Limiting capillary pressure

Finally, before leaving our exploration of the dusty gas model, we must compare the large pore (or high pressure) limiting form of its flux relations with the corresponding results derived in Chapter 4 by detailed solution of the continuum equations in a long capillary. The relevant equations are (4,23) and (4,25), to be compared with the corresponding scalar forms of equations (5.23) and (5.24). Equations (4.25) and (5.24).are seen to be identical, while (4,23) and (5.23) differ only in the pressure diffusion term, which takes the form... 

Coimectivity is a term that describes the arrangement and number of pore coimections. For monosize pores, coimectivity is the average number of pores per junction. The term represents a macroscopic measure of the number of pores at a junction. Connectivity correlates with permeability, but caimot be used alone to predict permeability except in certain limiting cases. Difficulties in conceptual simplifications result from replacing the real porous medium with macroscopic parameters that are averages and that relate to some idealized model of the medium. Tortuosity and connectivity are different features of the pore structure and are useful to interpret macroscopic flow properties, such as permeability, capillary pressure and dispersion. 

It has been established (P8, R5) that when the value of S exceeds about 0.25, the liquid bridges begin to coalesce with one another and the bonding mechanism changes over from the pendular to the funicular state. When S exceeds 0.8, the existence of discrete liquid bridges is no longer possible and now the capillary pressure state alone exists. Thus, the funicular state lies in a range of saturation bounded by the lower and upper critical limits denoted by Sp and Sc, respectively. 

In this case, as stated earlier, the limiting capillary pressure... 

Although the approach is theoretically sound, both the proposed relationships between capillary pressure and saturation (Equations 6.23 and 6.24) are highly nonlinear and limited in practicality by the requirement of multiparameter identification. In addition, due to the inherent soil heterogeneities and difference in LNAPL composition, the identified parameters at one location cannot be automatically applied to another location at the same site, or less so at another site. For example, Farr et al. (1990) has reported the Brooks-Corey and van Genutchen parameters, X, ii, and o.a0, for seven different porous media based on least-square regression of laboratory data. The parameters are found to vary about one order of magnitude and do not show any specific correlation for a particular soil type. 

In a subsequent theoretical study, Stamenovic [60] obtained an expression for the shear modulus independent of foam geometry or deformation model. The value of G was reported to depend only on the capillary pressure, which is the difference between the gas pressure in the foam cells and the external pressure, again for the case of <)> ca 1. Budiansky et al. [61] employed a foam model consisting of 3D dodecahedral cells, and found that the ratio of shear modulus to capillary pressure was close to that obtained by Princen, but within the experimental limits given by Stamenovic and Wilson. 

The method of standard porosimetry (MSP)41-43 was one of the first approaches used to obtain air-water capillary pressure curves for GDMs.16 In this test, a GDM sample is initially saturated with water and contacted with a water-saturated porous disc, which is a standard with known Pc(Sw) behavior. The capillary pressure of the sample-standard system is varied by allowing the liquid to evaporate from the standard and sample while in contact. If the two media can be assumed to be in capillary equilibrium, their capillary pressures are equal. Saturation is determined by measuring the weights of the sample and standard periodically. The capillary pressure of the system is found by reference to the known capillary pressure curve of the standard. This method is limited to scanning... 

For a suitably high critical value of A, this theoretical model predicts a lower limit on the equilibrium thickness that can be observed. This lower limit on Z, Z n, is defined by the conditions F= 0 and <1F/<1L = 0 since for a stable film F= 0 and dF/dL > 0 (Clarke, 1987). Various solutions to these conditions have been examined by Knowles and Turan (2000). In the absence of capillary pressure and external pressure, Zmin = 2.58. Using reasonable estimates for Knowles and Turan estimate Zmin to be >6.50 A. That in practice the observed intergranular film thicknesses are typically of the order of 1-2 nm in non-oxide engineering ceramics indicates that the relevant Hamaker constants for ceramics are significantly lower than the critical value. 

The concept of the limiting capillary pressure of foam collapse in porous medium has been introduced by Khatib, Hirasaki and Fall [174] and are based on studies of Kristov, Exerowa and Kruglyakov on the critical capillary pressure in static foams [12,179-182] (see Chapters 6 and 7). 

Fig. 10.20. Dependence of film rupture pressure on salt concentration black circles - 0.03% NaDoS blank squares - 0.5% NaDoS the arrow indicates that the film is stable up to the limiting capillary pressure for the porous glass frit.

There are three important ways by which capillary pressure affects the dynamics of dispersion formation and disappearance the lower and upper limits on the range of capillary numbers over which capillary snap-off can occur in homogenous media, and an upper limit on the capillary number above which lamellae are unstable and droplets quickly coalesce. 

A third, related limit on the capillary pressure is created by the existence of an upper critical capillary pressure above which the life times of thin films become exceedingly short. Values of this critical capillary number were measured by Khistov and co-workers for single films and bulk foams (72). The importance of this phenomenon for dispersions in porous media was confirmed by Khatib and colleagues (41). Figure 5 shows the latter authors plot of the capillary pressures required for capillary entry by the nonwetting fluid and for lamella stability versus permeability of the porous medium. 

As discussed above, the upper limiting capillary pressure for lamella formation must be considered, as well as the limiting capillary pressure for lamella stability. 

Capillary pressure effects appear to explain the very important 1964 discovery of Bernard and Holm that dispersions could make the mobility of the nonwetting phase essentially independent of the absolute permeability of the porous medium (52). (See above.) Indeed, the theoretical analysis of Khatib, et al., which was corroborated by experiments, gave dispersed-phase mobilities at the upper limiting capillary pressure (for coalescence) that were nearly constant for absolute permeabilities ranging from 7D to ca. 1,000D (41). 

Relationship between capillary entry pressure, limiting capillary pressure, and permeability of the medium. (Reproduced with permission from Ref. 41. Copyright 1986 SPE-AIME.)... 

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