5 Darcy’s Law: Porosity, Permeability, Transmissibility, specific yield, specific
Hydrology, Porosity, Permeability, Transmissibility, specific yield, specific retention and hydraulic conductivity.
Introduction:
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.
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’s Law in 1856.
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. “The 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’s Law.“
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.
Hydrology
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.
“Hydrology 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”. (Chow et al. 1998)
AboutHenry P. G. Darcy
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’s Law in 1856. He diedunexpectedly on January 3, 1858 from pneumonia during a trip to Paris.
Why Henry Darcy was motivated in the field of hydrology?
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.
Assignment of Henry Darcy;
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.
Objective:
Cleaning of contaminated groundwater and surface waterof Dijon City, France.
Assumptions of Henry Darcy:
The following assumptions of Darcy’s law.
1.-The soil is saturated.
2.-The flow through soil is laminar.
3.-The flow is continuous and steady.
4.-The total cross sectional area of soil mass is considered.
5. -The 27 C temperature was considered at the time of testing.
Condition of Darcy’s Law:
1. Saturated flow and unsaturated flow
2. Steady-state and transient flow
3. Flow in aquifers and aquitards
4. Flow in homogeneous and heterogeneous systems
5. Flow in isotropic or anisotropic media
6. Flow in rocks and granular media
First Experiment of Henry Darcy.
Fig-1: Darcy Experiment(1886):Flow rate determined by head lossdh=h -h
1 2
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.
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.
The constant of proportionality is called the hydraulic conductivity (K).
Fig-2: The cross-sectional meters at various flow rates.
Where;
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.
The meaning of pressurized system to supply surface water without use of pumps and without filters.
Observations of Henry Darcy;
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.
Darcy, in search of suitable filtering media, conducted experiments with sand-packed filters.
Henri Darcy established empirically that the energy lost ∆h in water flowing through a permeable formation is proportional to the length of the sediment column ∆L. The constant of proportionality K is called the hydraulic conductivity. The Darcy Velocity VD:
VD = – K (∆h/∆L)
where Q = VD A ( where A = total area)
Q = – KA (dh/dL)
Important Contributions;
1. Flow and friction loss in pipes
2. Designed and created improved pilot tubes
3. He was the first who postulate the existence of a boundary layer in fluid flow.
A- What is Permeability?
• Permeability is defined as the property of soil which permits flow of water through it.
• A soil is highly pervious when water can flow through it easily. E.g. Gravels.
• In an impervious soil, the permeability is very low and water cannot easily flow through it. E.g. Clays.
• Rocks are impermeable
• Permeability is a very important engineering property of soils. A knowledge of Permeability is essential for:-
Ø Settlement of building
Ø Yield of wells
Ø Seepage through and below the earth structures
Ø Earth pressure
Ø Uplift pressure under hydraulic structure.
Factors affecting permeability of Soils;
The following factors affect the permeability of soils:-
1. Particle size
2. Properties of pore fluid.
3. Void ratio of soil.
4. Shape of particles.
5. Structure of soil mass.
6. Degree of saturation.
7. Absorbed water.
8. Impurities in water.
Aquifer:
a. Properties: Porosity, specific yield, specific retention.
b. Potential: Transmissivity, storativity.
c. Types: confined, unconfined.
d. Hydraulic conductivity, Physical Laws controlling water transport
Fig-3: Experiment:Constant Head of permeability Test
Object:-
To determine the coefficient of permeability of a soil specimen by constant head method.
Equipment’s:-
Permeability mould, internal diameter = 100mm, effective height = 127.3mm, capacity = 1000ml, complete with all accessories
Constant head tank.
Graduated cylinder, stop water, thermo meter. Filter paper, vacuum pump.
Weighing balance, 0.1 gm accuracy.
Test Procedure:-
1. Measure internal dimensions of the mould and apply a little grease on the inside to the mould.
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.
3. Remove collar and base plate and trim the excess soil level with the top of the mould.
4. 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.
5. Saturate the porous stones.
6. Place the porous stone on the drainage base and keep a filter paper on the porous stone.
7. Place the mould with soil on the drainage base.
8. Place a filter paper and a porous stone on the top of specimen.
9. Connect the constant head tank to the drainage cap inlet.
10. Open the stop cock, and allow the water downward so that all the air is removed then close the stop cock.
11. 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.
12. Measure the difference of head (h) in levels between the constant head tank and the outlet in the base.
B- What is Porosity?
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 “passage.”
Principle of Ground Water Flow
Porosity and effective porosity
Total porosity is the part of rock that’s void space;
nT = Vv/VT = (VT – Vs)/VT
Where Vv is Void volume, VT total volume and Vs is solid volume.
Void ratio (e )=Vv/Vs
Primary Porosity: It is original in rock. Primary porosity is range from 26 percent to 47 percentin different arrangements and in ideal spheres.
Secondary Porosity: It is in fracture or solution porosity.
Total Porosity: It is an amount of pore space. It does not require pore connection.
Effective Porosity: It is the percentage of interconnected pore space available for groundwater flow. Porosity is one order of magnitude smaller than total porosity.
Porosity in Sediments and Rocks: 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.
Darcy’s Law and FlowDarcy’s law provides an accurate description of the flow of ground water in almost all hydrogeological environments.
Darcy allows an estimate of:
• the velocity or flow rate moving within the aquifer
• the average time of travel from the head of the aquifer to a point located downstream.
Fig-4: Downstream aquifer
Darcy’s Experiment (1856):Flow rate determined by Head loss dh = h1 – h2
• Henri Darcy established empirically flux of water through a permeable formation is proportional to the distance between top and bottom of the soil column.
