Darcy's_law - Wikiparency

# Darcy's law **Darcy's law** is the equation that describes the flow of a fluid through a porous medium, such as [[Water|water]] through sand, soil or rock. It says that the rate of flow is proportional to the drop in [[Hydraulic_head|hydraulic head]] along the flow path and to a property of the material called its hydraulic conductivity. The French engineer Henry Darcy found the relationship in 1856 from experiments on the sand filters of the public fountains of Dijon.[^woessner41][^ochsner] It is "one of the most important relationships in soil physics, hydrology, and hydrogeology,"[^ochsner] the basis for calculating how fast [[Groundwater|groundwater]] moves through an [[Aquifer|aquifer]] and how much oil or gas a well can draw from a reservoir. On the Thury spine, Darcy's law is the fluid dynamics of the ground. The speed at which water moves through the aquifers under the state is set by this one relation. ## History Henry Darcy was an engineer working on the water supply of Dijon. In his 1856 report *Les Fontaines publiques de la ville de Dijon* he described experiments meant to improve the flow of water through the sand filter beds of the city's water system.[^darcy1856][^woessner41] He varied the type of sand, the area and thickness of the filter bed, and the force driving water through it, and found that the flow through a sand-packed column was a linear function of the loss of hydraulic head across it.[^woessner41][^cohen] Darcy's result is the porous-medium counterpart of the law Poiseuille had found for [[Laminar_flow|laminar flow]] through narrow tubes.[^ochsner] ## Description In its simplest form, Darcy's law gives the discharge *Q* through a cross-section of area *A*: *Q* = −*K A* (*h*₂ − *h*₁) / Δ*L* = −*K i A* where *K* is the hydraulic conductivity, *h*₂ − *h*₁ is the change in head over the distance Δ*L*, and *i* is the hydraulic gradient. The minus sign means that water flows from higher head to lower.[^cohen] Hydraulic conductivity measures "the ease with which water flows through a material" and has units of length per time.[^woessner41] Steven Earle's *Physical Geology* gives a worked example of the groundwater version, *V* = *K i*. With a hydraulic conductivity of 0.00001 m/s and a gradient of 0.08, water moves about 0.069 m a day, and would take about 1,450 days, nearly four years, to travel 100 m.[^earle14] Groundwater is slow. ## Derivation Darcy's law can be understood by picturing a porous rock as a bundle of tiny tubes. Flow through each tube follows Poiseuille's law, and adding up the tubes gives a flow proportional to the pressure gradient and inversely proportional to the fluid's [[Viscosity|viscosity]]. Mehdi Zeidouni's *Petroleum Reservoir Dynamics* introduces the concept of permeability this way.[^zeidouni2] The permeability *k* belongs to the rock alone; the hydraulic conductivity *K* combines it with the fluid's density ρ and viscosity μ and with [[Gravity|gravity]]: *K* = *k*ρ*g*/μ.[^freeze2] Because viscosity changes with temperature, the same soil conducts water faster when the water is warm.[^ochsner] ## Applications ### Petroleum engineering In [[Petroleum_engineering|petroleum engineering]], Darcy's equation is the most basic relation for calculating production rates from a reservoir.[^zeidouni1] Reservoir properties vary enormously: permeability can range from 10⁻⁶ to 10 darcy and viscosity from 0.01 to 10⁶ centipoise.[^zeidouni1] The unit of permeability, the darcy, is defined as the permeability that gives a specific discharge of 1 cm/s for a fluid with a viscosity of 1 centipoise under a gradient of 1 atmosphere per centimetre. One darcy is about 10⁻⁸ cm², or 9.869233 × 10⁻¹³ m².[^freeze2][^sizes] Consolidated sands typically fall between 0.01 and 1 darcy.[^britannica] ## Additional forms ### Differential expression For flow in three dimensions, Darcy's law is written as a [[Differential_equation|differential equation]] relating the flux at each point to the local gradient of head. Combined with conservation of mass, it gives the groundwater flow equations on which regional groundwater [[Mathematical_model|models]] are built.[^freeze2] ### Quadratic law At higher flow rates, inertia adds an extra pressure drop that grows faster than the velocity. The Forchheimer equation adds a quadratic term to Darcy's law to account for it.[^zeidouni25] ### Correction for gases in fine media (Knudsen diffusion or Klinkenberg effect) A gas flowing at low pressure through fine pores slips along the pore walls, so its apparent permeability is higher than the rock's true, absolute permeability. This is the Klinkenberg effect: the gas permeability rises linearly with the reciprocal of the average pressure, so measurements at several pressures can be extrapolated back to the absolute permeability.[^zeidouni25] ## Validity Darcy's law assumes slow, [[Laminar_flow|laminar]] flow. Its limit is usually expressed with a [[Reynolds_number|Reynolds number]] based on the average grain diameter. Jacob Bear put it that "Darcy's law is valid as long as the Reynolds number, based on average grain diameter, does not exceed some value between 1 and 10."