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| '''Darcy's law''' is a [[Phenomenology (science)|phenomenologically]] derived [[constitutive equation]] that describes the flow of a [[fluid]] through a [[porous]] medium. The law was formulated by [[Henry Darcy]] based on the results of experiments<ref>H. Darcy, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris (1856).</ref> on the flow of [[water]] through beds of [[sand]]. It also forms the scientific basis of fluid [[Permeability (fluid)|permeability]] used in the [[earth science]]s, particularly in [[hydrogeology]].
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| == Background ==
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| Although Darcy's law (an expression of [[conservation of momentum]]) was determined experimentally by Darcy, it has since been derived from the [[Navier-Stokes equations]] via [[Homogenization (mathematics)|homogenization]]. It is analogous to [[Fourier's law]] in the field of [[heat conduction]], [[Ohm's law]] in the field of [[electrical networks]], or [[Fick's law]] in [[diffusion]] theory.
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| One application of Darcy's law is to water flow through an [[aquifer]]; Darcy's law along with the equation of [[conservation of mass]] are equivalent to the [[groundwater flow equation]], one of the basic relationships of [[hydrogeology]]. Darcy's law is also used to describe oil, water, and gas flows through petroleum reservoirs.
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| == Description ==
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| [[Image:Darcy's Law.png|thumb|right|300px|Diagram showing definitions and directions for Darcy's law.]]
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| Darcy's law is a simple proportional relationship between the instantaneous discharge rate through a porous medium, the [[viscosity]] of the fluid and the pressure drop over a given distance.
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| : <math>Q=\frac{-kA}{\mu} \frac{(P_b - P_a)}{L}</math>'
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| The total discharge, ''Q'' (units of volume per time, e.g., m<sup>3</sup>/s) is equal to the product of the intrinsic [[Permeability (fluid)|permeability]] of the medium, ''k'' (m<sup>2</sup>), the cross-sectional area to flow, ''A'' (units of area, e.g., m<sup>2</sup>), and the total pressure drop (P<sub>b</sub> - P<sub>a</sub>), (Pascals), all divided by the [[viscosity]], ''μ'' (Pa·s) and the length over which the pressure drop is taking place (m). The negative sign is needed because fluid flows from high pressure to low pressure. If the change in pressure is negative (where P<sub>a</sub> > P<sub>b</sub>), then the flow will be in the positive 'x' direction. Dividing both sides of the equation by the area and using more general notation leads
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| : <math>q=\frac{-k}{\mu} \nabla P</math>
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| where ''q'' is the flux (discharge per unit area, with units of length per time, m/s) and <math>\nabla P</math> is the [[pressure gradient]] vector (Pa/m). This value of flux, often referred to as the Darcy flux, is not the velocity which the fluid traveling through the pores is experiencing. The fluid velocity (''v'') is related to the Darcy flux (''q'') by the [[porosity]] (''n''). The flux is divided by porosity to account for the fact that only a fraction of the total formation volume is available for flow. The fluid velocity would be the velocity a conservative tracer would experience if carried by the fluid through the formation.
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| : <math>v=\frac{q}{n}</math>
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| Darcy's law is a simple mathematical statement which neatly summarizes several familiar properties that [[groundwater]] flowing in [[aquifer]]s exhibits, including:
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| * if there is no pressure gradient over a distance, no flow occurs (these are [[hydrostatics|hydrostatic]] conditions),
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| * if there is a pressure gradient, flow will occur from high pressure towards low pressure (opposite the direction of increasing gradient - hence the negative sign in Darcy's law),
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| * the greater the pressure gradient (through the same formation material), the greater the discharge rate, and
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| * the discharge rate of fluid will often be different — through different formation materials (or even through the same material, in a different direction) — even if the same pressure gradient exists in both cases.
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| A graphical illustration of the use of the steady-state [[groundwater flow equation]] (based on Darcy's law and the conservation of mass) is in the construction of [[flownet]]s, to quantify the amount of [[groundwater]] flowing under a [[dam]].
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| Darcy's law is only valid for slow, [[viscous]] flow; fortunately, most groundwater flow cases fall in this category. Typically any flow with a [[Reynolds number]] less than one is clearly laminar, and it would be valid to apply Darcy's law. Experimental tests have shown that flow regimes with Reynolds numbers up to 10 may still be Darcian, as in the case of groundwater flow. The Reynolds number (a dimensionless parameter) for porous media flow is typically expressed as
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| : <math>Re = \frac{\rho v d_{30}}{\mu}</math>.
