Darcy’s permeability

For heterogeneous media composed of solvent and fibers, it was proposed to treat the fiber array as an effective medium, where the hydrodynamic drag is characterized by only one parameter, i.e., Darcy s permeability. This hydrodynamic parameter can be experimentally determined or estimated based upon the structural details of the network [297]. Using Brinkman s equation [49] to compute the drag on a sphere, and combining it with Einstein s equation relating the diffusion and friction coefficients, the following expression was obtained ... 

Model Structure Description Darcy s Permeability Coefficient ft(m/s)... 

Keywords compressibility, primary-, secondary- and enhanced oil-recovery, drive mechanisms (solution gas-, gas cap-, water-drive), secondary gas cap, first production date, build-up period, plateau period, production decline, water cut, Darcy s law, recovery factor, sweep efficiency, by-passing of oil, residual oil, relative permeability, production forecasts, offtake rate, coning, cusping, horizontal wells, reservoir simulation, material balance, rate dependent processes, pre-drilling. 

Creeping flow (Re <- 1) through porous media is often described in terms or the permeability k and Darcy s Law ... 

Heterogeneity, nonuniformity and anisotropy are terms which are defined in the volume-average sense. They may be defined at the level of Darcy s law in terms of permeability. Permeability, however, is more sensitive to conductance, mixing and capillary pressure than to porosity. 

Heterogeneity, nonuniformity and anisotropy are based on the probability density distribution of permeability of random macroscopic elemental volumes selected from the medium, where the permeability is expressed by the one-dimensional form of Darcy s law. 

Permeability is the conductance of the medium and has direct relevance to Darcy s law. Permeability is related to the pore size distribution, since the distribution of the sizes of entrances, exits and lengths of the pore walls constitutes the primary resistance to flow. This parameter reflects the conductance of a given pore structure. 

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. 

In reality, the melt productivity, dF/dz (degree of melting per km decompression), is likely to be non-linear with depth (Asimow et al. 1997). Darcy s law can then be used to describe the flow of melt given a permeable matrix and a driving force ... 

Figure 26.9 illustrates Darcy s law, the basic equation used to describe the flow of fluids through porous materials. In Darcy s law, the coefficient k, hydraulic conductivity, is often called the coefficient of permeability by civil engineers. 

Understanding the basic hydraulic mechanisms for synthetic liners and clay liners is very important in appreciating the advantages of a composite liner. Clay liners are controlled by Darcy s law (Q = kiA). In clay liners, the factors that most influence liner performance are hydraulic head and soil permeability. Clay liners have a higher hydraulic conductivity and thickness than do synthetic liners. Additionally, leachate leaking through a clay liner will undergo chemical reactions that reduce the concentration of contaminants in the leachate. 

Transmissivity is simply the coefficient of permeability, or the hydraulic conductivity (k), within the plane of the material multiplied by the thickness (T) of the material. Because the compressibility of some polymeric materials is very high, the thickness of the material needs to be taken into account. Darcy s law, expressed by the equation Q = kiA, is used to calculate the rate of flow, with transmissivity equal to kT and i equal to the hydraulic gradient (see Figure 26.22) ... 

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. 

A schematic of the flow through the cake and filter medium is shown in Fig. 13-6. The slurry flow rate is Q, and the total volume of filtrate that passes through the filter is V. The flow through the cake and filter medium is inevitably laminar, so the resistance can be described by Darcy s law and the permeability of the medium (K) ... 

The rate at which a porous medium will allow water to flow through it is referred to as permeability. Henry Darcy was the engineer who performed the first time-rate studies of water flowing through a sand filter. Darcy determined that, for a given material, the rate of flow is directly proportional to the driving forces (head) applied (hence, Darcy s law). 

Darcy s work was confined to the quantity of water discharged from a sand filter. Four examples of the application of Darcy s law as applied through a sand filter (actually a permeameter) is shown in Figure 3.10. Notice that the orientation of the cylinder has no effect on permeability. An example calculation of hydraulic conductivity (K) is presented in Figure 3.10 using the equation below ... 

Darcy s equation can be used to describe flow in this region however, the value of permeability varies as a function of saturation. Also, the value of moisture potential is a function of saturation. The total potential for flow (hydraulic gradient in Darcy s equation) can be defined as the difference between the moisture potential (minus) and the elevation potential (plus). When the potential for flow is positive, flow can occur. 

With these data and Darcy s law, the in-plane viscous permeabilihes were determined. Only the viscous permeability coefficient was determined because it was claimed that the inertial component was undetectable within the error limits of measuremenf for fhese fesfs. If is imporfanf to mention fhaf fhis technique could also be used to measure fhe permeabilify of diffusion layers wifh different fluids, such as liquid wafer. 

Chang et al. [183] presented a similar design in which two discs (with orifices in the middle) were used to compress the sample material. Pressurized air (without any moisture) was then passed through the orifices of fhe discs toward the sample DL, which then flowed peripherally to the atmosphere. The two discs were compressed in order to see how the permeability of the DL changed as a function of the clamping pressure. The permeability coefficient was solved using Darcy s law thus, only the viscous in-plane permeability was taken into account. Other, similar techniques can be found in the literature [215-217]. 

Other methods to study the through-plane permeabilities were presented by Chang et al. [183] and Williams et al. [90]. However, these methods only determined the viscous permeability coefficient with Darcy s law and did not take into account the inertial component of the permeability. 

Relative permeability is defined as the ratio between the permeability for a phase at a given saturation level to the total (or single-phase) permeability of the studied material. This parameter is important when the two-phase flow inside a diffusion layer is investigated. Darcy s law (Equation 4.4) can be extended to two-phase flow in porous media [213] ... 

Instead of the dilute solution approach above, concentrated solution theory can also be used to model liquid-equilibrated membranes. As done by Weber and Newman, the equations for concentrated solution theory are the same for both the one-phase and two-phase cases (eqs 32 and 33) except that chemical potential is replaced by hydraulic pressure and the transport coefficient is related to the permeability through comparison to Darcy s law. Thus, eq 33 becomes... 

In eq 51, the first term represents a convection term, and the second comes from a mass flux of water that can be broken down as flow due to capillary phenomena and flow due to interfacial drag between the phases. The velocity of the mixture is basically determined from Darcy s law using the properties of the mixture. The appearance of the mixture velocity is a big difference between this approach and the others, and it could be a reason the permeability is higher for simulations based on the multiphase mixture model. 

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. 

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