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

• 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)

Kh avg

Kv avg

Average vertical conductivity:

Kv avg = b / m=1,n (bm /Kvm)

O

Y

X dh/dx

dh/dy

Grad h = [(dh/dx)2 + (dh/dy)2]0.5

θ = arctan ((dh/dy)/(dh/dx))

θ

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 kg-m2/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 kg-m2/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

(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.

Energy per unit mass

• Etm = v2/2 + gz + P/ρ. [(L/T)2]

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].

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

• P2 = ρfghf

• P1 = ρpghp

• P2 = P1

• ρfghf = ρpghp

• hf = (ρp/ρf )hp

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.

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

• 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).

Laminar flow (Small R < 10)

Turbulent flow (Large R)

Flow lines

Flow lines

Darcy’s Law: Yes

Darcy’s Law: No

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.