SI and English Units

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

3. 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

4. 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.

5. 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.

6. 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

7. 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.

8. 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)

10. Grad h = [(dh/dx)2 + (dh/dy)2]0.5 Y θ = arctan ((dh/dy)/(dh/dx)) dh/dy θ O dh/dx X

12. Forces • Gravity – pulls water downward. • External pressure - Vadose zone: atmospheric pressure - Saturation zone: atmospheric + water • Molecular attraction.

13. Resisting Forces • Shear stresses - shear resistance – viscosity. • Normal stresses. • Friction = Shear stresses + Normal stresses.

14. 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]

15. 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].

16. 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.

17. Energy per unit mass • Etm = v2/2 + gz + P/ρ. [(L/T)2]

18. 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].

19. Hydraulic head, h • h = z + P/gρ = z + hp. • z = elevation. • hp = P/gρ - pressure head – height of water column.

23. Head in water with variable density • P2 = ρfghf • P1 = ρpghp • P2 = P1 • ρfghf = ρpghp • hf = (ρp/ρf )hp

25. 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.

27. Darcy’s Law • Q = -KA(dh/dl). • dh/dl = Hydraulic gradient. • dh = change in head between two points separated by small distance dl.

28. 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).

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

30. Specific discharge • Q = vA • v = Q/A = -K dh/dl • Specific discharge is also called Darcy flux.

31. 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)

35. 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.

40. 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.

42. 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.

43. Constant head h = 40 feet

44. 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.