Darcy's Law and Hydraulic Head

1. Hydraulic Head

hh12− QK= A h L p1

h1 h2 h1 and h2 are hydraulic heads associated with hp2 points 1 and 2. Q

The hydraulic head, or z1 total head, is a measure z2 of the potential of the datum water fluid at the measurement point. “Potential of a fluid at a specific point is the work required to transform a unit of mass of fluid from an arbitrarily chosen state to the state under consideration.” Three Types of Potentials

A. Pressure potential work required to raise the water pressure 1 P 1 P ρm P W1 = VdP = dP = ∫0 ∫0 m m w ρ w

ρw : density of water assumed to be independent of pressure

V: volume z = z P = P v = v Current state z = 0 P = 0 v = 0 Reference state B. Elevation potential work required to raise the elevation

1 Z W ==mgdz gz 2 m ∫0 C. Kinetic potential work required to raise the velocity (dz = vdt)

2 11ZZdv vv W ==madz m dz == vdv 3 m ∫∫∫00m dt 02

Total potential: Total [hydraulic] head: P v 2 Φ P v 2 h == ++z Φ= +gz + g ρ g 2g ρw 2 w Unit [L2T-1] Unit [L] 2 Total head or P v hydraulic head: h =++z ρw g 2g Kinetic term pressure elevation [L] head [L] Piezometer

P1 P2 ρg ρg h1 h2

z1 z2 datum

A fluid moves from where the total head is higher to where it is lower. For an ideal fluid (frictionless and incompressible), the total head would stay constant. (Fetter, p141)

Low elevation to high elevation Low pressure head to high pressure head

Flow between points of same elevation Flow between points of same pressure head For Groundwater Flow

P v2 h = + z + Kinetic term ρ w g 2g negligible h = hydraulic head [L]

P ρw g = pressure head [L] z = elevation head [L] Important: h is relative to datum (reference state) piezometers

flow h1 h2 direction?

A B

datum Water pressure varies with the water height

The pressure exerted at each hole and the force of the jet depend on the height of water above that particular hole

A strong jet indicates that the recharge area may be far from the point!

Ways of water, p125 5 Example: Po: atmospheric pressure = 1 atm= 1.013 x 10 Pa

PA = ? absolute pressure F P =+P AoA H Relative (gage) PA F mg ()HAρ g P == =w =ρ gH A A A A w Given: H= 10 m g = 9.8 m/s2 kg m P =1000 ×9.8 ×10m 3 3 A  3   2  ρw= 1.0 g/cm = 1000 kg/m m  s   kgm N = 98000 → → Pa s2m2  m2

PA ρw gH h p == ==Hm10 Pressure head at A? A ρw g ρw g In the previous example, what is the hydraulic head at A? hA = ?

B H

hA how about PB? z datum hB = ? Piezometer

Depth to Groundwater Groundwater level

(Pressure P (ρ g) head) w h (hydraulic head)

A (elevation z head) Datum

http://www.env.gov.bc.ca/wat/gws/gwbc/C02_origin.html Piezometer (measure Observation well or standpipe water level at a point) (measure water level along a section)

Water enters the Water enters the pipe through a point pipe through a section Nested piezometers to measure vertical hydraulic gradients

Datum

http://www.env.gov.bc.ca/wat/gws/gwbc/C02_origin.html More on hydraulic head Nested P Piezometers h = z + hp = z + ρg ABC Surface Elevation 225 225 225 Depth to piezometer 150 100 75 Depth to water 80 77 60

AB C Head: hA = elevation hB = =225

h = 66

C 77

80 75 Pressure (h ) = p A 100 Head: (hp)B = 150 (hp)C = Gradient between: AB = BC = datum Coastal aquifers Density Effect

When comparing heads of fluids with different densities, pressure heads must be converted to the same reference density.

ρ Ps = Pf ρ s ghs = f gh f ρ  h =  s h f   s  ρ f 

By average ρ = 1.04 ×10 3 kg s m 3 ρ = 1.0 ×10 3 kg f m 3 Point-water heads for a system of three aquifers, each containing water with a different density ?

? Aquifer Water Density Elevation Head Point-Water Head A 999 kg/m3 50.00 m 55.00 m B 1040 kg/m3 31.34 m 54.67 m C 1100 kg/m3 7.95 m 51.88 m

Aquifer Point-Water ρs Fresh-Water Fresh-Water Pressure Head ρ f Pressure Head Head A 5.00 m 1.00 5.00 m 55.0 m B 23.33 m 1.04 24.3 m 55.5 m C 43.93 m 1.10 48.3 m 56.3 m Applicability of Darcy’s Law

laminar flow ρvd R = µ

R: Reynolds number [-] ρ : fluid density [ML-3] turbulent flow v: seepage velocity [LT-1] d: diameter of passageway for fluid [L] µ: viscosity [MT-1L-1]

Darcy’s Law is only applicable to laminar flow with small Reynolds numbers (<1~10). Groundwater behaves as laminar flow under most circumstances.