Engineering
Darcy's Law Groundwater Flow, Hydraulic Gradient & Pore Seepage Velocity
Compute signed Darcy discharge, hydraulic gradient, specific discharge and average pore-water velocity under a one-dimensional saturated-flow model.
Distinguish specific discharge from average pore-water velocity
This model assumes steady one-dimensional saturated Darcy flow with uniform scalar hydraulic conductivity, a constant bulk area perpendicular to flow and consistent hydraulic heads measured from one datum. Hydraulic conductivity K is not intrinsic permeability. Here K is in metres/day, head and length in metres, area in square metres, and effective porosity is dimensionless.
Define positive flow from head location 1 toward location 2: i=(h1-h2)/L. Reversing the head difference reverses the sign of gradient, discharge and velocity. This convention is equivalent to the negative-gradient form of Darcy’s law.
Specific discharge q averages flow over the bulk area. Dividing by effective porosity gives average pore-water velocity, not the speed of every particle or a guaranteed contaminant arrival time. Dispersion, sorption, reaction and heterogeneous flow paths can change solute transport. The speed-up factor is 1/ne, not a universal factor of three or four.
| Hydrogeology Flow Metric | Formula | Sand Aquifer (K=12 m/d, dh=3m/150m, A=200, ne=0.30) | Flat Water Table (h1 = h2) |
|---|---|---|---|
| Hydraulic Gradient i | (B3 - B4) / B5 | (48 - 45) / 150 = 0.0200 m/m | 0.0000 |
| Darcy Flux q (m/day) | B2 * ((B3 - B4) / B5) | 12 * 0.02 = 0.24 m/day | 0.00 m/day |
| Aquifer Discharge Q (m^3/day) | B2 * ((B3 - B4) / B5) * B6 | 0.24 * 200 = 48.00 m^3/day | 0.00 m^3/day |
| Average pore-water velocity | (B2 * ((B3 - B4) / B5)) / B7 | 0.24 / 0.30 = 0.80 m/day (3.3x q!) | 0.00 m/day |
Volumetric Aquifer Discharge Q = K * ((h1 - h2) / L) * A
Multiplies hydraulic conductivity B2, hydraulic head gradient (B3 - B4) / B5, and saturated cross-sectional area B6.
=IF(COUNT(B2:B7)<>6,"Invalid inputs",IF(OR(B2<0,B5<=0,B6<=0,B7<=0,B7>1),"Invalid inputs",ROUND(B2 * ((B3 - B4) / B5) * B6, 2)))Average pore-water velocity
Divide specific discharge by effective porosity. K may be zero, heads may have either sign relative to their datum, and length and area must be positive. Porosity must be greater than zero and no greater than one.
=IF(COUNT(B2:B7)<>6,"Invalid inputs",IF(OR(B2<0,B5<=0,B6<=0,B7<=0,B7>1),"Invalid inputs",ROUND((B2 * ((B3 - B4) / B5)) / B7, 4)))Signed hydraulic gradient
Positive means head decreases from location1 toward location2.
=IF(COUNT(B2:B7)<>6,"Invalid inputs",IF(OR(B2<0,B5<=0,B6<=0,B7<=0,B7>1),"Invalid inputs",(B3-B4)/B5))Darcy specific discharge
K times the signed gradient, in metres/day. This is not volumetric flow.
=IF(COUNT(B2:B7)<>6,"Invalid inputs",IF(OR(B2<0,B5<=0,B6<=0,B7<=0,B7>1),"Invalid inputs",ROUND(B2 * ((B3 - B4) / B5), 4)))Try the example
Use the original values or change them to test your understanding.