Porous media capillary models

At the present time there exist no flux relations wich a completely sound cheoretical basis, capable of describing transport in porous media over the whole range of pressures or pore sizes. All involve empiricism to a greater or less degree, or are based on a physically unrealistic representation of the structure of the porous medium. Existing models fall into two main classes in the first the medium is modeled as a network of interconnected capillaries, while in the second it is represented by an assembly of stationary obstacles dispersed in the gas on a molecular scale. The first type of model is closely related to the physical structure of the medium, but its development is hampered by the lack of a solution to the problem of transport in a capillary whose diameter is comparable to mean free path lengths in the gas mixture. The second type of model is more tenuously related to the real medium but more tractable theoretically. 

A porous medium is modeled as made up of uniformly distributed straight circular capillaries of the same diameter. The flow through each capillary is an inertia free Poiseuille flow. By comparing the Poiseuille pressure drop and the Darcy pressure drop formulas, deduce an expression for the permeability. Discuss the difference between the result obtained and the Kozeny-Carman permeability. 

In Kozeny theory, the porous medium is modeled as a system of parallel capillaries with different diameters, but constant length. Kozeny assumed no tangential flow at any cross section perpendicular to the direction of bulk flow. The permeability, derived directly from the equations of motion, is ... 

The Washburn model is consistent with recent studies by Rye and co-workers of liquid flow in V-shaped grooves [49] however, the experiments are unable to distinguish between this and more sophisticated models. Equation XIII-8 is also used in studies of wicking. Wicking is the measurement of the rate of capillary rise in a porous medium to determine the average pore radius [50], surface area [51] or contact angle [52]. 

A microscopic description characterizes the structure of the pores. The objective of a pore-structure analysis is to provide a description that relates to the macroscopic or bulk flow properties. The major bulk properties that need to be correlated with pore description or characterization are the four basic parameters porosity, permeability, tortuosity and connectivity. In studying different samples of the same medium, it becomes apparent that the number of pore sizes, shapes, orientations and interconnections are enormous. Due to this complexity, pore-structure description is most often a statistical distribution of apparent pore sizes. This distribution is apparent because to convert measurements to pore sizes one must resort to models that provide average or model pore sizes. A common approach to defining a characteristic pore size distribution is to model the porous medium as a bundle of straight cylindrical or rectangular capillaries (refer to Figure 2). The diameters of the model capillaries are defined on the basis of a convenient distribution function. 

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. 

Redman JA, Grant SB, Olson TM, Estes MK (2001) Pathogen filtration, heterogeneity, and the potable reuse of wastewater. Environ Sci Technol 35 1798-1805 Redman JA, Walker SL, Elimelech M (2004) Bacterial adhesion and transport in porous media Role of the secondary energy minimum. Environ Sci Technol 38 1777-1785 Reeves CP, CeUa MA (1996) A functional relationship between capillary pressure, saturation, and interfacial area as revealed by a pore-scale network model. Water Resour Res 32 2345-2358 Richards LA (1931) Capillary conduction of liquids through porous mediums. Physics 1 318-333... 

Pressure-driven convective flow, the basis of the pore flow model, is most commonly used to describe flow in a capillary or porous medium. The basic equation covering this type of transport is Darcy s law, which can be written as... 

The simplest capillaric model is the one representing a porous medium by a bundle of straight parallel capillaries of average diameter <5, as shown in Fig. 5.14. The equivalent voidage a can be related to the averaged diameter by... 

The displacement flows can be miscible (brine after polymer solution, C02 after oil, steam after water) or immiscible (water after oil). In the former case, it is the mixing process itself which has to be understood and modeled steam recovery requires the thermal transport problem to be accurately modeled. In the latter case, the two fluid phases coexist within the porous medium their relative proportions are determined not only by flow and mixing processes, but equally by interfacial and surface tensions between the three phases (matrix material included). Local (capillary) variations in pressure between the two fluid phases become important. The overall flow field is determined by large-scale pressure gradients. 

