{"id":80,"date":"2019-06-10T06:14:11","date_gmt":"2019-06-10T06:14:11","guid":{"rendered":"http:\/\/geop16.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=80"},"modified":"2019-06-10T06:46:59","modified_gmt":"2019-06-10T06:46:59","slug":"darcys-law-porosity-permeability-transmissibility-specific-yield-specific","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/geop16\/chapter\/darcys-law-porosity-permeability-transmissibility-specific-yield-specific\/","title":{"rendered":"Darcy\u2019s Law: Porosity, Permeability, Transmissibility, specific yield, specific"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
<\/div>\r\n
<\/div>\r\n
<\/div>\r\n
<\/div>\r\n
Objectives<\/strong><\/div>\r\n
<\/div>\r\n
Cleaning of contaminated groundwater and surface water of Dijon City, France.<\/div>\r\n
<\/div>\r\n
<\/div>\r\n
Keywords<\/strong><\/div>\r\n
\r\n\r\nHydrology, Porosity, Permeability, Transmissibility, specific yield, specific retention and hydraulic conductivity.\r\n
\r\n\r\n \r\n\r\nIntroduction:<\/strong>\r\n\r\n \r\n

Water is essential to the life of the earth surface either organic or inorganic life. Population growth, urbanization and expansion of the economic activities leadto increasing demand of water for various uses. There are various uses of water. Domestic, industrial, irrigational, hyro-power, thermal power, recreational are the major users of water resources on the earth surface. Quality and quantity is the major concern of the water. The crisis about water resources development and management arises due to rapid growth of population and urbanization. The crisis arises because most of the available water on the earth is not for use without filtration. Quality water problem arises in the initial phase of urbanization in the world.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

Dijon City of France was facing an acute problem of drinking water due to mustard pollution during 18th century. Darcy carried out experiments while researching sand filters to full fill the water demand of Dijon city that lead to Darcy\u2019s Law in 1856.<\/p>\r\n \r\n

Darcy's law is a simple proportional relationship between instantaneous discharge rate through a porous medium, the viscosity of the fluid and the pressure drop over a given distance. Application of Darcy's law is to analyze water flow through an aquifer. \u201cThe flow rate through porous media is proportional to the head loss and inversely proportional to the length of the flow path is known universally Darcy\u2019s Law.\u201c<\/p>\r\n \r\n

Darcy's law is an 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 on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of earth sciences. Ground water hydrology works is in the natural state. The flow of water through aquifers is depending upon the porosity and permeability of geological structure.<\/p>\r\n \r\n\r\nHydrology<\/strong>\r\n\r\n \r\n

Darcy's law along with the equation of conservation of mass is equivalent to the groundwater flow equation, one of the basic relationships of hydrogeology.<\/p>\r\n \r\n

\u201cHydrology is a subject of great importance for people and their environment which treats all phases of the earth's water. Applied applications of hydrology are found in such way as the design, operation of hydraulic structures, water supply, wastewater treatment, disposal, irrigation, drainage, hydropower generation, flood control, navigation, erosion, salinity control, pollution, recreational use of water, fish cultivation and wildlife protection. The role of applied hydrology is to help analyze the problems involved in these tasks and to provide guidance for the planning and management of water resources\u201d. (Chow et al. 1998)<\/p>\r\n \r\n\r\nAboutHenry P. G. Darcy<\/strong>\r\n\r\n \r\n

Henry PhilibertGaspard Darcy was a French engineer. He was born on 10th June 1803 in Dijon, France. He studied at the French School of Bridges and Roads in Paris. This study was parts of the corps of Bridges and Roads. He was assigned by the corps to a position in Dijon after completion of graduation.He made important contributions to flow and friction loss in pipes, created an improved pitot tube design, and was the first to postulate the existence of a boundary layer in fluid flow. He carried out experiments while researching sand filters that lead to Darcy\u2019s Law in 1856. He diedunexpectedly on January 3, 1858 from pneumonia during a trip to Paris.<\/p>\r\n \r\n\r\nWhy Henry Darcy was motivated in the field of hydrology?<\/strong>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

