SI and English Units SI: - Mass = kilogram - Length = meter - time = second English - Mass = slug - Length = foot

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) 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 θ = arctan ((dh/dy)/(dh/dx)) dh/dy θ O dh/dx X

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 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) Flow lines Darcy’s Law: Yes Laminar flow (Small R < 10) Flow lines Darcy’s Law: No Flow lines Turbulent flow (Large R)

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.