Porous media consolidated

Most authors who have studied the consolidation process of solids 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 applied to sedimentation, thickening, cake filtration, and expression. 

Magnetic resonance imaging (MRI) has been applied to the study of the distribution of fluid components (i.e., water or a polymer used as consolidant) in a porous material such as stone or waterlogged wood by a direct visualization of the water or fluid confined in the opaque porous medium [13]. 

In the past, various resin flow models have been proposed [2,15-19], Two main approaches to predicting resin flow behavior in laminates have been suggested in the literature thus far. In the first case, Kardos et al. [2], Loos and Springer [15], Williams et al. [16], and Gutowski [17] assume that a pressure gradient develops in the laminate both in the vertical and horizontal directions. These approaches describe the resin flow in the laminate in terms of Darcy s Law for flow in porous media, which requires knowledge of the fiber network permeability and resin viscosity. Fiber network permeability is a function of fiber diameter, the porosity or void ratio of the porous medium, and the shape factor of the fibers. Viscosity of the resin is essentially a function of the extent of reaction and temperature. The second major approach is that of Lindt et al. [18] who use lubrication theory approximations to calculate the components of squeezing flow created by compaction of the plies. The first approach predicts consolidation of the plies from the top (bleeder surface) down, but the second assumes a plane of symmetry at the horizontal midplane of the laminate. Experimental evidence thus far [19] seems to support the Darcy s Law approach. 

The Dave model considers a force balance on a porous medium (the fiber bed). The total force from the autoclave pressure acting on the medium is countered by both the force due to the springlike behavior of the fiber network and the hydrostatic force due to the liquid resin pressure within the porous fiber bed. Borrowing from consolidation theories developed for the compaction of soils [23,24], the Dave model describes one-dimensional consolidation... 

In the second group, the solid-phase concentration is high, and solids particles are either loose but in contact, or consolidated. In this case, the solid phase is the matrix, while the liquid phase is the dispersed phase. In this group, electrical conductivity is used to measure the effective porosity of the porous medium (64, 65). Also, if two immiscible fluids, for example, oil and water, are present in a porous medium, the electrical conductivity can be employed to measure the relative saturations of the two fluids and to give an indication of the wettability of the porous medium (66, 67). 

As shown in Figure 1, the filtrate (liquid) passes through porous media of two types the consolidated layer (the cast) and the porous mold. The flow of liquid through a porous medium is described by Darcy s law which, in one dimension, can be written... 

Porous medium is a material consisting of a solid matrix with interconnected pores. The interconnected pores are responsible for allowing a fluid to traverse through the material. For the simplest situation, the medium is saturated with a single fluid ( single fluid flow ). In multiphase fluid flow, several fluids (liquids and/or gas) share the open pores. Porous media are classified as unconsolidated and consolidated. 

Permeability for a Rock Formation. For natural consolidated porous medium, however, the definitions of the equivalent spherical diameter and the specific surface area per unit volume are not widely used because of its difficulty in determination and relation to other measurable quantities. Just to serve as a comparison, we give the permeability equation based on the previous passage model with the tortuosity given by equation 61 and assuming that the areal porosity equation 54 still holds. The permeability can then be given by... 

The usage of the flow equations can be summarized as follows. For the case of a one-dimensional single fluid flow, either equation 106 or 108 can be used to predict the normalized pressure drop factor in a porous medium. The determined normalized pressure drop factor is related to the pressure drop by equation 11. For the simple case of packed spherical beads, ds and e are known a priori. The Reynolds number is evaluated using equation 93. For random packs of nonspherical particles, the particle s sphericity needs to be known. Equation 73 can be used to estimate ds. For the case of consolidated porous medium, one can estimate ds from the knowledge of the intrinsic permeability using equation 14. 

Porous Medium A solid containing voids or pore spaces. Normally such pores are quite small compared with the size of the solid and well-distributed throughout the solid. In geologic formations, porosity may be associated with unconsolidated (uncemented) materials, such as sand, or a consolidated material, such as sandstone. 

In this equation, d is the tortuosity of the porous medium (between 0 and 1 0.5 would be typical for a consolidated sandstone), Z>nuid is the diffusion coefficient in pure fluid and i is a retardation factor, defined as... 

The fifth term represents water retention due to the consolidation of the porous medium... 