He was in search of fresh and pure water because People of Dijon city was under stress of contaminated water. It was the cause of motivation in the field of hydrology. He thought that hydrology needs an accurate prediction with quantitative analysis. It requires scientific and mathematical solution.<\/strong><\/p>\r\n \r\n\r\nAssignment of Henry Darcy;<\/strong>\r\n\r\n \r\n

He was assigned to find out a solution of contaminated water for cleaning. The assignment was to supply pure water to Dijon city in the year 1856. Supply water was contaminated by the waste of mustard industry. His law provides an accurate description of the flow of ground water. It explains hydro-geologic environments.<\/p>\r\n \r\n\r\nObjective:<\/strong>\r\n\r\n \r\n\r\nCleaning of contaminated groundwater and surface waterof Dijon City, France.\r\n\r\n \r\n\r\nAssumptions of Henry Darcy<\/strong>:\r\n\r\n \r\n

The following assumptions of Darcy\u2019s law.<\/p>\r\n

1.-The soil is saturated.<\/p>\r\n

2.-The flow through soil is laminar.<\/p>\r\n

3.-The flow is continuous and steady.<\/p>\r\n

4.-The total cross sectional area of soil mass is considered.<\/p>\r\n

5.\u00a0 -The 27 C temperature was considered at the time of testing.<\/p>\r\n \r\n\r\nCondition of Darcy\u2019s Law:<\/strong>\r\n\r\n \r\n\r\n1.\u00a0\u00a0\u00a0 Saturated flow and unsaturated flow\r\n\r\n \r\n\r\n2.\u00a0\u00a0\u00a0 Steady-state and transient flow\r\n\r\n \r\n\r\n3.\u00a0\u00a0\u00a0 Flow in aquifers and aquitards\r\n\r\n \r\n\r\n4.\u00a0\u00a0\u00a0 Flow in homogeneous and heterogeneous systems\r\n\r\n \r\n\r\n5.\u00a0\u00a0\u00a0 Flow in isotropic or anisotropic media\r\n\r\n \r\n\r\n6.\u00a0\u00a0\u00a0 Flow in rocks and granular media\r\n\r\n \r\n\r\n \r\n\r\nFirst Experiment of Henry Darcy.<\/strong>\r\n\r\n \r\n

Fig-1: Darcy Experiment(1886):Flow rate determined by head lossdh=h -h<\/p>\r\n1\u00a0\u00a0\u00a0 2\r\n\r\n<\/div>\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

Water passes through a sand column and the volumetric flow rate Q .Sand measured at the area of the sand column was known, as was the length of the sand in the column. Darcy measured the distance between the water levels in the two mano outlet during the experiment.<\/p>\r\n \r\n

a Henri Darcy established empirically that the flux of water through a permeable formation is proportional to the distance between top and bottom of the soil column.<\/p>\r\n \r\n\r\nThe constant of proportionality is called the hydraulic conductivity<\/em> (K).\r\n\r\n\"\"\r\n

\r\n\r\nFig-2: The cross-sectional meters at various flow rates<\/strong>.\r\n\r\n<\/div>\r\n<\/div>\r\n\"\"\r\n\r\nWhere;\r\n\r\n\"\"\r\n

Henry Darcy successfully designed 12.7 km of aqueducts water supply system with surface water to Dijon City in the age of 25 in 1828. The system was to provide pressurized surface water supply lines. 28,000 meters pipe line was developed.<\/p>\r\n \r\n

The meaning of pressurized system to supply surface water without use of pumps and without filters.<\/strong><\/p>\r\n\r\n

\r\n\r\n\u00a0 \u00a0 <\/strong>\r\n\r\n \r\n\r\nObservations of Henry Darcy;<\/strong>\r\n\r\n \r\n

Henry Darcy was assigned to study public water supply development of Dijon city of France in 1856. He worked out to improve sand filters for purification of water. He used apparatus to measure water level at both ends and discharge at the rate of flow L3\/T through a vertical column filled with Sand.<\/p>\r\n \r\n\r\n \r\n

Darcy, in search of suitable filtering media, conducted experiments with sand-packed filters.<\/p>\r\n \r\n

