4.3 Darcy’s Law
As previously described, Darcy’s law is shown again in Equation (33).
| [latex]Q = - KA\frac{dh}{dl}[/latex] | (33) |
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Go through the four steps and try to understand what the equation means.
Steps 1 and 2. The variables and constants are as follows.
[latex]Q[/latex] = volumetric discharge: variable with dimensions of L3T-1
[latex]K[/latex] = hydraulic conductivity: constant with dimensions of LT-1
[latex]A[/latex] = cross‑sectional area perpendicular to flow: variable with dimensions of L2
[latex]dh[/latex] = change in head: variable with dimensions of L
[latex]dl[/latex] = change in length along flow path: variable with dimensions of L
Step 3. The numerator‑denominator relationships to discharge ([latex]Q[/latex]) are:
- cross‑sectional area (A), directly proportional;
- change in head ([latex]dh[/latex]), directly proportional;
- change in length ([latex]dl[/latex]), inversely proportional;
- hydraulic gradient ([latex]dh/dl[/latex]), directly proportional; and
- hydraulic conductivity, directly proportional.
Step 4. There are no modifiers on the variables so the relationships are linear.
Using the above, we can express Darcy’s law in a sentence as “The volumetric discharge is directly proportional to the cross‑sectional area and the hydraulic gradient, and the proportionality is regulated by a linear constant called the hydraulic conductivity” (Exercise 6⮧).