POROSITY, SPECIFIC YIELD & SPECIFIC RETENTION. Physical properties of

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1 POROSITY, SPECIFIC YIELD & SPECIFIC RETENTION Porosity is the the ratio of the voids to the total volume of an unconsolidated or consolidated material. Physical properties of n = porosity as a decimal fraction Primary Porosity: original Vt = the total volume of a material porosity in the rocks (i.e., initial Vs = the volume of the solids in the empty space between/within the material particles that make up the rock) Vv = the volume of the voids Secondary Porosity: porosity in a rock derived from external processes (i.e., fracturing, dissolution, etc.) Often, porosity is expressed as a percentage by multiplying the ratio by 100. Porosity also depends on the range of grain size (sorting) and shape of the subject material, but not on the size. Finegrained materials tend to be better sorted than coarsegrained materials, thereby exhibiting greater porosities. Source: Kasenow,
2 Specific yields of different formations Relation between Sp.Yield, Porosity and Specific retention HYDRAULIC HEAD & GRADIENTS HYDRAULIC HEAD & GRADIENTS water entering an unconfined or confined well will stand at a particular level. This level is often termed as the hydraulic head and is actually the sum of three components  the pressure head, elevation head and velocity head. The velocity head is often disregarded because ground water movement in most cases is relatively slow.
3 In practical applications, a depth to ground water measurement is obtained and subtracted from the top of the well casing elevation to measure total head. Note that the datum plane illustrated below is often calibrated to sea level. The direction of ground water movement can be understood in the fact that ground water always flows in the direction of decreasing head. The rate of movement on the other hand is dependent on the hydraulic gradient, which is the change in head per unit distance. The change in head measurement is ideally in the direction where the maximum difference of head decrease occurs. Hydraulic gradient In the example below, the hydraulic gradient is determined to be ft./ft. (the change in head divided by the change in distance). Notice the units are foot by foot but can be described in more inconsistent units such as foot per mile. Sand Tank Model for showing Hydraulic gradient
4 DARCY'S LAW & HYDRAULIC CONDUCTIVITY (K) DARCY'S LAW & HYDRAULIC CONDUCTIVITY (K) In the mid1800s the French engineer Henry Darcy successfully quantified several factors controlling ground water movement. These factors are expressed in an equation that is commonly known as Darcy's Law. Units for Hydraulic conductivity By rearranging Darcy's Law and solving for hydraulic conductivity (K) in common units we can get a sense of what hydraulic conductivity really represents. Velocity of ground water Another important factor controlling ground water movement is its velocity. The ground water velocity equation can be derived from a combination of the velocity equation of hydraulics and from, Darcy's Law.
5 Hydraulic conductivity in different rock types
6 Isotropic and heterogeneous The aquifer is homogeneous if hydraulic conductivity is the same and heterogeneous if different in different rock formations. If the value of hydraulic conductivity is the same in all directions, then the aquifer is said to be isotropic. If hydraulic conductivity is different in different directions, the aquifer is said to be anisotropic. Validity of Darcy s s law Darcy s law basically states that V H V is the average velocity Passing though the area of cross section and The factor of proportionality is K which is called as Hydraulic conductivity. As V is moving through the entire porous material, solids as well as pores, the volume rate of flow is calculated as Q= V * A. But the actual velocity in the soil is many times higher than the obtained at the cross section as water moves in the pores and can be assumed as though water is moving through capillary tubes.
7 Validity of Darcy s s law The relation between the actual velocity vm and the darcy s velocity v is as under Vm=(A/Acap)*v A= Total area normal to the flow Acap= sum of the cross sectional areas of capillarity tubes Vm=v/n where n= Porosity The Darcy s law is applicable only for laminar flow, for very small velocities and water molecules travel smooth paths and more or less parallel to the solid boundaries of the pores. If the velocities are increased, the flow becomes turbulent, and water moves in irregular manor and the Darcy s law is not applicable. Validity of Darcy s s law Groundwater is normally slow and the darcy s law is acceptable except is fractured/ cavernous flow. The flow regime is normally expressed as Reynolds no in fluid mechanics as under Nr=ρvD/µ v= velocity of the fluid D= dimension of conduit ( Diameter of pipe) ρ= density of fluid(gm/cm3) µ= viscosity of fluid (g/sms) or p µ/ ρ= kinematic viscosity of velocity If Nr= < 2100 the flow is laminar If Nr is in between 2100 to 4000, it is intermediate If Nr>4000 the flow is turbulent Factors affecting hydraulic conductivity Temperature Kt= (µ20/ µt) K20 Kt= K at temperature t µ t= Absolute viscosity at t K20= K at 20deg C µ 20= Absolute viscosity at 20 Quality of water K also has depends on the water quality (SAR= Na/sqrt((ca+mg)/2) Rainfall for unconfined aquifers Intrinsic permeability The intrinsic permeability of the porous medium is the property of the medium only and independent of the density and viscosity of the fluid. This can be expressed as K= K µ/ ρg (cm/sec) k= intrinsic permeability (Darcy=0.987*10^8 cm2) ρ=density of fluid(g/cm2) g=acceleration due to gravity(cm/s2) The preferred units are m/day 1 Darcy= 864/m/day
8 Intrinsic permeability Hydraulic conductivity of sands K=Cd^2 Where C= Constant depending on temp, packaging, grain size distribution and shape K= 1 m[(1α^2)(θ/100 P/dm)^2] α ^3 Where m= packing factor(=5) θ= Sand shape factor varying from 6 for spherical to 7.7 to angular grains, P= % of sand held between adjacent sieves dm= is the geometric mean of rated size of adjacent sieves. Sand analysis Hazen method The Hazen equation was used for sediments with a uniformity of less than 5 and an effective grain size (de, which is equal to d10) between 0.1 mm and 3 mm. The formula for the Hazen equation is: K = g/v Ch*f(n)d10^2 where: Ch = 6 x 104 f(n) = [1 + 10(n0.26)] the function of porosity, n. g = m/s2 Gravity acceleration. v = 1.14 x 106 the kinematic viscosity.
