Capillary pressure/saturation, effect

The different capillary pressure, saturation, and permeability relationships of the different models can be compared. To do this, the effective permeabilities from some of the different models are plotted as a function of the capillary pressure in Figure 8. In the figure, the capillary pressure at which the effective permeability no longer changes is where the medium is fully saturated. Also, the values of the effective permeability are dependent on the diffusion media being tested. Furthermore, the value of the effective permeability at the right end of the curves corresponds to the saturated permeability, except for the model of Weber and Newman, who use a gas-phase residual saturation. 

Ferrand, L.A., and M.A. Celia. 1992. The effect of heterogeneity on the drainage capillary pressure-saturation relation. Water Resour. Res. 28 859-870. 

Mafiiematical models of DNAPL subsurface movement require capillary pressure/saturation and relative permeability/saturation constitutive relationships for DNAPL/water/soil systems. Wettability alterations may have a considerable effect on ciqiillary pressure/saturation relationships as discussed in the previous section. They may also have an intact on relative permeability/saturation relationships yet this has received less attention in file literature in large part due to file experimental difficulties associated with measuring these relationships. 

Predictive models based on the capillary pressure/saturation relationships are commonly used to estimate relative permeability in the absence of experimental data 37, 38). One such predictive model, that of Bradford et al. 37), is used in Figure 6 to illustrate the effects of varying wettability on relative... 

Conventionally, the sample is initially saturated with one fluid phase, perhaps including the other phase at the irreducible saturation. The second fluid phase is injected at a constant flow rate. The pressure drop and cumulative production are measured. A relatively high flow velocity is used to try to negate capillary pressure effects, so as to simplify the associated estimation problem. However, as relative permeability functions depend on capillary number, these functions should be determined under the conditions characteristic of reservoir or aquifer conditions [33]. Under these conditions, capillary pressure effects are important, and should be included within the mathematical model of the experiment used to obtain property estimates. 

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... 

The values of the effective permeabilities vary over orders of magnitude, and this corresponds to the different results of the models. Furthermore, as discussed in various papers,the effective permeability of Natarajan and Nguyen (curve e) varies significantly over a very small pressure range, although they state that their capillary-pressure equation mimics data well. With respect to the various equations, the models that use the Leverett J-function all have a similar shape except for that of Berning and Djilali (curve a), who used a linear variation in the permeability with respect to the saturation. The differences in the other curves are due mainly to different values of porosity and saturated permeability. As mentioned above, only the models of Weber and Newman (curve d) and Nam and Kaviany (curve f) have hydrophobic pores, which is why they increase for positive capillary pressures. For the case of Weber and Newman, the curve has a stepped shape due to the integration of both a hydrophilic and a hydrophobic pore-size distribution. 

Divisek et al. presented a similar two-phase, two-dimensional model of DMFC. Two-phase flow and capillary effects in backing layers were considered using a quantitatively different but qualitatively similar function of capillary pressure vs liquid saturation. In practice, this capillary pressure function must be experimentally obtained for realistic DMFC backing materials in a methanol solution. Note that methanol in the anode solution significantly alters the interfacial tension characteristics. In addition, Divisek et al. developed detailed, multistep reaction models for both ORR and methanol oxidation as well as used the Stefan—Maxwell formulation for gas diffusion. Murgia et al. described a one-dimensional, two-phase, multicomponent steady-state model based on phenomenological transport equations for the catalyst layer, diffusion layer, and polymer membrane for a liquid-feed DMFC. 

Fig. 4 Effective saturation (a normalized saturation value) as a function of capillary pressure, as described by the van Genuchten equation. The curves shown are based on data reported by Schroth et al. (From Refs. " . )...

Our first task is to evaluate the validity of the conventional concept about the mobility control requirement using a simulation approach. This model uses the UTCHEM-9.0 simulator (2000). The dimensions of the two-dimensional XZ cross-section model are 300 ft x 1 ft x 10 ft. One injection well and one production well are at the two extreme ends in the X direction, and they are fully penetrated. The injection velocity is 1 ft/day the initial water saturation and oil saturation are 0.5. The displacing fluid is a polymer solution. The purpose of using the polymer solutuion in the model is to change the viscosity of the displacing fluid. Therefore, polymer adsorption, shear dilution effect, and so on are not included in the model. To simplify the problem, it is assumed that the oil and water densities are the same that the capillary pressure is not included that the relative permeabilities of water and oil are straight lines with the connate water saturation and residual oil saturation equal to 0 and that the water and oil viscosity is 1 mPa s. Under these assumptions and conditions, we can know the fluid mobilities at any saturation. The model uses an isotropic permeability of 10 mD. 

The next question regards the means by which values of any of these quantities to be used in reservoir-engineering calculations may be obtained. There is a continuing history of theoretical attempts to calculate the mobility of foam starting from known quantities and familiar principles of two-phase flow in porous rocks. One of these principally considers the effect of capillary pressure and concludes that this quantity is a principal determinant of the stability and therefore of the population of lamellae. Presuming equilibrium conditions in which the radii of curvature of the Plateau boundaries determines the excess of absolute pressures in the gas over that in the liquid, Khatib et al. (16) computed a limiting value of the capillary pressure. Above this value, the lamellae become too thin for the surfactant to stabilize. Increasing the gas fractional flow decreases the water saturation and raises the capillary pressure. 

The coupled fluid-solid effects was also studied under non-isotheimal conditions for reservoir engineering, such as reported in Liu and Liu (2001c) for mathematical models of coupled multiphase fluid flow, heat transfer and solid deformation considering capillary pressure and both saturated and unsaturated conditions. Similar works was also reported in Wang and Du (2001). 

This example is to test the swelling effects under capillary pressures up to 10 Pa occurring in extremely low-permeable bentonite materials. For this purpose, a simple 1-D case is set up. A one meter long bentonite column is heated on the left hand side. Element discretization length is 0.01m. The initial conditions of the system are atmospheric gas pressure, full liquid saturation and a temperature of 12°C. The heater has a constant temperature of 1(X) C. Flow boundary conditions on the left side are gas pressure of 10 Pa and 15% liquid saturation. On the right side we have atmospheric pressure, full liquid saturation and no diffusive heat flux. As a consequence, a typical desaturation process of bentonite is triggered. The complete set of initial and boundary conditions and the material properties for this example was described in detail by Kolditz De Jonge (2003). 

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