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| where ''ρ'' is the [[density]] of [[water]] (units of mass per volume), ''v'' is the specific discharge (not the pore velocity — with units of length per time), ''d<sub>30</sub>'' is a representative grain diameter for the porous media (often taken as the 30% passing size from a [[grain size]] analysis using sieves - with units of length), and ''μ'' is the [[viscosity]] of the fluid.
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| == Derivation ==
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| For stationary, creeping, incompressible flow, i.e. <math>D\left(\rho u_i\right)/Dt\approx0</math>, the Navier-Stokes equation simplify to the [[Stokes flow|Stokes equation]]:
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| : <math> \mu\nabla^2 u_i +\rho g_i -\partial_i P=0</math>, | |
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| where <math>\mu</math> is the viscosity, <math>u_i</math> is the velocity in the i direction, <math>g_i</math> is the gravity component in the i direction and P is the pressure.
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| Assuming the viscous resisting force is linear with the velocity we may write:
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| : <math>-\left(k_{ij}\right)^{-1}\mu\phi u_j+\rho g_i-\partial_i P=0</math>,
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| where <math>\phi</math> is the [[porosity]], and <math>k_{ij}</math> is the second order permeability tensor. This gives the velocity in the <math>n</math> direction,
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| : <math>k_{ni}\left(k_{ij}\right)^{-1} u_j= \delta_{nj} u_j = u_n = -\frac{k_{ni}}{\phi\mu}\left(\partial_i P-\rho g_i\right)</math>, | |
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| which gives Darcy's law for the volumetric flux density in the <math>n</math> direction,
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| : <math>q_n=-\frac{k_{ni}}{\mu}\left(\partial_i P -\rho g_i\right)</math>.
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| In isotropic porous media the off-diagonal elements in the permeability tensor are zero, <math>k_{ij}=0</math> for <math>i\neq j</math> and the diagonal elements are identical, <math> k=k_{ii}</math>, and the common form is obtained
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| : <math>\boldsymbol{q}=-\frac{k}{\mu}\left(\boldsymbol{\nabla} P -\rho \boldsymbol{g}\right)</math>.
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| == Additional forms of Darcy's law ==
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| For very short time scales, a time derivative of flux may be added to Darcy's law, which results in valid solutions at very small times (in heat transfer, this is called the modified form of [[Fourier's law]]),
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| : <math>\tau \frac{\partial q}{\partial t}+q=-K \nabla h</math>,
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| where ''τ'' is a very small time constant which causes this equation to reduce to the normal form of Darcy's law at "normal" times (> [[nanosecond]]s). The main reason for doing this is that the regular [[groundwater flow equation]] ([[diffusion equation]]) leads to [[Mathematical singularity|singularities]] at constant head boundaries at very small times. This form is more mathematically rigorous, but leads to a [[hyperbolic]] groundwater flow equation, which is more difficult to solve and is only useful at very small times, typically out of the realm of practical use.
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| Another extension to the traditional form of Darcy's law is the Brinkman term, which is used to account for transitional flow between boundaries (introduced by Brinkman in 1949 <ref>{{cite journal
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| |last = Brinkman
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| |first = H. C.
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| |title = A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles
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| |journal = Applied Scientific Research
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| |volume = 1
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| |pages = 27–34
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| |url = http://dx.doi.org/10.1007/BF02120313
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| |doi = 10.1007/BF02120313
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| |year = 1949}}</ref>),
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| : <math>\beta \nabla^{2}q +q =-K \nabla P</math>,
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| where ''β'' is an effective [[viscosity]] term. This correction term accounts for flow through medium where the grains of the media are porous themselves, but is difficult to use, and is typically neglected.
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| Another derivation of Darcy's law is used extensively in [[petroleum engineering]] to determine the flow through permeable media - the most simple of which is for a one dimensional, homogeneous rock formation with a fluid of constant [[viscosity]].
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| : <math>Q= \frac {k A}{\mu} \left( \frac{\partial P}{\partial L} \right)</math>,
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| where Q is the [[flowrate]] of the formation (in units of volume per unit time), k is the relative [[Permeability (earth sciences)|permeability]] of the formation (typically in [[millidarcies]]), A is the cross-sectional [[area]] of the formation, ''μ'' is the [[viscosity]] of the fluid (typically in units of [[centipoise]], and L is the [[length]] of the porous media the fluid will flow through. <math>\partial P/ \partial L</math> represents the pressure change per unit length of the formation. This equation can also be solved for permeability, allowing for [[relative permeability]] to be calculated by forcing a fluid of known viscosity through a core of a known length and area, and measuring the pressure drop across the length of the core.