• The constant of proportionality is called the hydraulic conductivity (K).
• V = Q/A, V – ∆h, and V 1/∆L
V = – K (∆h/∆L) and since
Q = VA (A = total area)
Q = – KA (dh/dL)
Darcy & Seepage Velocity
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
From the Continuity Equation:
Q = A vD = AVVs
Where: Q = flow rate
A = total cross-sectional area of material
AV = area of voids
Vs = seepage velocity
VD = Darcy velocity
Therefore: VS = VD ( A/AV)
Multiplying both sides by the length of the medium (L) VS = VD ( AL / AVL ) = VD ( VT / VV )
Where: VT = total volume
VV = void volume
By Definition, Vv/ VT = n, the soil porosity
Thus VS = VD / n
Equations of Groundwater Flow
Description of ground water flow describes conservation of fluid mass partial differential equation of flow. is based on Darcy’s LawContinuity Equation–It during flow through a porous medium; results in a
Laplace’s Equation;
Mass In – Mass Out = Change in Storage
C- Transmissivity
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.
- T = bK
- T = transmissivity.
- b = saturated thickness.
- K = hydraulic conductivity.
- Multilayer =>T + T + … + T
1 2 n
Steady State Flow to Well
Darcy’s Law states that the velocity q in a porous medium is calculated from the head gradient the change in head per unit length in the direction of flow in an isotropic aquiferand hydraulic conductivity K .where K can be calculated from the transmissivity T and thickness b .
K = T / b.
Simplifying by assuming K = constant in all dimensionsand assuming that Transmissivity;
T = Kb and Q = flow rate to well at point (x,y) yields
E- Specific Yield and Retention
• Specific yield – S : ratio of volume of water that drains from a saturated rock owing to
y the attraction of gravity to the total volume of the rock.
• Specific retention – S : ratio of the volume of water in a rock can retain against gravity
r drainage to the total volume of the rock.
• n = S + S .
y r
• S increases with decreasing grain size.
r
F- Specific Storage:
- Specific storage S = amount of water per unit volume stored or expelled owing to s compressibility of mineral skeleton and pore water per unit change in head (1/L).
S = ρ g(α+nβ)
s w
α = compressibiliy of aquifer skeleton.
n = porosity.
β = compressibility of water.
Solutions to Ground Water Flow Equations.
Solutions for only a few simple problems can be obtained directly – generally need to apply numerical methods to address complex boundary conditions.
2 h0 called Laplace Eqn
Transient Saturated Flow
Simplifying by assuming K = constant in all dimensions And assuming that S = Ssb, and that T = Kb yields
Steady State Flow to Well
Simplifying by assuming K = constant in all dimensions and assuming that Transmissivity T = Kb and Q = flow rate to well at point (x,y) yields
Example of Darcy’s Law
1- A confined aquifer has a source of recharge.
2- K for the aquifer is 50 m/day, and n is 0.2.
3- The piezometric head in two wells 1000 m apart is 55 m and 50 m respectively, from a common datum.
4- The average thickness of the aquifer is 30 m, and the average width of aquifer is 5 km.
Calculate:
a) the rate of flow through the aquifer
(b) the average time of travel from the head of the aquifer to a point 4 km downstream
*assume no dispersion or diffusion
Fig-5: A Confined aquifer
The solution:
- Cross Sectional area=30×(5)×(1000)=15×104 2
- Hydraulic gradient=(55-50)/1000=5×10−3
- Rate of flow for K=50m/day Q=(50m/day)(75×10 2) =37,500 3/day
- Darcy velocity :V=Q/A=(37000 3/ )/15 × 104 2)=0.25m/day.
- Seepage Velocity:Vs=V/n=(0.25)/(0.2)=1.25m/day
- Time to travel 4 km downstream T=4(1000m)/(1.25m/day) 3200 days or 8.77years.
This example shows that water moves very slowly underground.
Example 2: Confined Aquifer; Where water flows through extremely fine-grained materials (colloidal clay
- A channel runs almost parallel to a river, and they are 2000 ft apart
- The water level in the river is at an elevation of 120 ft and 110ft in the channel.
- A pervious formation averaging 30 ft thick and with K of 0.25 ft/hr joins them.
- Determine the rate of seepage or flow from the river to the channel.
Fig-6: Confined Aquifer of Channel and River
Consider a 1-ft length of river (and channel)
Q=KA[(h1-h2)/L]
Validity of Henry Darcy Law;
- Darcy’s law is valid if the flow through soils is laminar:
a- The flow of water through soils depends upon the dimension of particles. In fine grained soils the dimensions of the interstices (voids) are very small and flow is necessarily laminar.
b- In course- grained soil, the flow is also laminar. However, in very coarse grained soils, such as gravels, the flow may be turbulent.
c- For flow through soils, the flow is laminar if the Reynolds number is less than unity.
- As per Allen Hazen, the maximum diameter of the particle for the flow to be laminar is about 0.50 mm.
3. It is valid for flow in clays, slits and fine sands. In coarse sands, gravels and boulders, the flow may be turbulent and Darcy’s law may not be applicable.
4. For Darcy’s law to be valid, the relationship between velocity (v) and hydraulic gradient (i) should be linear.
5. In extremely fine-grained soils, such as collodial clay, the interstices are very small. The velocity is therefore very small. In such soils, the Darcy’s law is not valid.
He died of pneumonia unexpectedly on 3rd January 1858 during a trip to Paris
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