The phenomenon of capillary condensation provides a method for measuring pore-size distribution. Nitrogen vapor at the temperature of liquid nitrogen for which cos(0) = 1 is universally used. To determine the pore size distribution, the variation in the amount of nitrogen inside the porous particle is measured when the pressure is slightly increased or decreased. This variation is divided into two parts one part is due to true adsorption and the other to capillary condensation. The variation due to adsorption is known from adsorption experiments with nonporous substances of known surface area, so the variation due to condensation can be calculated. The volume of this amount of nitrogen is equal to the volume of pores with the size as determined by the Kelvin equation. Once a certain model has been selected for the complicated pore geometry, the size of the pores can be calculated. Usually it is assumed that an array of cylindrical capillaries of uniform but different radii, and randomly oriented represents the porous medium. So the Kelvin equation in the form of Equation 3.9 is used. Since condensation is combined with adsorption, the thickness of the adsorption layer... 

After the single-capillary model discussed in the previous section, the next most complex "porous medium" is a bundle of unconnected capillaries of different radii and/or cross-sectional shapes. Porous media of this type have been much studied for other types of problems, but they appear too simplistic for dispersion flow (65). 

In actual use for mobility control studies, the network might first be filled with oil and surfactant solution to give a porous medium with well-defined distributions of the fluids in the medium. This step can be performed according to well-developed procedures from network and percolation theory for nondispersion flow. The novel feature in the model, however, would be the presence of equations from single-capillary theory to describe the formation of lamellae at nodes where tubes of different radii meet and their subsequent flow, splitting at other pore throats, and destruction by film drainage. The result should be equations that meaningfully describe the droplet size population and flow rates as a function of pressure (both absolute and differential across the medium). 

A novel flow cell has been developed to observe on a microscopic level the steady state, cocurrent flow of two pre-equilibrated phases in a porous medium. It consists of a rectangular capillary tube packed with a bilayer of monodisperse glass beads 109 microns in diameter. The pore sizes in the model are of the order of magnitude of those in petroleum reservoirs. An enhanced videomicroscopy and digital imaging system is used to record and analyze the flow data. 

The mechanisms of steady state, cocurrent, two-phase flow through a model porous medium have been established for the complete range of capillary numbers of interest in petroleum recovery. A fundamental understanding of the mobile ganglia behavior observed requires a knowledge of how phases break up during flow through porous media. Several mechanisms have been reported in the literature and two have been observed in this flow cell. 

The theoretical basis of the Hg-injection method is defined by Laplace law. By using a capillary model where the porous medium is assimilated to a bundle of cylindric capillary tubes the capillary pressure is Pc = y(l/Rci+l/Rc2) = 2y cos0 /Rc (3) where Pc is the capillary pressure Rd and Rc2 are mutually perpendiculcir radii of a surface segment R is the average pore-throat size (pm) 0is the angle between mercury menisc and pore wall (for mercury 0=140°) y is the interfacial tension (for mercury y = 0.480 N/m). 

IMarkin (Ml) made an extensive study of capillary equilibrium in porous solids, and formulated a model of a porous medium in which the... 

For the tortuous and irregular capillaries of porous media, it has been reported theoretically and experimentally that a minimum in the permeability of adsorbates at low pressures is not expected to appear. In our study of n-hexane in activated carbon, however, a minimum was consistently observed for n-hexane at a relative pressure of about 0.03, while benzene and CCI4 show a monotonically increasing behavior of the permeability versus pressure. Such an observation suggests that the existence of the minimum depends on the properties of permeating vapors as well as the porous medium. In this paper a permeation model is presented to describe the minimum with an introduction of a collision-reflection factor. Surface diffusion permeability is found to increase sharply at very low pressure, then decrease modestly with an increase in pressure. As a result, the appearance of a minimum in permeability was found to be controlled by the interplay between Knudsen diffusion and surface diffusion for each adsorbate at low pressures. 

The method that has received the most attention belongs to the first category. Specifically, the particular model is the capillary tube model for porous medium and the power law model for the emulsion (J6). The shear-stress (t) rate relationship for a power law fluid is given by... 