Henri Darcy established empirically that the energy lost \u2206h<\/em> in water flowing through a permeable formation is proportional to the length of the sediment column \u2206L<\/em>. The constant of proportionality K is called the hydraulic conductivity<\/em>. The Darcy Velocity VD:<\/p>\r\n \r\n\r\nV<\/em>D<\/em> = \u2013 K (\u2206h\/\u2206L)<\/em>\r\n\r\n<\/div>\r\n

\r\n

\u00a0 \u00a0 where\u00a0 Q = V<\/em>D<\/em> A ( where A = total area)<\/em><\/p>\r\n \r\n\r\nQ = \u2013 KA (dh\/dL)<\/em>\r\n\r\n \r\n\r\nImportant Contributions;<\/strong>\r\n\r\n \r\n

1. Flow and friction loss in pipes<\/p>\r\n

2. Designed and created improved pilot tubes<\/p>\r\n

3. He was the first who postulate the existence of a boundary layer in fluid flow.<\/p>\r\n \r\n\r\n \r\n\r\nA- What is Permeability?<\/strong>\r\n\r\n \r\n

\u2022 Permeability is defined as the property of soil which permits flow of water through it.<\/p>\r\n \r\n

\u2022 A soil is highly pervious when water can flow through it easily. E.g. Gravels.<\/p>\r\n \r\n

\u2022 In an impervious soil, the permeability is very low and water cannot easily flow through it. E.g. Clays.<\/p>\r\n \r\n

\u2022 Rocks are impermeable<\/p>\r\n \r\n

\u2022 Permeability is a very important engineering property of soils. A knowledge of Permeability is essential for:-<\/p>\r\n \r\n\r\n\u00d8\u00a0\u00a0 Settlement of building\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0\u00a0 Yield of wells\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0\u00a0 Seepage through and below the earth structures\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0\u00a0 Earth pressure\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0\u00a0 Uplift pressure under hydraulic structure.\r\n\r\n \r\n\r\nFactors affecting permeability of Soils;<\/strong>\r\n\r\n \r\n\r\nThe following factors affect the permeability of soils:-\r\n\r\n \r\n\r\n1.\u00a0\u00a0\u00a0\u00a0 Particle size\r\n\r\n2.\u00a0\u00a0\u00a0\u00a0 Properties of pore fluid.\r\n\r\n3.\u00a0\u00a0\u00a0\u00a0 Void ratio of soil.\r\n\r\n4.\u00a0\u00a0\u00a0\u00a0 Shape of particles.\r\n\r\n5.\u00a0\u00a0\u00a0\u00a0 Structure of soil mass.\r\n\r\n6.\u00a0\u00a0\u00a0\u00a0 Degree of saturation.\r\n\r\n7.\u00a0\u00a0\u00a0\u00a0 Absorbed water.\r\n\r\n8.\u00a0\u00a0\u00a0\u00a0 Impurities in water.\r\n\r\n \r\n\r\nAquifer:<\/strong>\r\n\r\n \r\n

a. Properties: Porosity, specific yield, specific retention.<\/p>\r\n

b. Potential: Transmissivity, storativity.<\/p>\r\n

c. Types: confined, unconfined.<\/p>\r\n

d. Hydraulic conductivity, Physical Laws controlling water transport<\/p>\r\n\r\n<\/div>\r\n \r\n\r\nFig-3: Experiment:Constant Head of permeability Test<\/strong>\r\n\r\n \r\n\r\n\"\"\r\n

\r\n\r\n\u00a0 \u00a0 Object:-<\/strong>\r\n\r\n \r\n

To determine the coefficient of permeability of a soil specimen by constant head method.<\/p>\r\n \r\n\r\nEquipment\u2019s:-<\/strong>\r\n\r\n \r\n

Permeability mould, internal diameter = 100mm, effective height = 127.3mm, capacity = 1000ml, complete with all accessories<\/p>\r\n \r\n\r\nConstant head tank.\r\n\r\n \r\n

Graduated cylinder, stop water, thermo meter. Filter paper, vacuum pump.<\/p>\r\n \r\n

