Darcy flow model

Using the Darcy flow model and the Boussinesq approximation, the governing equations are ... 

In addition to the modifications to the Darcy flow model, it is also sometimes necessary to note that the apparent thermal conductivity. ka, as used above arose from conduction through the fluid and solid media. This was the basis for Eq. (10.36). However, in fact, as tfie fluid flows through the tortuous paths between the solid... 

Numerical models for electrochemical process performance assessment or dimensioning generally assume uniform properties or one-dimensional property variations. For example, plug flow with axial dispersion is usually assumed within fdter-press electrolysers [1], whereas a Darcy flow model is commonly used within the gas diffusion layer of PEM electrolysers and fuel cells [2],... 

We used the data of nine gas fields located over the North German part of the European Upper Carboniferous Basin (see Fig. 1, black dots named A-G and M and N) to simulate the vertical flow of methane over the last 1(X) myr, using a diffusion model and a Darcy flow model. From the results obtained we determined the physical process driving gas through rocks. 

Calibration of the Darcy flow model the resulting rock salt seal permeabilities and their significance... 

Aerodynamic theory and Darcy flow modeling in porous media are similar in one respect only both derive from the Navier-Stokes equations governing viscous flows (Milne-Thomson, 1958 Schlichting, 1968 Slattery, 1981). We emphasize this because the great majority of our new solutions derive from the classical aerodynamics literature, but in a subtle manner. Very often, the superficial claim is made that, because petroleum pressure potentials satisfy p/9 + 9 p/9y = 0, the analogy to aerodynamic flowfields, which satisfy Laplace s equation + S cj/Sy = 0 for a similar velocity potential, can be... 

In a steady-state situation when gas flows through a porous material at a low velocity (laminar flow), the following empirical formula, Darcy s model, is valid ... 

In the fluid flow model, simulation is based on Darcy s law for the steady flow of Newtonian fluids through porous media. This law states that the average... 

The physical process of melt ascent during two-phase flow models is typically based on the separation of melt and solid described by Darcy s Law modified for a buoyancy driving force. The melt velocity depends on the permeability and pressure gradients but the actual microscopic distribution of the melt (on grain boundaries or in veins) is left unspecified. The creation of disequilibria only requires movement of the fluid relative to the solid. 

At 10 MPa and 35 °C, C02 has a density of approximately 700kg/m3. Under these conditions, a cubic meter of sandstone with 10% porosity contains approximately 70 kg of C02 if the pore space is completely filled by C02. However, saturation of C02 is not complete, and some brine remains in the invaded pore spaces (Saripalli McGrail 2002 Pruess et al. 2003). In addition, non-uniform flow of C02 bypasses parts of the aquifer entirely. Darcy-flow based analytical and numerical solutions are used to evaluate some of these effects by simulating the advance of the C02 front over time-scales of decades to hundreds of years and over lateral distances of tens to hundreds of kilometers. To account for the extreme changes in density and viscosity of C02 with pressure and temperature, these models must incorporate experimentally constrained equations of state (Adams Bachu 2002). 

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. 

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... 

The difference between the solution-diffusion and pore-flow mechanisms lies in the relative size and permanence of the pores. For membranes in which transport is best described by the solution-diffusion model and Fick s law, the free-volume elements (pores) in the membrane are tiny spaces between polymer chains caused by thermal motion of the polymer molecules. These volume elements appear and disappear on about the same timescale as the motions of the permeants traversing the membrane. On the other hand, for a membrane in which transport is best described by a pore-flow model and Darcy s law, the free-volume elements (pores) are relatively large and fixed, do not fluctuate in position or volume on the timescale of permeant motion, and are connected to one another. The larger the individual free volume elements (pores), the more likely they are to be present long enough to produce pore-flow characteristics in the membrane. As a rough rule of thumb, the transition between transient (solution-diffusion) and permanent (pore-flow) pores is in the range 5-10 A diameter. 

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 Springer15), Williams et al.16) and Gutowski17) 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... 

Simple one-dimensional reservoir models (two phase Darcy flow) indicate that, in general, the flow rates across permeable fault seds will be too high to sustain high pressure gradients or corresponding differences in hydrocarbon column lengths over geo-... 

To illustrate these top seal leakage dynamics, model calculations of simple Darcy flow were per-... 

Results of modeling case histories with Darcy flow... 