9 Kozeny method The Kozeny equation is applicable to course sand samples with a low uniformity of less than 2 and an effective grain size between 0.5 mm and 4 mm. The formula for the Kozeny equation is: K = g/v Ckf(n)d10^2 where: Ck = 8.3 x 103 f(n) = n3/(1n)2 Breyer method The Breyer equation is used for poorly sorted samples. The equation can be used for samples with uniformity values from 1 to 20 and effective grain sizes between 0.06 mm and 0.6 mm. The formula for the Breyer equation is: K = g/v Cbde^2 where: Cb = 6 x 104 log(500/u) DupuitForchheimer Assumptions Darcy s law can be used to solve one dimensional flow equations by assuming that the flow is purely horizontal and also uniformly distributed with depth. The vertical flow is negligible. These assumptions are called DupuitForchheimer assumptions One dimension problems Seepage from open channels Subsurface runoff Uniform infiltration and drainage to a stream Recharge rate of Leaky aquifer Height of perched water table Effect of river stage on water table in flood plain Aquifer parameters
10 TRANSMISSIVITY (T) Transmissivity (T) is the volume of water flowing through a crosssectional area of an aquifer that is 1 ft. x the aquifer thickness (b), under a hydraulic gradient of 1 ft./ 1 ft. in a given amount of time (usually a day). If we think about our definition of hydraulic conductivity, we can conclude that transmissivity (T) is actually equal to hydraulic conductivity(k) times aquifer thickness (b). Or otherwise denoted as T = Kb. We can also conclude that transmissivity is expressed as ft2/day because if T = Kb, then T = (ft./day)(ft./1). Storage Coefficient The "S" is used to represent the storage coefficient of an aquifer which is the volume of water released from an aquifer per 1 foot/ m surface area per 1 foot /m change in head. Notice that we are not speaking of water flowing through an aquifer, rather we are referring an aquifer's ability to store water. Mathematically, the storage coefficient is dimensionless as the equation below illustrates. The size of the storage coefficient is dependent whether the aquifer is unconfined or confined. In regards to a confined aquifer, water derived from storage is relative to; (1) the expansion of water as the aquifer is depressurized (pumped) and, (2) compression of the aquifer. In a confined aquifer setting, the load on top of an aquifer is supported by the solid rock skeleton and the hydraulic pressure exerted by water (the hydraulic pressure acts as a support mechanism). Because of these variables, the storage coefficient of most confined aquifers range from 105 to 103 ( to 0.001). Conversely, in an unconfined aquifer setting, the predominant source of water is from gravity drainage and the expansion of water and compaction of the rock skeleton is negligible. Thus, the storage coefficient is approximate to value of specific yield and ranges from 0.1 to about 0.3.
11 Diffusivity = T/S Specific Capacity Specific capacity is defined as the discharge in cum/day per m drawdown of a well C= Q/s Where C= Specific capacity of well(m^2/day) Q= discharge (cum/day) s= drawdown (m)
12 CONES OF DEPRESSION As water is withdrawn from a well, the water level in the well begins to decline as water is removed from storage in the well. The head in the well will fall below the level of the surrounding aquifer and water begins moving from the aquifer into the well. The water level will continue to decline and the flow rate of water into the well will increase until the inflow rate is equal to withdrawal rate. Water from the aquifer must converge on the well from all directions and the hydraulic gradient must get steeper near the well. For this reason the resultant 3D shape of water withdrawal is a called a cone of depression. End
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