2.2 Properties of Porous Media
2.2.4 Fluid-Matrix Interaction Properties
A multiphase system is characterized by multiple phases (α) which may coexist at the same time in an REV. The distribution of the fluid within the REV is expressed by the volume fractionSαwhich is the fluid occupied within the pore space of the respective volume:
Sα= volume occupied by phaseα
volume of pore space within the REV, 0Sα1. (2.7) Evidently the sum of all fluid phases has to be unity:
∑
αSα=1.0. (2.8)When a fluid phase drains from a porous medium it reaches a point where it becomes discontinuous and stops flowing. Theresidual saturationof the phaseα(Sαr) is the fluid that cannot be mobilized and recovered from the porous media. In the literature there are many definitions and methods for determining the residual saturation in principle depending on the scale of the experiment, the laboratory or field conditions, and type of engineering branch (e.g. petroleum, environmental). A short review of these methods is given in Adamski et al. [2003]. The residual saturation is dependent on the properties of the porous medium and of the fluids and on the system’s history of displacement processes (e.g. which may lead to hysteretic behavior Helmig [1997]).
2.2.4.2 Interfacial Tension and Wettability
A fluid phase in contact with another fluid or solid phase has a free interfacial energy between the phases caused by the net attractive forces that exist in those molecules that are near the interface. The interfacial energy manifests itself asinterfacial tension σi,k[J/m2] or [N/m], and it is constant for any pair of phasesiandk. The equilibrium between the wetting and non-wetting phase is given by the Young formula of Helmig [1997]:
σGLcosθ=σSG−σSL (2.9)
cosθ=σSG−σSL
σGL (2.10)
whereθis the contact angle between the two fluids.
The contact angles reflects the affinity of a fluid to be wetting or non-wetting for the solid surface: whenθis less than 90◦, the fluid is wetting; whenθis greater than 90 the fluid is non-wetting.
2.2.4.3 Capillary Pressure
Figure 2.2 depicts an interface between two fluids that coexist in a porous medium.
The capillary tube has radiusrand connects the two fluids. At equilibrium the surface tensionσof the free liquid surface causes capillary forcesFcapthat are in equilibrium with the gravitational forcesFgravof the wetting-phase column.
2πrσcos(θ) =πr2hgρw, (2.11)
wherehis the height of wetting fluid,rthe radius of the capillary tube. Hence,hcan be formulated:
h=2σcos(θ)
rgρw (2.12)
pAw=pAn=pA (2.13)
pBn=pA−ρngh
pBw=pA−ρwgh (2.14)
Figure 2.2: Capillary tube connecting two immiscible fluids with interfacial tensions between solid and non-wetting fluidσsn, between solid and wetting fluid σsw, between wetting and non-wetting fluidσwn. At equilibrium the capillary forcesFcapare equal to the gravitational forcesFg, the fluid height ishand the contact angleθ.
Writing the differences between the two pressures results:
pBn−pBw= (ρn−ρw)gh=pBc (2.15)
The capillary pressure can therefore be expressed with Laplace equation:
pBc=2σcos(θ)
r (2.16)
We have shown the derivation of the capillary pressure for a single capillary tube.
Equation (2.16) defines a microscopic capillary pressure between two fluid phases that are at equilibrium and expresses pressure relationships at a microscopic point on the interface between the fluid phases.
For practical applications at macroscale or field scale, however, the microscopic capillary pressure has very little use. In an actual porous medium the capillary pressure, as well as, the pressures of the wetting and the non-wetting fluids are considered as astatistical average taken over the void space in the vicinity of a considered point in the porous medium(Bear [1988]). Then, thecapillary pressure(pc) is defined as the difference between the pressure in the nonwetting phase (pnw) and the pressure in the wetting phase (pw):
pc=pnw−pw (2.17)
Two of the most common formulations for describing the capillary pressure on the macroscale are given by the semi-empirical formulas of Brooks and Corey [1964] (BC)
pc=pdS−e1λ, (2.18)
where:
Se=S1−Sw−Swrwr effective saturation [-], Swr residual water saturation [-],
λ BC-parameter [-],
pd BC-parameter, entry pressure [Pa].
and Van Genuchten [1980] (VG) pc=1
α Se−1/m−1 1/n (2.19)
where:
n,m VG-parameter [-],
α VG-parameter [1/Pa].
Equations (2.18) and (2.19) describe the capillary pressure as a function of the wetting phase saturationpc=pc(Sw). They are based on parameters that characterize the pore space geometry and are determined by fitting to experimental data. Their analytical determination is practically impossible considering the complexity of the porous media.
Theλ- BC-parameter usually lies between 0.2 and 3.0. Very smallλvalues indicate a material composed of grains of similar size, whereas a largeλvalue shows a highly non-uniform material. The entry pressure is the capillary pressure required to displace the wetting fluid from the largest occurring pore (Figure 2.3). The m-parameter-VG is usually defined asm=1−1/n.
2.2.4.4 Relative Permeability
In a multiphase fluid system the hydraulic conductivity defined in 2.6 is : Kf=Kkrαραg
µα (2.20)
wherekrαis therelative permeabilitywhich accounts for the increased resistance to flow for a given phase due to the presence of the other phase. Even though the
Figure 2.3: Capillary pressure-saturation curves fitted to the experimental measure-ments of Kazemi [1976] withBrooks-CoreyandVan Genuchtenmodels relative permeability,krα, depends on the phase saturation, saturation history, pore structure and wettability, the common equation ofkrαincludes only the dependency on phase saturation,Sα:
0nα=1phases
∑
krα(Sα)1. (2.21)Thehydraulic conductivityof phaseαis defined:
Kα=Kkrα, (2.22)
For a wetting fluid saturation decrease, the cross-sectional area available for flow also decreases and the wetting fluid must flow around the areas filled with non-wetting fluid. This results in increasing the tortuosity of the flow channels and longer flow paths for the wetting fluid. For saturations below the residual saturation the relative permeability of the phase is zero which means that the phase is immobile.
The relative saturation of the phase is strictly monotonously increasing between krα(Sα=Srα) =0andkrα(Sα=1) =1.
The relative permeability-saturations functions were developed based on purely em-pirically approaches or on quasi-analytical ones from simplified pore network models.
Like in the capillary pressure-saturation case, the most widely used approaches are Brooks-Corey(BC) in conjunction withBurdinetheorem:
krw= (Se)(2+3λ)/λ krn= (1−Se)2 1−Se2+λλ
(2.23)
andVan Genuchten(VG) in conjunction with the approach ofMualem:
krw=Sεe 1− 1−Sem1 m 2
krn= (1−Se)γ 1−Sem1
2m (2.24)
Parametersεandγare form parameters which describe the connectivity of the pores.
Typical values areε=1/2andγ=1/3(Helmig [1997]). Figure 2.4 illustrates the plot of the relative permeability-saturation curves based on (BC) and (VG) approaches.
The relative-permeability - saturation relationship for fractures is described in Sec.
2.5.6.