Porous media dispersion

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. 

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. 

Presence of Autocorrelation in Flow Paths Dispersion can result because all pores in tlie porous medium are not accessible to a fluid element after it has entered a particular flow path. In order words, the connectivity of the medium is not complete. 

Edwards, DA, Charge Transport Through a Spatially Periodic Porous Medium Electrokinetic and Convective Dispersion Phenomena, Philosophical Transactions of the Royal Society of London A 353, 205, 1995. 

Foam generated in porous media consists of a gas (or a liquid) dispersed in a second interconnected wetting liquid phase, usually an aqueous surfactant solution (1). Figure 1 shows a micrograph of foam flowing in a two-dimensional etched-glass porous medium micromodel (replicated from a Kuparuk sandstone, Prudhoe Bay, Alaska (2)). Observe that the dispersion microstructure is not that of bulk foam. Rather discontinuous... 

Regardless of the transport equation considered, the major effect of sorption on contaminant breakthrough curves is to delay the entire curve on the time axis, relative to a passive (nonsorbing) contaminant. Because of the longer residence time in the porous medium, advective-diffusive-dispersive interactions also are affected, so that longer (non-Fickian) tailing in the breakthrough curves is often observed. 

Diw is the molecular diffusion coefficient of the chemical in water, x is tortuosity, and aL is the (longitudinal) dispersivity (dimension L). The first term describes molecular diffusion in a porous medium (Eq. 18-57), the second the effect of dispersion (Eq. 22-52). Typical values of the dispersivity aL for field systems with flow distances of up to about 100 m lie between 1 and 100 m. Since aL depends strongly on the scale... 

The transport of TCE through a porous medium, such as soil, can be described by the following ID advection-dispersion equation (Fetter, 1993). 

Dispersion arises from the fact that, even in a relatively homogenous porous medium, small-scale heterogeneities exist which cause airflow to proceed along various channels at different rates. Barometric pumping causes a significant increase in the coefficient of hydrodynamic dispersion over a pure diffusion-based transport model, thus increasing the overall transport rate. 

Achdou, Y. and Avellaneda, M. (1992) Influence of pore roughness and poresize dispersion in estimating the permeability of a porous medium from electrical measuraments, Phys Fluids 4 (12), 2561... 

The flow through porous media of emulsions, foams, and suspensions can be important in a number of applications ranging from fixed-bed catalytic reactors in the chemical process industries, to flows through soil environments, to flow in underground reservoirs. To understand the flow of dispersions in porous media one needs a knowledge not only of the properties of the dispersion, but also of the porous medium. Pore characterization itself has been reviewed elsewhere [30,416]. 

Jang, Jiin Yuh and Chen, Jiing Lin. Thermal Dispersion and Inertia Effects on Vortex Instability of a Horizontal Mixed Convection Flow in a Saturated Porous Medium . Int. J. Heat and Mass Transfer. Vol. 36. No. 2, pp. 383-389. 1993. 

Dispersivity is a property of the porous medium that has been shown to be proportional to the scale of the system under consideration [4]. Thus, the dispersivity of a porous medium in a laboratory column will be considerably smaller than the dispersivity of an aquifer through which contaminated ground-water is flowing over distances of hundreds of meters. Presumably, this scale-effect is due to the increased spreading caused by variations in velocities due to larger scale heterogeneities [2]. 

No exact general criterion is available when it is necessary to include the relaxation terms in the equations of change however, relaxation terms are necessary for viscoelastic fluids, dispersed systems, rarefied gases, capillary porous mediums, and helium, in which the frequency of the fast variable transients may be comparable to the reciprocal of the longest relaxation time. 

Eddies If tlie flow within tlie individual flow chaimels of Uie porous medium (soil) becomes turbulent, dispersion results from eddy migration. 

The dispersion of a liquid that flows inside a porous medium is the macroscopic result of some individual motions of the liquid determined by the pore network of the solid structure. These motions are characterised by the local variations of the velocity magnitude and direction. Accepting the simplified structure of a porous structure shown in Fig. 4.31, the liquid movement can be described by the motion of a liquid element in a +x direction (occurring with the probability p) compared to the opposite motion or —x displacement (here q gives the probability of evolution and Ax represents the length portion of the pore which is not in contact with the nearby pores). Indeed, the balance of probability that shows the chances for the liquid element to be at time t in x position can be written as follows ... 