Weighing balance, 0.1 gm accuracy.<\/p>\r\n \r\n\r\nTest Procedure<\/strong>:-\r\n\r\n \r\n

1. Measure internal dimensions of the mould and apply a little grease on the inside to the mould.<\/p>\r\n \r\n

2. Take approximately 2.5kg of the soil, form a thoroughly mixed wet soil, in the mould. Compact the soil at the required dry density using a suitable compacting device.<\/p>\r\n \r\n

3. Remove collar and base plate and trim the excess soil level with the top of the mould.<\/p>\r\n \r\n

4.\u00a0 Clean the outside of the mould and Find the mass of the soil in the mould. Take a small specimen of the soil in container for the water constant determination.<\/p>\r\n \r\n\r\n5.\u00a0 Saturate the porous stones.\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

6.\u00a0 Place the porous stone on the drainage base and keep a filter paper on the porous stone.<\/p>\r\n \r\n

7.\u00a0 Place the mould with soil on the drainage base.<\/p>\r\n \r\n\r\n8.\u00a0 Place a filter paper and a porous stone on the top of specimen.\r\n\r\n \r\n\r\n9.\u00a0 Connect the constant head tank to the drainage cap inlet.\r\n\r\n \r\n

10.\u00a0 Open the stop cock, and allow the water downward so that all the air is removed then close the stop cock.<\/p>\r\n \r\n

11.\u00a0\u00a0 Open the stop cock and at the same time start the stopwatch and start collecting the water flowing out of the base in a measuring flask for some time interval.<\/p>\r\n \r\n

12.\u00a0\u00a0 Measure the difference of head (h) in levels between the constant head tank and the outlet in the base.<\/p>\r\n \r\n\r\n \r\n\r\nB- What is Porosity?<\/strong>\r\n\r\n \r\n

Porosity is the quality of being porous, or full of tiny holes. Liquids go right through things that have porosity. Porosity stems from the Greek word poros for \"pore,\" which means \u201cpassage.\u201d<\/p>\r\n \r\n\r\nPrinciple of Ground Water Flow<\/strong>\r\n\r\n \r\n\r\nPorosity and effective porosity<\/strong>\r\n\r\nTotal porosity is the part of rock that's void space;\r\n\r\n \r\n\r\nn<\/strong>T<\/strong> = V<\/strong>v<\/strong>\/V<\/strong>T<\/strong> = (V<\/strong>T<\/strong> \u2013 V<\/strong>s<\/strong>)\/V<\/strong>T<\/strong>\r\n\r\n \r\n\r\nWhere V<\/em>v<\/em> is Void volume, V<\/em>T<\/em> total volume and Vs is solid volume.\r\n\r\n \r\n\r\nVoid ratio (e )=Vv\/Vs<\/em><\/strong>\r\n\r\n \r\n

Primary Porosity<\/strong>: It is original in rock. Primary porosity is range from 26 percent to 47 percentin different arrangements and in ideal spheres.<\/p>\r\n \r\n

Secondary Porosity<\/strong>: It is in fracture or solution porosity.<\/p>\r\n \r\n

Total Porosity<\/strong>: It is an amount of pore space. It does not require pore connection.<\/p>\r\n \r\n

Effective\u00a0 Porosity<\/strong>: It is the percentage of interconnected pore space available for groundwater flow.\u00a0\u00a0\u00a0\u00a0 Porosity is one order of magnitude smaller than total porosity.<\/p>\r\n \r\n

Porosity in Sediments and Rocks<\/strong>: It is depending on grain size, particle shape, arrangement, and actual values of porosity ranges from zero and near zero to more than 60 percent.The smaller the particle size and the higher the porosity is in the sedimentary rocks.<\/p>\r\n \r\n

Darcy\u2019s Law and FlowDarcy\u2019s law provides an accurate description of the flow of ground water in almost all hydrogeological environments.<\/p>\r\n \r\n\r\nDarcy allows an estimate of:<\/strong>\r\n\r\n \r\n

\u2022\u00a0 the velocity or flow rate moving within the aquifer<\/p>\r\n

\u2022\u00a0 the average time of travel from the head of the aquifer to a point located downstream.<\/p>\r\n\r\n<\/div>\r\nFig-4: Downstream aquifer<\/strong>\r\n\r\n\"\"\r\n