The AECL team used an in-house MOTIF finite-element code (Guvanasen and Chan 2000), which is based on an extension of the classical poroelastic theory of Biot (1941). This code has undergone extensive verification and validation (Chan et al. 2003). The CTH team employed the commercially available, general-purpose finite-element code ABAQUS/Standard 6.3 (ABAQUS manuals). This code adopts a macroscopic thermodynamic approach. The porous medium is considered as a multiphase material, and an effective stress principle is used to describe its behaviour. ABAQUS allows the value of bulk modulus of the mineral grains as an input parameter. In order to select an appropriate value for this low-permeability, low-porosity rock, the CTH team compared the ABACjus solution with Biot s (1941) analytical solution for ID consolidation in the form presented by Chan et al. 2003). 

In light of the new 57 km AlpTransit Gotthard base tunnel, which is currently under construction and whose trajectory passes close to several dams, the assessment of possible surface subsidence is of great interest. In this paper we present the results of a two-dimensional finite-element study which includes coupled fluid pressure diffusion and consolidation with the rock mass treated as an equivalent isotropic porous medium. 

Biot (1941) derived the 3-D consolidation theory, which describes the coupled hydraulic and mechanical transient response of a linear elastic, isotropic, homogeneous porous medium. One aspect of his theory is the effective stress law for elastic deformation ... 

Analysis of time-dependent consolidation requires the solution of Biot s consolidation equations coupled to the equations describing flow. The transient hydro-mechanical coupling between pore pressure and volumetric strain for a linear elastic, mechanically isotropic porous medium and fully saturated with a single fluid phase (i.e. water), is given by the fluid continuity equation ... 

The major problem experienced in the field to date in chemical flooding processes has been the inability to make contact with residual oil. Laboratory screening procedures have developed micellar-polymer systems that have displacement efiiciencies approaching 100% when sand packs or uniform consolidated sandstones are used as the porous medium. When the same micellar-polymer system is applied in an actual reservoir rock sample, however, the efficiencies are usually lowered significantly. This is due to the heterogeneities in the reservoir samples. When the process is applied to the reservoir, the efiiciencies become even worse. Research is being conducted on methods to reduce the effect of the rock heterogeneities and to improve the displacement efficiencies. 

In this section we reconstruct the theory of consolidation by introducing the concept of a finite strain and a nominal stress rate, which are given in Chap. 2. Note that in Chap. 5 a mixture theory was developed for a porous medium with multiple... 

We assume that in Biot s consolidation equations the porous medium is fully saturated, and the fluid phase is incompressible. The excess pore pressure Ap is an increase (or a decrease) of pore pressure from the hydrostatic pressure therefore the equation of equilibrium (6.36) is given in an incremental form. 

Consolidated porous media here, the contact (cohesive) forces between elementary grains maintain the mechanical stability of the material. Usual examples of consolidated porous media include calcareous rocks, clays, vegetable, and animal tissues. The deformation or mechanical equilibrium of the porous medium is not a concern, unless its mechanical breakdown under the effect of strong forces is considered. 

Prediction of the permeability reduction from properties of the porous rock and the polymer is not possible at this time. Experimental measurement with the rock and polymer of interest is necessary. It is often possible, however, to correlate permeability reduction for the same polymer in the same type of porous medium and use the resulting correlation for interpolation and extrapolation. Gogarty37 correlated the permeability of consolidated porous rocks after contact with polyacrylamide by use of an empirical relationship. 

There are various conceptual ways of describing a porous medium. One concept is a continuous solid with holes in it. Such a medium is referred to as consolidated, and the holes may be unconnected (impermeable) or connected (permeable). Another concept is a collection of solid particles in a packed bed, where the fluid can pass through the voids between the particles, which is referred to as unconsolidated. Both of these concepts have been used as the basis for developing the equations which describe fluid flow behaviour. ... 

The term porosity refers to the fraction of the medium that contains the voids. When a fluid is passed over the medium, the fraction of the medium (i.e., the pores) that contributes to the flow is referred to as the effective porosity of the media. In a general sense, porous media are classified as either unconsolidated and consolidated and/or as ordered and random. Examples of unconsolidated media are sand, glass beads, catalyst pellets, column packing materials, soil, gravel and packing such as charcoal. 

Figure 13-1 Porous media, (a) Consolidated medium (b) unconsolidated medium.

Mauran S., Rigaud L., and CoudeviUe O., Application of the Carman-Kozeny correlation to a high-porosity anisotropic consolidated medium The compressed expanded natural graphite. Transport in Porous Media 43 2001 355-376. 

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