SI and English Units
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![Page 1: SI and English Units](https://reader035.fdocuments.net/reader035/viewer/2022081417/56812f93550346895d950f76/html5/thumbnails/1.jpg)
SI and English Units
• SI:
- Mass = kilogram
- Length = meter
- time = second• English
- Mass = slug
- Length = foot
- time = second
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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 => T1 + T2 + … + Tn
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Specific Storage
• Specific storage Ss = amount of water per unit volume stored or expelled owing to compressibility of mineral skeleton and pore water per unit change in head (1/L).
• Ss = ρwg(α+nβ)• α = compressibiliy of aquifer skeleton.• n = porosity.• β = compressibility of water.
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Storativity of confined Unit
S = b Ss
• Ss = specific storage.
• b = aquifer thickness.
• All water released in confined, saturated aquifer comes from compressibility of mineral skeleton and pore water.
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Storativity in Unconfined Unit
• Changes in saturation associated with changes in storage.
• Storage or release depends on specific yield Sy and specific storage Ss.
• S = Sy + b Ss
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Volume of water drained from aquifer
• Vw = SAdh
• Vw = volume of water drained.
• S = storativity (dimensionless).
• A = area overlying drained aquifer.
• dh = average decline in head.
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Average horizontal conductivity: Kh avg = m=1,n (Khmbm/b)
Kh avg
Kv avg
Average vertical conductivity:
Kv avg = b / m=1,n (bm /Kvm)
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O
Y
X dh/dx
dh/dy
Grad h = [(dh/dx)2 + (dh/dy)2]0.5
θ = arctan ((dh/dy)/(dh/dx))
θ
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Forces
• Gravity – pulls water downward.
• External pressure
- Vadose zone: atmospheric pressure
- Saturation zone: atmospheric + water
• Molecular attraction.
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Resisting Forces
• Shear stresses - shear resistance – viscosity.
• Normal stresses.
• Friction = Shear stresses + Normal stresses.
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Mechanical Energy
• Kinetic energy:
• Ek = ½ m v2 [ML2/T2; slug-ft2/s2 or kg-m2/s2]
• m = mass [M; slug or kg]
• v = velocity [L/T; ft/s or m/s]
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Mechanical Energy
• Gravitational potential energy:
• W = Eg = mgz. [ML2/T2; slug-ft2/s2 or kg-m2/s2].
• z = elevation [L; ft or m].
• g = gravitational acceleration [L/T2; ft/s2 or m/s2].
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Pressure
• Pressure P = F/A.• P = pressure [M/LT2; slug/ft/s2 or
(kg-m/s2)/m2].• A is cross-sectional area perpendicular to
the direction of the force (L2; ft2 or m2).• F is force (ML/T2; slug-ft/s2 or kg-m/s2).• P unit is Pascal (N/m2).• P => potential energy per unit volume.
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Energy per unit mass
• Etm = v2/2 + gz + P/ρ. [(L/T)2]
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Hydraulic head, h
• Hydraulic head is energy per unit weight.
• h = v2/2g + z + P/gρ. [L].
• Unit: (L; ft or m).
• v ~ 10-6 m/s or 30 m/y for ground water flows.
• v2/2g ~ 10-12 m2/s2 / (2 x 9.8 m/s2) ~ 10-13 m.
• h = z + P/gρ. [L].
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Hydraulic head, h
• h = z + P/gρ = z + hp.
• z = elevation.
• hp = P/gρ - pressure head – height of water column.
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Head in water with variable density
• P2 = ρfghf
• P1 = ρpghp
• P2 = P1
• ρfghf = ρpghp
• hf = (ρp/ρf )hp
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Force potential and hydraulic head
• Force potential• Ф = gz + P/ρ = gz + ρ ghp/ ρ = g(z+hp)• h = z + hp
• Ф = gh.• g can be considered a constant ~ head can
be used to represent the force potential.• Head controls the movement of ground
water.
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Darcy’s Law
• Q = -KA(dh/dl).
• dh/dl = Hydraulic gradient.
• dh = change in head between two points separated by small distance dl.
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Reynolds number
• R = ρqd/μ.
• R - the Reynolds number (dimensionless).
• ρ – fluid density (M/L3; kg/m3).
• μ – fluid viscosity (M/T-L; kg/s-m).
• q – discharge velocity (L/T; m/s).
• d – diameter of the passageway through which the fluid moves (L; m).
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Laminar flow (Small R < 10)
Turbulent flow (Large R)
Flow lines
Flow lines
Darcy’s Law: Yes
Darcy’s Law: No
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Specific discharge
• Q = vA
• v = Q/A = -K dh/dl
• Specific discharge is also called Darcy flux.
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Seepage (average linear) velocity
• vx = Q/(neA) = -K/ne dh/dl
• vx = average linear velocity (L/T; ft/s; m/s).
• ne = the effective porosity (dimensionless)
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Dupuit assumptions
• Hydraulic gradient is equal to the slope of the water table.
• For small water-table gradients, the streamlines are horizontal and equipotential lines are vertical.
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Flow lines and flow nets
• A flow line is an imaginary line that traces the path that a particle of ground water would flow as it flows through an aquifer.
• A flow net is a network of equipotential lines and associated flow lines.
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Boundary conditions
• No-flow boundary – flow line – parallel to the boundary. Equipotential line - intersect at right angle.• Constant-head boundary – flow line – intersect at right angle. Equipotential line - parallel to the boundary.• Water-table boundary – flow line – depends. Equipotential line - depends.
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Constant head
h = 40 feet
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Estimate the quantity of water from flow net
• q’ = Kph/f.• q’ – total volume discharge per unit width of aquifer
(L3/T; ft3/d or m3/d).• K – hydraulic conductivity (L/T; ft/d or m/d).• p – number of flowtubes bounded by adjacent pairs of
flow lines.• h – total head loss over the length of flow lines (L; ft
or m).• f - number of squares bounded by any two adjacent
flow lines and covering the entire length of flow.
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