Porous medium equation

L. A. Peletier, The Porous Media Equation in Application of Nonlinear Analysis in the Physical Sciences, H. Amann, N. Baxley, and K. Kirchgassner, eds., Pitman, Boston, 1981, pp. 229-241. 

In this chapter we shall treat some particular instances of the system (3.1.15) and the related phenomena. Thus in 3.2, we shall concentrate upon binary ion-exchange and discuss the relevant single nonlinear diffusion equation. It will be seen that in a certain range of parameters this equation reduces to the porous medium equation with diffusivity proportional to concentration. Furthermore, it turns out that in another parameter range the binary ion-exchange is described by the fast diffusion equation with diffusivity inversely proportional to concentration. It will be shown that in the latter case some monotonic travelling concentration waves may arise. 

In this section we shall study still another peculiarity of (3.2.3)—the occurrence of uniformly bounded, monotonic travelling waves. These waves, very common in reaction-diffusion (see, e.g., [25]—[27]), seem fairly unexpected in the reactionless diffusion under discussion. Their occurrence here is directly related to the singularity of diffusivity in (3.2.3) and thus can be viewed as the fast diffusional counterpart of the aforementioned peculiarities of the porous medium equation. 

It was observed in the previous section that a certain limit case of non-reactive binary ion-exchange is described by the porous medium equation with m = 2 in other words, a weak shock is to be expected at the boundary of the support. Recall that this shock results from a specific interplay of ion migration in a self-consistent electric field with diffusion. Another source of shocks (weak or even strong in the sense to be elaborated upon below) may be fast reactions of ion binding by the ion-exchanger. 

By introducing the equations of state for the groundwater and the porous medium (Equations 1.18, and 1.21 and 1.23, respectively) into the continuity Equation 1.13 gives (e.g. Walton, 1970)... 

Last but not least, we would like to point out a very recent investigation [64] in which we show that the change of detached monomers with time is governed by a differential equation that is equivalent to the nonlinear porous medium equation. 

Thiele, E.W., 1939. Relation between catalytic activity and size of particle. Ind. Eng. Chem. 31, 916-920. Vasquez, J.L., 2006. The Porous Medium Equations Mathematical Theory. Oxford University Press, Oxford. Yablonsky, G.S., Constales, D., Gleaves, J.T., 2002. Multi-scale problems in the quantitative characterization of complex catalytic materials. Syst. Anal. Model. Simul. 42, 1143-1166. 

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]. 

The Stefan-Maxwell equations have been presented for the case of a gas in the absence of a porous medium. However, in a porous medium whose pores are all wide compared with mean free path lengths it is reasonable to guess that the fluxes will still satisfy relations of the Stefan-Maxwell form since intermolecular collisions still dominate molecule-wall collisions. 

One of Che earliest examples of a properly conceived experimental investigation of the flux relations for a porous medium is provided by the work of Gunn and King [53] on the dusty gas model equations, and the following discussion is based largely on their work. Since all their experiments were performed on binary mixtures, the appropriate flux relations are (5.26) and (5,27). Writing... 

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

The differential material balances contain a large number of physical parameters describing the structure of the porous medium, the physical properties of the gaseous mixture diffusing through it, the kinetics of the chemical reaction and the composition and pressure of the reactant mixture outside the pellet. In such circumstances it Is always valuable to assemble the physical parameters into a smaller number of Independent dimensionless groups, and this Is best done by writing the balance equations themselves in dimensionless form. The relevant equations are (11.20), (11.21), (11.22), (11.23), (11.16) and the expression (11.27) for the effectiveness factor. 

Most authors who have studied the consohdation process of soflds in compression use the basic model of a porous medium having point contacts which yield a general equation of the mass-and-momentum balances. This must be supplemented by a model describing filtration and deformation properties. Probably the best model to date (ca 1996) uses two parameters to define characteristic behavior of suspensions (9). This model can be potentially appHed to sedimentation, thickening, cake filtration, and expression. 

The governing flow equation describing flow through as porous medium is known as Darcy s law, which is a relationship between the volumetric flow rate of a fluid flowing linearly through a porous medium and the energy loss of the fluid in motion. 

