Porous media modeling flow

In order to understand the nature and mechanisms of foam flow in the reservoir, some investigators have examined the generation of foam in glass bead packs (12). Porous micromodels have also been used to represent actual porous rock in which the flow behavior of bubble-films or lamellae have been observed (13,14). Furthermore, since foaming agents often exhibit pseudo-plastic behavior in a flow situation, the flow of non-Newtonian fluid in porous media has been examined from a mathematical standpoint. However, representation of such flow in mathematical models has been reported to be still inadequate (15). Theoretical approaches, with the goal of computing the mobility of foam in a porous medium modelled by a bead or sand pack, have been attempted as well (16,17). 

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. 

Let s take the following model. The mesh bundle is considered as a highly permeable porous medium. Gas flows through microchannels whose effective diameter is... 

From the micro-scale simulations, information will be obtained about the permeability as a function of fibre stmcmre and flow velocity, which can be fed into the porous medium model used in all the meso-scale simulations. The range of velocities of interest for the micro-scale simulations is obtained from the meso-scale DNS and RANS simulations. Thus, there is a two-way information exchange... 

To predict a fiber-mafrix separation, the second force required to squeeze the resin through the bed of fibers at the rib entrance is now considered. The fiber bed has a permeability K which is expressed in terms of D Arcy s Law. Bakharev and Tucks- [5] used this law to describe flow through a porous medium. The flow of the resin through the fiber bed is modeled as one dimensional flow of a Non-Newtonian fluid through a porous matsial... 

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. 

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... 

Compared with the use of arbitrary grid interfaces in combination with reduced-order flow models, the porous medium approach allows one to deal with an even larger multitude of micro channels. Furthermore, for comparatively simple geometries with only a limited number of channels, it represents a simple way to provide qualitative estimates of the flow distribution. However, as a coarse-grained description it does not reach the level of accuracy as reduced-order models. Compared with the macromodel approach as propagated by Commenge et al, the porous medium approach has a broader scope of applicability and can also be applied when recirculation zones appear in the flow distribution chamber. However, the macromodel approach is computationally less expensive and can ideally be used for optimization studies. 

By making use of these analogies, electrical analog models can be constructed that can be used to determine the pressure and flow distribution in a porous medium from measurements of voltage and current distribution in a conducting medium, for example. The process becomes more complex, however, when the local permeability varies with position within the medium, which is often the case. 

Bolton EW, Lasaga AC, Rye DM (1996) A model for the kinetic control of quartz dissolution and precipitation in porous media flow with spatially variable permeability Eormulation and examples of thermal convection. J Geophys Res 101 22,157-22,187 Bolton EW, Lasaga AC, Rye DM (1997) Dissolution and precipitation via forced-flux injection in the porous medium with spatially variable permeability Kinetic control in two dimensions. J Geophys Res 102 12,159-12,172... 

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. 

Resin flow models are capable of determining the flow of resin through a porous medium (prepreg and bleeder), accounting for both vertical and horizontal flow. Flow models treat a number of variables, including fiber compaction, resin viscosity, resin pressure, number and orientation of plies, ply drop-off effects, and part size and shape. An important flow model output is the resin hydrostatic pressure, which is critical for determining void formation and growth. 

Two matrix flow submodels have been proposed the sequential compaction model [15] and the squeezed sponge model [11], Both flow models are based on Darcy s Law, which describes flow through porous media. Each composite layer is idealized as a fiber sheet surrounded by thermoset resin (Fig. 13.9). By treating the fiber sheet as a porous medium, the matrix velocity iir relative to the fiber sheet is given as (Eq. 13.5) ... 

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... 

Assume that the fibers in a filter are cylindrical they are parallel to each other and are uniformly assembled. Consider cake filtration in which particles are collected with the deposited particles forming a layer of porous structure as shown in Fig. 7.13(a). Thus, to account for the total pressure drop, three basic flow modes are pertinent (1) flow is parallel to the axis of fibers (2) flow is perpendicular to the axis of the cylinder and (3) flow passes through a layer of a homogeneous porous medium. In the analysis of the first two modes given later, Happel s model [Happel, 1959] is used, while for the third mode, Ergun s approach [Ergun, 1952] is used. 

In the Darcy model of flow through a porous medium, it is assumed that the flow velocities are low and that momentum changes and viscous forces in the fluid are consequently negligible compared to the drag force on the particles, i.e., if flow through a control volume of the type shown in Fig. 10.5 is considered, then ... 

Unidirectional flow through a porous medium can often be approximately modeled as flow through a series of parallel plane channels as show n in Fig. P10.1. Using this model derive an expression for the permeability. K, in terms of the channel size. W, and the porosity, . 

An approximate model of the flow in a vertical porous medium-filled enclosure assumes that the flow consists of boundary layers on the hot and cold w alls with a stagnant layer between the two boundary layers, this layer being at a temperature that is the average of the hot and cold wall temperatures. Use this model to find an expression for the heat transfer rate across the enclosure and discuss the conditions under which this model is likely to be applicable. 

Very briefly, 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 spring-like 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 22 23), the Dave model describes one-dimensional consolidation with three-dimensional Darcy s Law flow. Numerical solutions were in excellent agreement with closed-form solutions for one- and two-dimensional resin flow cases in which the fiber bed permeabilities and compressibility, as well as the autoclave pressure, are all held constant21). 

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