In the scientific literature, we can find a large quantity of experimental results where the flow characterization inside a porous medium has shown that the value of the dispersion coefficient is not constant. Indeed, for the majority of porous structures the diffusion is frequently a function of the time or of the concentration of the diffusing species. As far as simple stochastic models cannot cover these situations, more complex models have been built to characterize these dependences. One of the first models that gives a response to this problem is recognized as the modd of motion with states having multiple vdodties. 

Films of wetting fluid that extend across pores and may cause dispersion formation are called lamellae. (See Figures 2 and 3.) Several mechanisms have been identified that collectively determine the number of lamallae and the distribution of droplet sizes of a dispersion in a porous medium. For noncondensible fluids they... 

Effects of Capillary Number, Capillary Pressure, and the Porous Medium. Since the mechanisms of leave-behind, snap-off, lamella division and coalescence have been observed in several types of porous media, it may be supposed that they all play roles in the various combinations of oil-bearing rocks and types of dispersion-based mobility control (35,37,39-41). However, the relative importance of these mechanisms depends on the porous medium and other physico-chemical conditions. Hence, it is important to understand quantitatively how the various mechanisms depend on capillary number, capillary pressure, interfacial properties, and other parameters. 

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. 

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). 

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). 

Core Floods. At present the strong coupling between droplet size and flow has major experimental consequences (1) flow experiments must be performed under steady-state conditions (since otherwise the results may be controlled by long-lived, uninterpretable transients) (2) in situ droplet sizes cannot be obtained from measurements on an injected or produced dispersion (because these can change at core faces and inside the core) and (3) care must be taken that pressure drops measured across porous media are not dominated by end effects. Likewise, since abrupt droplet size changes can occur inside a porous medium, if the flow appears to be independent of the injected droplet-size distribution, it is likely that a new distribution is quickly forming inside the medium (38). 

Because capillary snap-off produces a dispersion of the nonwetting fluid in the fluid that wets the porous medium, if the "wrong" fluid wets the solid, the "wrong" type of dispersion will be produced. As described by the Young-Dupre equation. 

Thus, depending on the mineral and also these variables, the system may be in any of the various regions of the isotherm. Hence, the porous medium may have greatly different wettabilities, and the chance of forming the desired type of dispersion may also change. 

The three dispersion types described by Wellington et al. are important mechanistically, in view of the apparent importance of capillary snap-off. Extant descriptions of the snap-off mechanism explicitly treat the first type of dispersion, and they should be able to accommodate the second dispersion type by addition of a second fluid that does not wet the porous medium. However, if the aqueous phase of the first two dispersion types wets the porous... 

Phase Behavior and Surfactant Design. As described above, dispersion-based mobility control requires capillary snap-off to form the "correct" type of dispersion dispersion type depends on which fluid wets the porous medium and surfactant adsorption can change wettability. This section outlines some of the reasons why this chain of dependencies leads, in turn, to the need for detailed phase studies. The importance of phase diagrams for the development of surfactant-based mobility control is suggested by the complex phase behavior of systems that have been studied for high-capillary number EOR (78-82), and this importance is confirmed by high-pressure studies reported elsewhere in this book (Chapters 4 and 5). 

In laboratory experiments and field applications the gas is delivered to the rock face either as continuous gas or as a course gas-liquid dispersion. In both cases, for the gas to move into the porous medium, the gas pressure at the rock face must be higher than the capillary entry pressure. For a gas finger or a bubble train to advance through the porous medium, the face pressure must be maintained at a level above the maximum capillary pressure that the gas finger or bubble train will experience along its path through the medium. 

If the foam phase is thought of as a pseudo continuous fluid with an apparent viscosity Vapp = it follows that Papp is greater than that of the aqueous liquid phase. (For the tests here, values of Uapp were on the order of 1 to 50 times that of water). Because of this, when foam and liquid move through a porous medium under an applied pressure drop, the foam, being the most viscous phase, must occupy a larger region of the pore space. Consequently, as observed, the gas saturation is increased over that of non-dispersed phase flow and the liquid permeability is correspondingly decreased. 

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