Transcript Flow Principles (3/28/05)

SI and English Units
• SI:
- Mass = kilogram
- Length = meter
- time
= second
• English
- Mass = slug
- Length = foot
- time
= second
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
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.
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.
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
Volume of water drained from
aquifer
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Vw = SAdh
Vw = volume of water drained.
S = storativity (dimensionless).
A = area overlying drained aquifer.
dh = average decline in head.
Average horizontal conductivity:
Kh avg = m=1,n (Khmbm/b)
Kv avg
Kh avg
Average vertical conductivity:
Kv avg = b / m=1,n (bm /Kvm)
Grad h = [(dh/dx)2 + (dh/dy)2]0.5
Y
dh/dy
θ
X dh/dx
O
θ = arctan ((dh/dy)/(dh/dx))
Forces
• Gravity – pulls water downward.
• External pressure
- Vadose zone: atmospheric pressure
- Saturation zone: atmospheric + water
• Molecular attraction.
Resisting Forces
• Shear stresses - shear resistance – viscosity.
• Normal stresses.
• Friction = Shear stresses + Normal stresses.
Mechanical Energy
• Kinetic energy:
• Ek = ½ m v2 [ML2/T2; slug-ft2/s2 or kgm2/s2]
• m = mass [M; slug or kg]
• v = velocity [L/T; ft/s or m/s]
Mechanical Energy
• Gravitational potential energy:
• W = Eg = mgz. [ML2/T2; slug-ft2/s2 or kgm2/s2].
• z = elevation [L; ft or m].
• g = gravitational acceleration [L/T2; ft/s2 or
m/s2].
Pressure
• Pressure P = F/A.
• P = pressure [M/LT2; slug/ft/s2 or (kgm/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.
Energy per unit mass
• Etm = v2/2 + gz + P/ρ. [(L/T)2]
Hydraulic head, h
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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].
Hydraulic head, h
• h = z + P/gρ = z + hp.
• z = elevation.
• hp = P/gρ - pressure head – height of water
column.
Head in water with variable
density
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P2 = ρfghf
P1 = ρpghp
P2 = P1
ρfghf = ρpghp
hf = (ρp/ρf )hp
Force potential and hydraulic
head
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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.
Darcy’s Law
• Q = -KA(dh/dl).
• dh/dl = Hydraulic gradient.
• dh = change in head between two points
separated by small distance dl.
Reynolds number
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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).
Darcy’s Law:
Yes
Darcy’s Law: No
Laminar flow (Small R < 10)
Flow lines
Turbulent flow (Large R)
Flow lines
Specific discharge
• Q = vA
• v = Q/A = -K dh/dl
• Specific discharge is also called Darcy flux.
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)
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.
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.
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.
Constant head
h = 40 feet
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.