[^woessner45] Zeidouni gives the same range for rock: flow is Darcy flow while Re < 1, and when Re exceeds 10 it is called non-Darcy flow, as can happen near a well where velocities are high.[^zeidouni25] ## Minnesota *This section is specific to Wikitube.* Minnesota's County Geologic Atlas program maps aquifers county by county. The Minnesota Geological Survey prepares Part A, on geology, and the Minnesota DNR prepares Part B, which covers the direction of groundwater flow, aquifer properties, groundwater chemistry and pollution sensitivity.[^mndnr-cga] Aquifer tests supply the Darcy parameters. At Prairie Island in southeastern Minnesota, the U.S. Geological Survey measured an average hydraulic conductivity of 10 feet per day in the Mt. Simon aquifer.[^usgs-mtsimon] Around Rochester, transmissivities in the St. Peter–Prairie du Chien–Jordan aquifer range from less than 5,000 to more than 20,000 square feet per day.[^usgs-rochester] ## See also - [[Groundwater]] - [[Aquifer]] - [[Hydraulic_head]] - [[Hydrostatics]] - [[Diffusion]] ## References [^darcy1856]: Darcy, Henry (1856). *Les Fontaines publiques de la ville de Dijon*. Paris: Victor Dalmont. https://gallica.bnf.fr/ark:/12148/bpt6k624312 [^woessner41]: Woessner, William W.; Poeter, Eileen P. (2020). *Hydrogeologic Properties of Earth Materials and Principles of Groundwater Flow*. The Groundwater Project. §4.1 "Darcy's Law." https://doi.org/10.21083/978-1-7770541-2-0 [^woessner45]: Woessner, William W.; Poeter, Eileen P. (2020). *Hydrogeologic Properties of Earth Materials and Principles of Groundwater Flow*. The Groundwater Project. §4.5 "Applicability of Darcy's Law," quoting Bear (1972). https://books.gw-project.org/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow/chapter/applicability-of-darcys-law/ [^cohen]: Cohen, R. M.; Cherry, J. A. (2020). *Conceptual and Visual Understanding of Hydraulic Head and Groundwater Flow*. The Groundwater Project. §2.1 "Darcy's Law." https://books.gw-project.org/conceptual-and-visual-understanding-of-hydraulic-head-and-groundwater-flow/chapter/darcys-law/ [^freeze2]: Freeze, R. Allan; Cherry, John A. (1979). *Groundwater*. Prentice-Hall; Groundwater Project web edition. Chapter 2. https://fc79.gw-project.org/english/chapter-2/ [^ochsner]: Ochsner, Tyson (2019). *Rain or Shine: An Introduction to Soil Physical Properties and Processes*. Oklahoma State University Libraries. §4.4 "Darcy's Law," p. 103. https://open.library.okstate.edu/rainorshine/ [^earle14]: Earle, Steven (2015). *Physical Geology*. BCcampus. §14.2 "Groundwater Flow," p. 383. https://opentextbc.ca/geology/ [^zeidouni1]: Zeidouni, Mehdi (2025). *Petroleum Reservoir Dynamics*. LSU Scholarly Repository, Louisiana State University. Chapter 1, pp. 4–5. https://repository.lsu.edu/etext [^zeidouni2]: Zeidouni, Mehdi (2025). *Petroleum Reservoir Dynamics*. Chapter 2, "Darcy Equation and its Limitations," §2.2 "Permeability." https://repository.lsu.edu/etext [^zeidouni25]: Zeidouni, Mehdi (2025). *Petroleum Reservoir Dynamics*. §2.5 "Complications": §2.5.1 "Inertia" and §2.5.2 "Slippage." https://repository.lsu.edu/etext [^sizes]: Sizes.com. "darcy." https://www.sizes.com/units/darcy.htm [^britannica]: Encyclopaedia Britannica. "darcy." https://www.britannica.com/science/darcy [^mndnr-cga]: Minnesota Department of Natural Resources. "County Geologic Atlas (CGA) Series." https://www.dnr.state.mn.us/waters/groundwater_section/mapping/county-geo-atlas.html [^usgs-mtsimon]: Winterstein, T. A. (2002). *Hydraulic properties of Mt. Simon aquifer, Prairie Island Indian Community, southeastern Minnesota, 2001*. U.S. Geological Survey Water-Resources Investigations Report 2002-4263. https://pubs.usgs.gov/publication/wri024263 [^usgs-rochester]: Lindgren, R. J. (1997). *Hydraulic properties and ground-water flow in the St. Peter-Prairie du Chien-Jordan aquifer, Rochester area, southeastern Minnesota*. U.S. Geological Survey Water-Resources Investigations Report 97-4015. https://pubs.usgs.gov/publication/wri974015  **Part of the [[WT!Thury_Hydrodynamics_Compendium|Thury Hydrodynamics Compendium]]** — main article for section 35, *Flow through the ground*. Related sections: [[Hydrology]] · [[Viscosity]] · [[Lake_Agassiz]].   **Microsim — three.js (Wikitube framework):** *Darcy's law* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/thury/Darcy's_law.html" data-title="Darcy's law"></div> *Built from `MICROSIM_GUIDE/specs/sims/Darcy's_law.json`; part of the [[WT!Thury_Hydrodynamics_Compendium|Thury Hydrodynamics Compendium]] set.*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Darcy%27s_law) : [Wikitube](https://en.wikitube.io/wiki/Darcy%27s_law) · pinned revision [1366832167](https://en.wikipedia.org/w/index.php?oldid=1366832167) · 2026-09-10 ## Previous hub tags Hubs: `Life_Physics`, `Systems`. Portals: [[PORTAL_Thury_Hydrodynamics_Apex_Spine]], [[PORTAL_Stock_and_flow]]. --- *Thury main articles, wave 2 · 2026-09-10 · drafted · Compendium section 35 · related three.js microsim live (Porous_medium); own sim pending THY-039.*