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| For very high velocities in porous media, [[inertial]] effects can also become significant. Sometimes an [[inertial]] term is added to the Darcy's equation, known as [[Philipp Forchheimer|Forchheimer]] term. This term is able to account for the [[non-linear]] behavior of the pressure difference vs velocity data.<ref>A. Bejan, Convection Heat Transfer, John Wiley & Sons (1984)</ref>
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| :<math>\nabla P=-\frac{\mu}{k}q-\frac{\rho}{k_1}q^2</math>,
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| where the additional term <math>k_1</math> is known as inertial permeability.
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| Darcy's law is valid only for flow in [[Continuum mechanics|continuum]] region. For a flow in transition region, where both [[viscous]] and [[Knudsen]] friction are present a new formulation is used, which is known as binary friction model <ref>{{cite journal
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| |last = Pant
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| |first = Lalit M.
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| |coauthors = Sushanta K. Mitra, Marc Secanell
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| |title = Absolute permeability and Knudsen diffusivity measurements in PEMFC gas diffusion layers and micro porous layers
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| |journal = Journal of Power Sources
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| |url = http://dx.doi.org/10.1016/j.jpowsour.2012.01.099
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| |doi=10.1016/j.jpowsour.2012.01.099
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| |year = 2012}}</ref>
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| <math>\nabla P=-\left(\frac{k}{\mu}+D_K\right)^{-1}q</math>,
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| where <math>D_K</math> is the [[Knudsen]] [[diffusivity]] of the fluid in porous media.
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| ==See also==
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| *The [[darcy]] unit of fluid permeability
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| *[[Hydrogeology]]
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| *[[Groundwater flow equation]]
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| == Notes ==
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| <references/>
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| [[Category:Water]]
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| [[Category:Civil engineering]]
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| [[Category:Soil mechanics]]
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| [[Category:Soil physics]]
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| [[Category:Hydrology]]
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Are you always having problems with the PC? Are you constantly seeking ways to heighten PC performance? Next this is the post you have been shopping for. Here we will discuss several of the many asked questions with regards to having you PC serve we well; how could I make my computer faster for free? How to create my computer run faster?
Document files let the user to input data, images, tables and different elements to improve the presentation. The only issue with this structure compared to different file kinds including .pdf for illustration is its ability to be easily editable. This means that anyone watching the file could change it by accident. Additionally, this file format will be opened by different programs yet it refuses to guarantee which what we see inside the Microsoft Word application may nonetheless be the same whenever we view it using another program. However, it is actually nonetheless preferred by many computer consumers for its ease of employ plus features.
Your PC might furthermore have a fragmented difficult drive or the windows registry may have been corrupted. It may moreover be as a result of the dust and dirt which must be cleaned. Whatever the issue, you can always find a answer. Here are some strategies on how to create your PC run quicker.
Handling intermittent errors - whenever there is a content to the impact which "memory or hard disk is malfunctioning", you may put inside brand-new hardware to substitute the faulty part till the actual issue is found out. There are h/w diagnostic programs to identify the faulty portions.
These are the results which the tuneup utilities found: 622 incorrect registry entries, 45,810 junk files, 15,643 unprotected privacy files, 8,462 bad Active X goods that have been not blocked, 16 performance features which were not optimized, plus 4 changes that the computer needed.
Software errors or hardware errors that happen whenever running Windows and intermittent mistakes are the general reasons for a blue screen bodily memory dump. New software or motorists that have been installed or changes inside the registry settings are the typical s/w causes. Intermittent mistakes refer to failed program memory/ difficult disk or over heated processor plus these too may cause the blue screen physical memory dump error.
The System File Checker (SFC) can aid in resolving error 1721 as it, by its nature, scans the program files for corruption and replaces them with their original versions. This requires you to have the Windows Installation DVD ROM for continuing.
All of these problems usually be easily solved by the clean registry. Installing the registry cleaner can allow we to employ the PC without worries behind. You might capable to use we system without being scared that it's going to crash inside the middle. Our registry cleaner can fix a host of errors on the PC, identifying missing, invalid or corrupt settings in a registry.