Capillaric model

For the purpose of illustration, in this paper we use a viscosity-capillarity model (Truskinovsky, 1982 Slemrod, 1983) as an artificial "micromodel",and investigate how the information about the behavior of solutions at the microscale can be used to narrow the nonuniqueness at the macroscale. The viscosity-capillarity model contains a parameter -Je with a scale of length, and the nonlinear wave equation is viewed as a limit of this "micromodel" obtained when this parameter tends to zero. As we show, the localized perturbations of the form x /-4I) can influence the choice of attractor for this type of perturbation, support (but not amplitude) vanishes as the small parameter goes to zero. Another manifestation of this effect is the essential dependence of the limiting solution on the... 

The simplest example of a theory which incorporates both dispersion and dissipation is the so called viscosity-capillarity model (Truskinovsky, 1982, Slemrod, 1983). It combines van der Waals correction to the energy with Kelvin viscoelasticity, which in the present context amounts to the following additional constitutive assumption... 

Figure 6. Set ofjump discontinuities (kinks and shocks) compatible with the isothermal viscosity-capillarity model W=2.5.

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... 

With the aid of the straight capillaric model and according to the Hagen-Poiseuille equation and Darcy s law, a is proportional to the viscosity of the fluid. Thus, Eq. (5.322) becomes... 

Figure 4.4. Capillaric model of a packed column. The flow space is assumed to consist of a bundle of parallel capillaries.

The apparent permeability, a function of the proportion of pores penetrated by the mud filtrate, can be determined from the pore size distribution curve, absolute permeability and porosity of the rock material (Tan et al., 1996a). The methodology is based on a simple capillaric model which represents a porous medium by a bundle of parallel capillaries (Scheideggar, 1974). From the law of Hagen-Poiseuille, the absolute permeability of such a model is given by ... 

Tables 1, 2, and 3 list the inputs for the geomechanics model and the capillarity model, and include relative permeability data.

The simplest models that can be constructed are based on the idea that the porous medium is like a bundle of capillaries. Early capillaric models have been discussed by a number of authors in the context of various physical problems (Kawakami, 1932 Smith, 1932 Rainard, 1947 Henderson, 1949 Purcell, 1949 Burdine et ai, 1950 Calhoun, 1953). There are a number of variants of this type of model, the simplest being the linear case representing a porous medium by a bundle of capillaries of uniform radius (see, for example, Scheidegger, 1953) this model is represented in Figure 6.1(a). It is easily shown that for such a model the porosity, (/>, permeability, /c, and pore radius, R, are related by ... 

The porous medium may be assumed to consist of a bundle of straight capillaries of equal size (capillaric model). A particle follows a certain trajectory and is captured when it hits the wall. 

In the development of an expression for Et, a more detailed model of the porous medium is needed. Of the three types of models of a granular filter as a porous medium, the capillaric model is not preferred. For the sake of simplicity, we will consider one of the other two, namely the spherical collector model. In this model, the filter grain is assumed to be a sphere. There are a number of alternative approaches based on a spherical collector. We will illustrate the approach by Happel (1958). In Happel s model, the granular porous medium is assumed to consist of a large collection of identical cells, where each cell consists of a spherical particle of radius ((dgr)/2) (i.e. half of the average grain diameter) surrounded by a liquid envelope of radius b, such that the void volume of this cell is identical to the void volume of the porous medium ... 

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