Question 5 of 9: Darcy's Law and Groundwater Flow

Question 5: Darcy's Law and Groundwater Flow (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Darcy's law: $q = -K\,\dfrac{dh}{dl}$, where $q$ = specific discharge (Darcy velocity, an apparent/bulk velocity as if flow occupied the full cross-section), $K$ = hydraulic conductivity, and $dh/dl$ = hydraulic gradient (head loss per unit flow-path length).

Find. (a) the equation and its terms; (b) which soil and fluid properties control $K$ and their direction of effect; (c) how to convert $q$ to the true (seepage) velocity.

Approach. Expand $K$ into its intrinsic-permeability and fluid-property components, then apply the effective-porosity correction that converts bulk Darcy velocity into true pore velocity.

1. (a) State Darcy's law and define terms. $q=-K\,dh/dl$: $q$ [L/T] is the volumetric flow rate per unit total cross-sectional area (bulk/apparent velocity, not the true fluid speed in the pores); $K$ [L/T] is hydraulic conductivity, a measure of how easily the specific fluid moves through the specific medium; $dh/dl$ [dimensionless] is the hydraulic gradient, the drop in hydraulic head $h$ per unit distance $l$ along the flow path; the minus sign shows flow moves down-gradient (from high to low head).
2. (b) The soil/fluid-coupled parameter is $K$ itself. $K = k\,\rho g/\mu$, where $k$ [L$^2$] is the intrinsic permeability (a property of the solid medium alone – grain size, sorting, pore connectivity) and $\rho,\mu$ are the fluid's density and dynamic viscosity. Soil properties: larger, better-sorted, better-connected pores (coarser/well-sorted sand or gravel) ↑ increase $k$ and hence $K$; finer grains, poor sorting, and higher clay content ↓ decrease $K$. Fluid properties: higher fluid density $\rho$ ↑ increases $K$ (more driving weight per unit volume); higher fluid viscosity $\mu$ ↓ decreases $K$ (thicker fluid flows less readily) – this is why oil (higher $\mu$, similar $\rho$ to water) has a lower hydraulic conductivity than water through the identical medium.
3. (c) Seepage (true average linear) velocity. Because water only moves through the connected pore space, not the full bulk cross-section, the true average velocity is the Darcy velocity divided by the effective porosity: $v_s = q/n_e$. Since $n_e < 1$, the seepage velocity is always larger than the specific discharge – this is the speed that should be used for contaminant travel-time and capture-zone calculations, not $q$ itself.

Worked numeric example (illustrative, not from the source data): a medium sand with $K=12\text{ m/day}$ carries a head drop of $\Delta h = 3.5$ m over a flow path $\Delta l = 250$ m, and $n_e = 0.30$. The gradient is $i=3.5/250=0.014$, so $q = 12 \times 0.014 = \boxed{0.168\text{ m/day}}$, and the seepage velocity is $v_s = 0.168/0.30 = \boxed{0.56\text{ m/day}}$ – more than three times the specific discharge.