Tortuosity t is basically a correction factor applied to the Kozeny equation to account for the fact that in a real medium the pores are not straight (i.e., the length of the most probable flow path is longer than the overall length of the porous medium) ... 

The most straightforward porous media model which can be used to describe the flow in the multichannel domain is the Darcy equation [117]. The Darcy equation represents a simple model used to relate the pressure drop and the flow velocity inside a porous medium. Applied to the geometry of Figure 2.26 it is written as... 

The simple pore structure shown in Figure 2.69 allows the use of some simplified models for mass transfer in the porous medium coupled with chemical reaction kinetics. An overview of corresponding modeling approaches is given in [194]. The reaction-diffusion dynamics inside a pore can be approximated by a one-dimensional equation... 

The development of the theory of solute diffusion in soils was largely due to the work of Nye and his coworkers in the late sixties and early seventies, culminating in their essential reference work (5). They adapted the Fickian diffusion equations to describe diffusion in a heterogeneous porous medium. Pick s law describes the relationship between the flux of a solute (mass per unit surface area per unit time, Ji) and the concentration gradient driving the flux. In vector terms. 

The airflow equations presented above are based on the assumption that the soil is a spatially homogeneous porous medium with constant intrinsic permeability. However, in most sites, the vadose zone is heterogeneous. For this reason, design calculations are rarely based on previous hydraulic conductivity measurements. One of the objectives of preliminary field testing is to collect data for the reliable estimation of permeability in the contaminated zone. The field tests include measurements of air flow rates at the extraction well, which are combined with the vacuum monitoring data at several distances to obtain a more accurate estimation of air permeability at the particular site. 

This equation defines the permeability (K) and is known as Darcy s law. The most common unit for the permeability is the darcy, which is defined as the flow rate in cm3/s that results when a pressure drop of 1 atm is applied to a porous medium that is 1 cm2 in cross-sectional area and 1 cm long, for a fluid with viscosity of 1 cP. It should be evident that the dimensions of the darcy are L2, and the conversion factors are (approximately) 10 x cm2/darcy C5 10-11 ft2/darcy. The flow properties of tight, crude oil bearing, rock formations are often described in permeability units of millidarcies. 

Flow in a porous medium in two or three dimensions is important in situations such as the production of crude oil from reservoir formations. Thus, it is of interest to consider this situation briefly and to point out some characteristics of the governing equations. 

Consider the flow of an incompressible fluid through a two-dimensional porous medium, as illustrated in Fig. 13-2. Assuming that the kinetic energy change is negligible and that the flow is laminar as characterized by Darcy s law, the Bernoulli equation becomes... 

In Equation 6.22, Sa at Zow is maximum with the limiting value of 1 - Sr, where ST is the irreducible aqueous-phase saturation in the porous medium under study. 

Ideally, cross-flow microfiltration would be the pressure-driven removal of the process liquid through a porous medium without the deposition of particulate material. The flux decrease occurring during cross-flow microfiltration shows that this is not the case. If the decrease is due to particle deposition resulting from incomplete removal by the cross-flow liquid, then a description analogous to that of generalised cake filtration theory, discussed in Chapter 7, should apply. Equation 8.2 may then be written as ... 

Richards equation (Richards 1931), based on a mass conservation balance, together with Eq. 9.1, can be used to describe the transient flow of water through a partially saturated porous medium. In one dimension (vertically), Richards equation is given as... 

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. 

The model proposed above is analogous to a continuous, unsteady state filtration process, and therefore may be called "Filtration Model". In this model, the concentration of the filtrate, viz. the concentration of the solute remained in the treated solid is one s major concern. This is given by the rate of Step 3, which may be expressed by an equation similar to Pick s Law including a transmission coefficient D for the porous medium, viz. the P.S.Z. and the concentration difference Aw across the P.S.Z. as the driving force, and the thickness of the P.S.Z. as the distance Ax. 

In addition to the similarity between the heat conduction equation and the diffusion equation, erosion is often described by an equation similar to the diffusion equation (Culling, 1960 Roering et al., 1999 Zhang, 2005a). Flow in a porous medium (Darcy s law) often leads to an equation (Turcotte and Schubert, 1982) similar to the diffusion equation with a concentration-dependent diffu-sivity. Hence, these problems can be treated similarly as mass transfer problems. 

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