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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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This is perfectly normal though, as many women have irregular cycles. Just prior to the release of egg from ovary, the level of your basal body temperature drops about a half degree, and then it rises again after ovulation. Sometimes just by using natural methods of eliminating stress and insomnia, infertility can be managed. So many women of all ages of just about each and every age category get totally changed virtually any infertility problems. It is an excellent source of selenium that is considered to boost men fertility and help to avoid the breaking of chromosome that occurs in early miscarriage.
Are you looking for some information on how to get pregnant step by step. Moderate intake of caffeine does not causes either infertility or miscarry, but over dose of caffeine liking drinking a lot of coffee will increase the risk of infertility and miscarry because caffeine is stimulant causing over production of certain hormones which affect ovulation and hamper conception. Counting from the initial spotting at the start of your period is wrong. You can direct Sims to fall in love and start their very own family. You might reason that both of you are young and healthy, so conceiving a child won't be a problem.
Once you look closer into this program additionally, you will read testimonials from ladies worldwide with similar stories. Just prior to ovulation, it's doable you'll see an increase in clear and slippery secretions that look rather similar to uncooked egg whites. When you are past the stage of fertility it's even harder to get pregnant. Many couples think that in order to get pregnant, all that has to be done is to merely stop the pill or any other form of contraception. Are there any changes you can make to improve how you feel.
In IVF, sperm and eggs are combined in a petri dish and then the resulting embryos are either transferred into the uterus or frozen for future use. It is caused by not enough estrogen to stimulate the production of cervical mucus which are friendly to sperm invasion. Failing to conceive after so many attempts at ferility treatment can takes its toll physically and emotionally. But even the researchers couldn't believe just how strong an effect this chemical was having. Herbs can help detoxify the liver, which is important to insulin regulation, and stimulate ovary functions.
- Vitamin C is an essential nutrient for humans in which it functions as an anti-oxidant, since it protects the body against oxidative stress It is also a cofactor in at least eight enzymatic reactions, including several collagen synthesis reactions that cause the most severe symptoms of scurvy when they are dysfunctional. As a matter of fact, the Office on Women's Health states that for most patients, clomiphene is usually tried first. When you are bearing in mind how you can best get pregnant rapidly, try to bear in mind that pregnancy is awfully tough to control or force. Merely performing intercourse isn't necessarily all there is to getting pregnant. My oldest daughter had some issues in high school with her joints and she was unable to do some of her cheer leading moves because of it, but for the most part she was and is functioning normally.
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