In the original Buckley-Leverett theory, gravitational, compressibility and capillarity are ignored. Devereux (36) presents the solution for the case of constant pressure, and the constant-velocity case was derived by Soo and Radke 12). The model requires a knowledge of the capillary retarding force per unit volume of the porous medium, and the relative permeabilities of the oil droplets in the emulsion and the continuous water phase. These relative permeabilities are assumed to be functions of the oil saturation in the porous medium. These must be determined before the model can be used. 

Re-entrainment of liquid droplets that are captured can also occur as a result of squeezing when the local pressure drop is increased to overcome the capillary resistance force. The shape of the liquid droplets depends on the wettability of the rock. On the basis of this physical picture, Soo and Radke 12) proposed a model to describe the flow of dilute, stable emulsion flow in a porous medium. The flow redistribution phenomenon and permeability reduction are included in the model. Both low and high interfacial tension were considered. 

As a model for this transport process, consider the axisymmetric spreading of a fluid of density p + Ap in a porous media containing a fluid of density/ ). Assume the fluid spreads out over an impermeable bottom and that the volume of the dense fluid or gravity current is given by Qta, where t is time. The viscosity of the gravity current is p. and the permeability of the porous medium is k. Also, neglect the effects of capillary forces and assume the flow is dominated by a balance between buoyancy and viscous forces. This balance of forces in a porous medium is described by Darcy s equations, which are given by... 

In our further description of transfer processes we will not address the complicated issues involved in porous medium structures, and hence our discussion will be based on a simple model of transfer through an isolated capillary. 

A capillary tube model can be used to estimate the permeability of the medium before fines deposition or release has occurred. The Car-man-Kozeny equation uses the diameter of the substrate particles, dg, and the tortuosity of the medium, r, to evaluate the effective permeability of the porous medium. 

The flow situation in the porous medium comprising the column of packed resin beads is a complex one. One approach long used to model flow through porous media has been to consider the medium as made up of bundles of straight capillaries or assemblages of randomly oriented straight pores or capillaries in which the flow is of Poiseuille type. 

Despite our use of a capillary model to characterize a porous medium, most porous beds employed for chromatographic purposes are random and generally the medium is isotropic. In such media, the effective solute dispersivity still arises from the nonuniform pore velocity coupled with molecular diffusion... 

To the extent that dispersion in an inertia free porous medium flow arises from a nonuniform velocity distribution, its physical basis is the same as that of Taylor dispersion within a capillary. Data on solute dispersions in such flows show the long-time behavior to be Gaussian, as in capillaries. The Taylor dispersion equation for circular capillaries (Eq. 4.6.30) has therefore been applied empirically as a model equation to characterize the dispersion process in chromatographic separations in packed beds and porous media, with the mean velocity identified with the interstitial velocity. In so doing it is implicitly assumed that the mean interstitial velocity and flow pattern is independent of the flow rate, a condition that would, for example, not prevail when inertial effects become important. 

To analyze the flow through a porous medium, we can, as before, model the medium as a collection of parallel cylindrical microcapillaries. As noted in Section 4.7, the actual sinuous nature of the capillaries may be accounted for by the introduction of an empirical tortuosity factor. The results for electroosmotic flow through a capillary are then readily carried over to the porous medium by using Darcy s law (Eq. 4.7.7) and, for example, the Kozeny-Carman permeability (Eq. 4.7.16). 

We observe here that in a capillary the volume flow rate due to a fixed pressure gradient is proportional to a Tra l8p. dpldx) for a circular capillary). The electroosmotic flow rate is proportional to U multiplied by the cross-sectional area TTa Therefore, the ratio of electroosmotic to hydraulic flow rate will be proportional to a. Thus, for example, if we employ a capillary model for a porous medium, it is evident that as the average pore size decreases electroosmosis will become increasingly effective in driving a flow through the medium, compared with pressure, provided... 

The phenomena and processes described can be modeled by convective diffusion equations with chemical reactions. In the simplest model, we may apply these equations in a cylindrical capillary and by means of a capillary model to a porous medium. Assuming dilute solutions, rapid chemical reactions, the double-layer thickness to the soil pore radius and the Peclet number based on the pore radius both small, the overall transport rate for the ith species in a straight cylindrical capillary is... 

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