Of course, in RTM process modelling one must combine the above kinetic and chemoviscosity models into mass, momentum and energy balances within a flow simulation. Specifically, the momentum balance must combine any flows induced by pressure and any flows into porous media (as characterized by Darcy s law). Simple onedimensional RTM flow modelling and two- and three-dimensional RTM simulations have been summarized by Rudd et al. (1997) and show the importance of kinetic, rheological and permeability coefficients to the simulation of pressure-profile and flow-front predictions. 

The output of a flow model consists of the head distribution in time and space. Darcy s Law is used to convert the head distribution to a velocity distribution suitable for input to a contaminant transport model. In a two-dimensional application, Darcy s Law is used to compute two sets of velocity components ... 

Fluid flow modelling was performed assuming two-phase flow (petroleum and water) based on the Darcy flow equation. This simplification is assumed valid as the petroleum phase encountered in Snorre field is undersaturated. 

Unsaturated flow formulation is necessary here as far as suction has to be known. The flow model used is based on works in relation with the problem of nuclear waste disposal (Collin et al. 2002a). For each fluid (Water and oil), balance equations and state equations are written. In partial saturation conditions, the permeability and the storage law have to be modified a generalised Darcy s law defines the fluid motion (Bear 1972). Numerous couplings existing between mechanics and flows are considered. 

In order to set up a two-phase flow model, Darcy s law for each individual phase is assumed. As the permeability for a given phase in presence of a second phase will decrease, a relative permeability is introduced ... 

For the dynamic simulation of the SMB-SFC process a plug-flow model with axial dispersion and linear mass-transfer resistance was used. The solution of the resulting mass-balance equations was performed with a finite difference method first developed by Rouchon et al. [69] and adapted to the conditions of the SMB process by Kniep et al. [70]. The pressure drop in the columns is calculated with the Darcy equation. The equation of state from Span and Wagner [60] is used to calculate the mobile phase density. The density of the mobile phase is considered variable. 

Table 11.5 contains the simulation parameters of the simplified 0-order axisym-metrical volume-average ferrohydrodynamic model. Note that three superficial velocities are chosen such that the ferrofluid flow is i) completely controlled by inertia, i.e. Burke-Plummer type of flow (Uq = 0.2 m/s) ii) mostly dominated by viscous forces, i.e. Darcy flow (Uo = 2x 10 m/s), iii) mixed with comparable inertial and viscous contributions, i.e. Forchheimer flow (Uq = 2 x 10 m/s). The Reynolds number values. Re, corresponding to these three cases are, respectively, pUodp/f] = 2110,0.211 and 21.1. The consensual values for the laminar and inertial Ergun constants are used, i.e. respectively, 150 and 1.75 (see Eqs. (11.56)-(11.58) linear momentum balance. Table 11.4). 

The model accounts for three different transport mechanisms, molecular diffusion, Knudsen diffusion, and viscous transport. The total diffusive flux in DGM results from molecular diffusion acting in series with Knudsen diffusion. The viscous porous media flow (Darcy flow) acts in parallel with diffusive flux. The DGM can be written as an implicit relationship among molar concentrations, fluxes, concentration gradients and pressure gradient as... 

The flow in the gas channels and in the porous gas diffusion electrodes is described by the equations for the conservation of momentum and conservation of mass in the gas phase. The solution of these equations results in the velocity and pressure fields in the cell. The Navier-Stokes equations are mostly used for the gas channels while Darcy s law may be used for the gas flow in the GDL, the microporous layer (MPL), and the catalyst layer [147]. Darcy s law describes the flow where the pressure gradient is the major driving force and where it is mostly influenced by the frictional resistance within the pores [145]. Alternatively, the Brinkman equations can be used to compute the fluid velocity and pressure field in porous media. It extends the Darcy law to describe the momentum transport by viscous shear, similar to the Navier-Stokes equations. The velocity and pressure fields are continuous across the interface of the channels and the porous domains. In the presence of a liquid phase in the pore electrolyte, two-phase flow models may be used to account for the interaction between the gas phase and the liquid phase in the pores. When calculating the fluid flow through the inlet and outlet feeders of a large fuel cell stack, the Reynolds-averaged Navier-Stokes (RANS), k-o), or k-e turbulence model equations should be used due to the presence of turbulence. 

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