [39] for relations of the relative permeabilities which are implied by Equations 3.16 and 3.17. This yields
kr,w=Sw
2+3λbc
λbc and (3.19)
kr,n= (1−Sw)2 1−S
2+λbc λbc
w
!
. (3.20)
3.4.3 Curve ofVAN GENUCHTEN
As an alternative to the curves proposed by BROOKS and COREY, we can also use the relation proposed byVANGENUCHTEN[87, 39]. Like the BROOKS-COREYcurves, theVAN
GENUCHTENapproach presumes that the value of capillary pressure depends solely on the saturation of the wetting phase, but in contrast to the BROOKS-COREYrelation, it does not use the concept of entry pressure. Instead, theVANGENUCHTENapproach assumes that the capillary pressure is zero in a medium which is fully saturated by the wetting phase. Also clearly visible in Figure 3.2 is the fact that forVANGENUCHTENcurves, the slope is infinite if the porous medium is fully saturated by the wetting phase. This property is thus similar to the entry pressure concept in the BROOKS-COREYapproach.
The concrete function proposed byVANGENUCHTEN[87] is given by pc,nw =pn−pw = 1
αvg
S−
1 mvg
w −1
nvg1 ,
whereαvg,mvg, andnvgare shape parameters. Like for the BROOKS-COREYrelation, at least two of those parameters must be obtained by fitting experimental data. For the third, it is
Capillary Pressure and Relative Permeability 39
often possible to use the relation
mvg= 1−1/nvg
as proposed byVANGENUCHTEN[87]. We note that instead of using this equation, we also may take advantage of others which better fit experimentally obtained capillary pressure curves [47].
Like for the BROOKS-COREY capillary pressure curves, we can evaluate Equations 3.16 and 3.17 to gain closed relative permeability functions. In contrast to the BROOKS-COREY
approach, the parameterization proposed by MUALEM is usually used [39],i.e., A = 0.5, B = 1, andC = 2. With these values, we get
kr,w=Swvg
1− 1−Sw
1 mvg
mvg2
and (3.21)
kr,n= (1−Sw)γvg
1− 1−Sw
1 mvg
mvg2mvg
(3.22) wherevgandγvgare parameters that depend on the microscopic properties of the porous medium. Often [39] we can assume their values to bevg=1/2andγvg =1/3.
3.4.4 Three-Phase Systems
The final capillary pressure relations which we will cover in detail were proposed by STONE[79, 39], and are concerned with three-phase flow in porous media.
For such systems, we assume three potentially present phases called “wetting liquid”, “non-wetting liquid”, and “gas”, indicated byh·iw,h·in, andh·igin the following. For such systems, it has been observed [52], that in many situations the relative permeability of the wetting liquid phasekr,wand the relative permeability of the gas phasekr,g primarily depend on the saturations of the wetting liquid and the gas, respectively. Also, the capillary pressures between the wetting liquid and the non-wetting liquidpc,nw has been shown to depend mainly on the saturation of the wetting liquid and we can assume that the only non-negligible dependency of the capillary pressure between the gas and the non-wetting liquid phasespc,gn
is the saturation of the gas phase [39].
The main idea of the two approaches proposed by STONE[79, 80] is to take advantage of the capillary pressure relations for the two-phase cases with the wetting and the non-wetting liquids—which we will indicate byh·iwn in the following—as well as the capillary pressure curves for the two-phase system featuring the non-wetting liquid and the gas—indicated byh·ing. Under these assumptions, we can use the appropriate two-phase relations to define the three-phase capillary pressure relations,i.e.,
pc,nw(Sw, Sn, Sg) =pcwn,nw(Sw) and (3.23) pc,gn(Sw, Sn, Sg) =pcnw,gn(1−Sg). (3.24)
Similarly, we define the wetting liquid and the gas relative permeabilities of the corresponding two-phase systems,i.e.,
kr,w(Sw, Sn, Sg) =krwn,w(Sw) and (3.25) kr,g(Sw, Sn, Sg) =krng,g(1−Sg). (3.26) For the relative permeability of the non-wetting liquidkr,n, STONEproposed two approaches:
The first [79] uses the relation
kr,n(Sw, Sn, Sg) =Sn
krwn,n(Sw) 1−Sw
krng,n(1−Sg) 1−Sg
, (3.27)
whilst the second [80] defineskr,nas
kr,n(Sw, Sn, Sg) =(krng,n(1−Sg) +krng,g(1−Sg))·(krwn,n(Sw) +krwn,w(Sw))
−(krwn,w(Sw) +krng,g(1−Sg)).
3.4.5 Advanced Concepts
We note that, the approaches which we covered so far do not take residual saturations [39]
into account in order to simplify the discussion. We can imagine the residual saturation of a phase as the saturation of the fluid that cannot be transported by advection – an effect that is caused by various mechanisms that trap fluids within a porous medium [39]. Having said that, this material can be dissolved by the other fluids and be transported this way.
Also, there are quite a few advanced concepts when it comes to capillary pressure: Amongst these are temperature [73] and fluid composition [27] dependence, approaches which dis-card the assumption of mechanical equilibrium [38], and methods to take hysteresis into account [65, 59, 10].