How to Calculate the Velocity of Ground Water Flow? | Geography

Ground water moves from levels of higher energy to levels of lower energy, its energy being essentially the result of elevation and pressure, the velocity heads being neglected since the flow is essentially laminar. The velocity is very small in the laminar range—of the order of 1 cm/sec—and the Reynolds number (Re) for ground water flow varies from 1 to 10 and is given by-

Re = ρv (d_m/µ) …(4.7)

Where v = velocity (seepage or bulk) of ground water flow; d_m = mean diameter of the soil grains (usually taken as -D₁₀); ρ = density of ground water and µ = dynamic viscosity of ground water.

Velocity of ground water flow, which is entirely laminar is given by Darcy’s law which states that ‘the velocity of flow in a porous medium is proportional to the hydraulic gradient’, Fig. 4.5.

v = K_i …(4.8)

Where K = coefficient permeability and I = hydraulic gradient

I = Δh/L, if a head Dh is lost in a length L

Where A = cross-sectional area of the aquifer; w = width of the aquifer; b = saturated thickness of the aquifer and T = transmissibility of the aquifer, i.e., capacity of a unit prism of aquifer to transmit water.

The actual velocity (v_a) at which the water is moving through an aquifer, i.e., on an average, the velocity at which a tracer would move through a permeable medium, is given by-

V_a = Q/A_act = Q/nA = v/n …(4.10)

Where v(= Q/A) is the apparent or seepage velocity given by Darcy’s law v – Ki.

Flow in coarse-grained aquifer under high drawdown, especially in the area adjacent to the pumping well is given by the non-Darcy regime of flow generally described by the Forchheimer equation-

i = av + bv² …(4.10(a))

Where a and b are constants depending upon the properties of porous media and the fluid, and have the dimensions-

[a] = 1/[v] = T/L, [b] = 1/[v²] = T₂/L₂.

Where [ ] means ‘dimensions of. That is, the flow is no longer laminar, particularly as it arrives at the well face due to high gradients and exhibits non-linear relationship between the velocity and hydraulic gradient. While the use of Darcy’s law is valid for low Reynolds number, typically R_e < 1, various flow situations have been observed where the Reynolds number of flow is likely to be greater than unity. For example in a gravel packed well (mean size of gravel ≈ 5 mm), R_e ≈ 45 and the flow would be transitional at a distance of about 5 to 10 times the well radius.

Example:

It was observed in a field test that 3 hr 20 min was required for a tracer to travel from one well to another 20 m apart, and the difference in their water surface elevations was 0.5 m. Samples of the aquifer between the wells indicated a porosity of 15%. Determine the permeability of the aquifer, seepage velocity, and the Reynolds number for the flow, assuming an average grain size of 1 mm and v_water at 27°C = 0.008 Stoke.

Solution:

Actual velocity of flow through the aquifer as indicated by the tracer-

Seepage velocity as given by Darcy’s law-

From. Eq. 4.10,

v_a – v/n

6 = K/40 (0.15)

K = 36 m/hr at the field temperature

= 864 m/day or m³/day/m²

= 8.64 × 10⁵ 1pd/m²

Seepage velocity (Darcy’s) v = K/40 = 864/40 = 21.6 m/day, or 0.025 cm/sec

Against v_a = 0.166 cm/sec

Note:

v_a > v

1 Stoke = 1 cm²/s ; 1 c. St. = stoke = 1/100 stoke

Reynolds number Re = vd_m/v = 0.025/100 × 1/1000 × 1/0.008 × 10^-4 = 0.3 < 1