Isotherms capillary pressure

Comparison of the proposed dynamic stability theory for the critical capillary pressure shows acceptable agreement to experimental data on 100-/im permeability sandpacks at reservoir rates and with a commercial a-olefin sulfonate surfactant. The importance of the conjoining/disjoining pressure isotherm and its implications on surfactant formulation (i.e., chemical structure, concentration, and physical properties) is discussed in terms of the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory of classic colloid science. 

We can conclude that the stability of static foam in porous media depends on the medium permeability and wetting-phase saturation (i.e., through the capillary pressure) in addition to the surfactant formulation. More importantly, these effects can be quantified once the conjoining/disjoining pressure isotherm is known either experimentally (8) or theoretically (9). Our focus... 

Figure 6. Evolution of the lamella thickness as it transports down the periodic pore for the conjoining/disjoining pressure isotherm of Figure 4. Three capillary pressures are considered in curves 1 through 3. These capillary-pressure values are also labelled in Figure 4. Curve 2 defines the critical or marginally...

Equation 3 can be rearranged to obtain the adsorption isotherm for pressures below those of the capillary condensation pressure, and for pressures above this limit, the pores can be considered as completely filled with the adsorbate. This leads to the following model MCM-41 nitrogen adsorption isotherm for pores with the capillary condensation pressure pc/po ... 

The dependence of the average excess pressure in foam (capillary pressure of bubbles) on its specific area is established by Derjaguin [105]. The mechanical work W done under isothermal compression (or decompression) of foam equals... 

The direct measurement of the various important parameters of foam films (thickness, capillary pressure, contact angles, etc.) makes it possible to derive information about the thermodynamic and kinetic properties of films (disjoining pressure isotherms, potential of the diffuse electric layer, molecular characteristics of foam bilayer, such as binding energy of molecules, linear tension, etc.). Along with it certain techniques employed to reveal foam film structure, being of particular importance for black foam films, are also considered here. These are FT-IR Spectroscopy, Fluorescence Recovery after Photobleaching (FRAP), X-ray reflectivity, measurement of the lateral electrical conductivity, measurement of foam film permeability, etc. 

At equilibrium film thickness hi the disjoining pressure equals the external (capillary) pressure, n = p This is a common thin film and its equilibrium is described by the DLVO-theory. If h < hcr, at which the film ruptures (see Section 3.2.2), the film is common black (schematically presented in Fig. 3.42). Such a film forms via black spots (local thinnings in the initially thicker non-equilibrium film). The pressure difference nmax - pa is the barrier which hinders the transition to a film of smaller thickness. According to DLVO-theory after nmax the disjoining pressure should decrease infinitely. Results from measurements of some thermodynamic parameters of foam films [e.g. 251,252] show the existence of a second minimum in the 17(6) isotherm (in direction of thickness decrease) after which the disjoining pressure sharply ascends. 

Films were formed by approaching the surfaces of a biconcave drop in a porous plate. The change in film thickness was achieved by gradually increasing the capillary pressure (reversibly and isothermally). 

The course of h(Cci) dependence indicating the decrease in equilibrium thickness up to the transition to NBF as well as the course of n(Ii) isotherm with a distinct barrier transition, reveal the electrostatic character of the forces acting in the film. Thus, double electric layer can be estimated, knowing that n / = pc+T vw The capillary pressure pa was measured experimentally while Tlvw was calculated from Eq. (3.89). The potential was determined within the electrolyte concentration range of 5-10 4 to 10 3 mol dm 3 (Fig. 3.48) in which the films were relatively thick, yielding a value of (po = 36 3 mV. In this respect films stabilised with the zwitterionic lipid DMPC are very similar to those stabilised with non-ionic surfactants [e.g. 100,186,189] (see also Section 3.4.1.1). The low ( -potential leads to the low barrier in the FI(Ii) isotherm which can easily be overcome at relatively low electrolyte concentrations and low pressure values. 

The increase in capillary pressure to 1.2T05 Pa does not alter the NBF thickness. If compared with the IT(/j) isotherm drawn in arbitrary scale that demonstrates the origination of CBF and NBF. It is clear that when (dTl/dh) > 0, films thicknesses cannot be measured since films are thermodynamically unstable (see Section 3.1). NBF does not change its thickness with the decrease in pressure down to p0 = 0.25-105 Pa (the dashed line on bottom of Fig. (3.57)). At this point, however, again with a jump, it transforms into a CBF with the... 

It is interesting to apply the method of n(h) isotherms at high capillary pressures to studying the multilayer structure of black films. The first results obtained refer to biostructures (see Chapter 11) and lamellar structures in solubilising solutions (see Section 3.4.2.5). 

As it is seen from Eq. (3.95) C ,cr depends on the capillary pressure pa. The values of Ceixr determined from the TI(/i) isotherms are lower than those obtained at pa = const. For instance, for NaDoS films Ceicr = 0.334 mol dm 3 (see Table 3.10) while the value found from the n(/i) isotherm is CeiiCr = 0.165 mol dm 3 (see Table 3.8). 