\r\n\r\nDarcy\u2019s Experiment (1856):Flow rate determined by Head loss dh = h<\/em><\/strong>1<\/em><\/strong> - h<\/em><\/strong>2<\/em><\/strong>\r\n\r\n \r\n

\u2022 Henri Darcy established empirically flux of water through a permeable formation is proportional to the distance between top and bottom of the soil column.<\/p>\r\n \r\n

\u2022 The constant of proportionality is called the hydraulic conductivity<\/em> (K).<\/p>\r\n \r\n\r\n\u2022\u00a0 \u00a0V = Q\/A, V\u00a0\u00a0 \u2013 \u2206h, and\u00a0 V\u00a0\u00a0 1\/\u2206L\r\n\r\n \r\n\r\nV = \u2013 K (\u2206h\/\u2206L)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 and since\r\n\r\n \r\n\r\nQ = VA (A = total area)\r\n\r\n \r\n\r\nQ = \u2013 KA (dh\/dL)\r\n\r\n \r\n\r\n \r\n\r\nDarcy & Seepage Velocity<\/strong>\r\n\r\n \r\n

Darcy velocity is a fictitious velocity since it assumes that flow occurs across the entire cross-section of the soil sample. Flow actually takes place only through interconnected pore channels<\/p>\r\n \r\n

From the Continuity Equation:<\/p>\r\n \r\n

Q = A v<\/em>D<\/em> = A<\/em>V<\/em>V<\/em>s<\/em><\/p>\r\n \r\n\r\nWhere: Q<\/em> = flow rate\r\n\r\n \r\n\r\nA <\/em>= total cross-sectional area of\u00a0 material\r\n\r\n \r\n\r\nA<\/em>V<\/em> = area of voids\r\n\r\n \r\n\r\nV<\/em>s<\/em> = seepage velocity\r\n\r\n \r\n\r\nV<\/em>D<\/em> = Darcy velocity<\/span>\r\n\r\n \r\n\r\nTherefore:\u00a0 <\/span>V<\/em>S<\/em> = V<\/em>D<\/em> ( A\/A<\/em>V<\/em>)<\/em>\r\n\r\n<\/div>\r\n

\r\n\r\n\u00a0 \u00a0 \u00a0Multiplying both sides by the length of the medium (L<\/em>) V<\/em>S<\/em> = V<\/em>D<\/em> ( AL \/ A<\/em>V<\/em>L ) = V<\/em>D<\/em> ( V<\/em>T<\/em> \/ V<\/em>V<\/em> )<\/em>\r\n\r\n \r\n\r\nWhere: V<\/em>T<\/em> = total volume\r\n\r\n \r\n\r\nV<\/em>V<\/em> = void volume\r\n\r\n \r\n\r\nBy Definition, V<\/em>v<\/em>\/ V<\/em>T<\/em> = n<\/em>, the soil porosity\r\n\r\n \r\n\r\nThus\u00a0\u00a0 V<\/em><\/strong>S<\/em><\/strong> = V<\/em><\/strong>D<\/em><\/strong> \/ n<\/em><\/strong>\r\n\r\n \r\n\r\n \r\n\r\nEquations of Groundwater Flow<\/strong>\r\n\r\n<\/div>\r\n
\r\n

<\/p>\r\n \r\n

Description of ground water flow describes conservation of fluid mass partial differential equation of flow. is based on Darcy\u2019s LawContinuity Equation\u2013It during flow through a porous medium; results in a<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\u00a0 \u00a0 \u00a0<\/strong>\r\n\r\nLaplace\u2019s Equation;<\/strong>\r\n\r\n \r\n\r\nMass In - Mass Out = Change in Storage<\/strong>\r\n\r\n<\/div>\r\n\u00a0<\/strong>\r\n\r\n\"\"\r\n\r\nC- Transmissivity<\/strong>\r\n\r\n \r\n

The amount of water that can be transmitted horizontally through a unit width by the full saturated thickness of the aquifer under a hydraulic gradient of 1.<\/p>\r\n\r\n