Darcys equation

The three-dimensional flow of groundwater through the subsurface can be described by a combination of Darcy s equation for groundwater flow with a continuity equation (or mass balance equation) and equations of state for the groundwater and the porous medium. A detailed theoretical overview of equations for groundwater flow is given by e.g. Barenblatt et al. (1990), De Marsily (1986), Domenico and Schwartz (1990) and Freeze and Cherry (1979). 

Sections 1.2.1 to 1.2.3 present the basic equations for groundwater flow through a representative elementary volume of the porous medium (Bear, 1972) under certain restrictive assumptions, such as isothermal and isochemical subsurface conditions and absence of changes in tectonic stress. 

Darcy s law (Darcy, 1856) is a phenomenological law that is valid for the viscous flow of a single-phase fluid (e.g. groundwater flow) through porous media in any direction. This basic law of fluid flow is a macroscopic law providing averaged descriptions of the actual microscopic flow behaviour of the fluids over some representative elementary volume of the porous medium. For isothermal and isochemical subsurface conditions, the law can be written as (Hubbert, 1953) 

By introducing the appropriate expression for the hydraulic head, i.e. Equation 1.3 or 1.4, into Darcy s equation 1.8, it describes the flow for compressible and incompressible groundwater, respectively. The flow of compressible groundwater can thus be described by a more general form of Darcy s equation, in which the gradient of the groundwater potential is given by Equation 1.5 

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For direct measurement from core samples, the samples are mounted in a holder and gas is flowed through the core. The pressure drop across the core and the flowrate are measured. Providing the gas viscosity (ji) and sample dimensions are known the permeability can be calculated using the Darcy equation shown below. 

For filter design and performance prediction it is necessary to predict the rate of filtration (velocity or volumetric flowrate) as a function of pressure drop, and the properties of the fluid and particulate bed. This can be achieved using the modified Darcy equation developed in Chapter 3. 

The single-phase friction loss (pressure drop) for these situations in chemical and petrochemical plants is still represented by the Darcy equation with specific limitations [3] ... 

If calculated pressure drop from inlet to outlet of line system (not including control or hand valves) is greater than approximately 10%, but less than about 40% of the inlet pre.s.sure Pi (pounds per. square inch gauge), the Darcy equation will yield reasonable accuracy when using a specific volume based on the average of upstream (inlet) and do vnstream (outlet)... 

The fnctio-nal resistance or pressure drop due to the flow of the fluid, hf, is expressed by the Darcy equation ... 

For the discharge of compressible fluids from the end of a short aiping length into a larger cross section, such as a larger pipe, vessel, or atmosphere, the flow is considered adiabatic. Corrections are applied to the Darcy equation to compensate for fluid property changes due to the expansion of the fluid, and these are known as Y net expansion factors [3]. The corrected Darcy equation is ... 

Because flashing steam-condensate lines represent two-phase flow, with the quantity of liquid phase depending on die system conditions, these can be designed following the previously described two-phase flow methods. An alternate by Ruskin [28] uses the concept but assumes a single homogeneous phase of fine liquid droplets dispersed in the flashed vapor. Pressure drop was calculated by the Darcy equation ... 

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... 

The flow velocity, pressure and dynamic viscosity are denoted u, p and fj and the symbol (...) represents an average over the fluid phase. Kim et al. used an extended Darcy equation to model the flow distribution in a micro channel cooling device [118]. In general, the permeability K has to be regarded as a tensor quantity accounting for the anisotropy of the medium. Furthermore, the description can be generalized to include heat transfer effects in porous media. More details on transport processes in porous media will be presented in Section 2.9. 

A generalized Darcy equation and equation of continuity for each fluid phase is used to describe the flow of multiple immiscible fluid phases ... 

Sometimes, however, A is expressed as [jtd2]/[4( 144)] with the ji/4(144) buried into an overall coefficient. For example, Crane14 has a solved problem that uses the Darcy equation form ... 

The TDE moisture module (of the model) is formulated from three equations (1) the water mass balance equation, (2) the water momentum, (3) the Darcy equation, and (4) other equations such as the surface tension of potential energy equation. The resulting differential equation system describes moisture movement in the soil and is written in a one dimensional, vertical, unsteady, isotropic formulation as ... 

Results from constant differential pressure filtration tests have been analyzed according to traditional filtration science techniques with some modifications to account for the cross-flow filter arrangement.11 Resistivity of the filter medium may vary over time due to the infiltration of the ultrafine catalyst particles within the media matrix. Flow resistance through the filter cake can be measured and correlated to changes in the activation procedure and to the chemical and physical properties of the catalyst particles. The clean medium permeability must be determined before the slurries are filtered. The general filtration equation or the Darcy equation for the clean medium is defined as... 

To determine the pressure gradient along the reactor, the Fanning or Darcy equation for flow in cylindrical tubes may be used (Knudsen and Katz, 1958, p. 80) ... 

Since chromatographic flow is laminar in nature, the specific permeability (Bq) of a packed column can be calculated using the Darcy equation [131,132]... 

Xu et al. (2006b) have successfully introduced Brinkman-Forchheimer-extended Darcy equation in order to solve the performance of molten carbonate fuel cell. As a verification of REV method, Poiseuille flow profiles in the porous media modifying LBM with... 

Equation (6.6) is notably named the Darcy equation [8]. It is applicable to both laminar flow, Re < 2,000, and to turbulent flow, Re > 10,000. Two restrictions govern this equation, however ... 

At this point, a review is in order. Please note again Eq. (6.6). The Darcy equation, also known as the Fanning equation, is applicable to any single-phase flowing fluid, liquid or gas. There are seven factors that make up the dimensional analysis of this equation ... 

Using IT lb/h, for fluid flow allows usage of this same equation for gas or liquid the density term p also applies to either. The velocity variable and density transport property factor must stay reasonably near a constant value for the Darcy equation to be valid. For a liquid fluid, p and v remain nearly constant. Thus, most any pipeline length may be calculated using a single application of Eq. (6.6). 

Please note that Eq. (6.9) is actually a part of the Darcy equation, Eq. (6.6). Adding the velocity head gives the following Darcy equation ... 

There are more useful Darcy equation forms than Eq. (6.10). Equation (6.13) is proposed, a more user-friendly equation that calculates the pressure loss AP in units of psi. 

Please note the resemblance of Eq. (6.26) to the earlier Darcy equation. Why is the factor 2 installed The answer is that Eq. (6.6) is based on the Moody friction factor f and Eq. (6.26) is based on the Fanning friction factor f. These two friction factors are identical, except that the Moody f is exactly four times larger than the Fanning f. Without a long story as to how these two research scientists came up with the same answer, with the exception of one being four times larger in every case, simply accept the fact that one perhaps wanted to be recognized as an independent developer. Nonetheless, the bottom line is that both Eqs. (6.6) and (6.26) are indeed the same equation. 

If the permeability of the material is very high, it may not be possible to ignore the viscous forces in setting up the momentum equation. In this case, if the effective viscosity is taken to be the viscosity of the fluid, the Darcy equations, i.e., Eqs. (10.12) to (10.14) become ... 

To calculate flow velocity within the bed, the momentum equation is solved. The pressure loss is calculated via the Darcy equation, implying low flow velocities and a linear dependence of the pressure loss on the flow velocity. Thus the momentum equation is given by... 

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