The CBF/NBF transition at pH discussed above was performed at constant ionic strength and capillary pressure. Obviously, such a transition can also be realised when the capillary pressure is altered, for instance, with the Thin Liquid Film-Pressure Balance Technique (see Section 2.1.8). Thus, it is possible to conduct the experiments at lower ionic strength which proves to be important when ri(/i) isotherms of Ci0(EO)4 and NP20 [285], and non-ionic sugar-based surfactants [260] are plotted with respect to pH (see Section 3.4.1). 

Scientific research would benefit if different phenomena are not confused, as this appears to be the case in [260] for Cw and Ceicr. Furthermore, it should be kept in mind that CBF/NBF transition can occur under various conditions change in electrolyte concentration up to reaching Ceicr, increase in capillary pressure (equal to the disjoining) up to reaching nmax at different Cel or pH, i.e. different isotherms, as well as via external disturbances in overcoming nmax-... 

Fig. 3.75 depicts a stepwise P(h) isotherm of system I. This is a summarised isotherm drawn from a number of measurements [358] performed in the two variants A and C of the measuring cell (see Fig. 2.3, Chapter 2). Variant A allows film formation at constant capillary pressure and observation of spontaneous film thickness transitions, i.e. stratified films at pc = const. At 4.5-103 Pa pressure, the film acquired thickness approximately equal to the doubled thickness of two surfactant monolayers, i.e. a bilayer film. Subsequent increase in pressure up to 104 Pa (variant C) did not affect the thickness, which is a good reason to consider this film as being at thermodynamic equilibrium.

The analysis of these P(h) isotherms emphasises that stratified foam films are formed from both systems (I and II). A phenomenon not revealed so far is that spontaneous (under constant capillary pressure) and forced (under various capillary pressures) stepwise thinning can occur in the same single foam film. A question arises as to whether the film that acquired such a thickness is in thermodynamic equilibrium or is kinetically stabilised. It should be noted that these transitions occur only in the direction of increasing pressure, i.e. the process... 

The n(fc) isotherms of different types of foam films are shown in Fig. 7.8. The surfactant (NaDoS) and electrolyte (NaCl) concentrations were the same as those used in the experiments with foams. The equilibrium thickness of thin films and CBF decreased with the increase in pa = II. Films ruptured in a definite range of capillary pressure (marked with arrows on curves 1 and 2). The thickness of NBF did not change and they ruptured at a definite capillary pressure (marked with an arrow on curve 3). 

Pore-radius distributions and ab-/ desorption isotherms are important structural characteristics of generic porous media [80, 88]. The absorption isotherm provides a relation for the liquid uptake of a porous medium under controlled external conditions, viz., the pressure of an external fluid. Within a bounded system, such as a cylindrical tube, a discontinuity of the pressure field across the interface between two fluid phases exists. The corresponding pressure difference is called capillary pressure, Pc. In the case of contact between gas phase, Pg, and liquid water phase, P1, the capillary pressure is given by... 

A close set of equations was formulated in Ref. 16, related to the capillary pressure isotherms determined by the method of standard porosimetry [60], In the latter procedure, the equilibrium amount of the wetting liquid is measured in the porous sample under study. Simultaneously, the amount of the wetting liquid is measured in the standard specimen with a genuine porous structure, in which the capillary equilibrium is established. The standards are kept in thermodynamic equilibrium with the sample. The comparison of the amount of wetting liquid in the membrane with the pore-radius distribution in the standards, enables one to record (with a minimum of theoretical assumptions), the volume-size and surface-size distribution curves, specific pore-space surface area, and absorption isotherm in the membrane of interest, for various wetting liquids. 

Deryaguin and Titijevskaya [37] measured the isotherms of disjoining pressure of microscopic foam films (common thin films) in a narrow range of pressures. At equilibrium, the capillary pressure in the flat horizontal foam film is equal to the disjoining pressure n in it,... 

We present now the extension of the constitutive equation (7) to partially saturated porous media. The material is assumed to be saturated by a liquid phase (noted by index w) and a gas mixture (noted by index g ). The gas mixture is a perfect mixture of dry air (noted by index da) and vapour (noted by index va). Based on most experimental data of unsaturated rocks and soils (Fredlund and Rahardjo 1993), and on the theoretical background of micromechanical analysis (Chateau and Dormieux 1998), the poroelastic behaviour of unsaturated material should be non-linear and depends on the water saturation degree. We consider here the particular case of spherical pores which are dried or wetted under a capillary pressure equal to the superficial tension on the air-solid interface. By adapting the macroscopic non-linear poroelastic model proposed by Coussy al. (1998) to unsaturated damaged porous media, the incremental constitutive equations in isothermal conditions are expressed as follows ... 

The evaluation of the adsorption isotherms in terms of the mesopore size distribution assumes that the sample is unchanged during analysis. However, capillary condensation is accompanied by a capillary pressure that is due to the surface tension trying to minimize the interfacial energy. The respective stress a exerted onto the skeleton of the aerogel can be calculated by [72]... 

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