Equation Sheet
Hydraulic Gradient:
\(i = \dfrac{dh}{dl} \quad \text{or} \quad \dfrac{\partial h}{\partial s}\)
Darcy's Law:
\(q = k \dfrac{dh}{dl} A = kiA\)
\(v_d = ki\)
\(v_s = \dfrac{v_d}{n_e}\)
Equation of Continuity:
\(k_x \dfrac{\partial^2 h}{\partial x^2} + k_y \dfrac{\partial^2 h}{\partial y^2} = 0\)
Flow Net:
\(q = k \dfrac{n_f}{n_e} \dfrac{b}{l} h\)
Dupuit Solution (Rectangular Section, Unconfined):
\(h = \sqrt{\dfrac{H_D^2 - H_o^2}{D} x + H_o^2}\)
\(q = k \dfrac{\left(H_o^2 - H_D^2\right)}{2D}\)
Well Equation (Confined Aquifer):
\(q = 2\pi k D \dfrac{H - H_w}{\ln(R/r_w)}\)
\(h = H_w + \dfrac{q}{2\pi k D}\ln(r/r_w)\)
\(h = H + \dfrac{q}{2\pi k D}\ln(r/R)\)
\(h = H - \dfrac{1}{2\pi k D}\sum_{i=1}^{n} q_i \ln(R/r_i)\)
Well Equation (Unconfined Aquifer):
\(q = \pi k \dfrac{H^2 - H_w^2}{\ln(R/r_w)}\)
\(h^2 = H^2 - \dfrac{q \ln(R/r)}{\pi k}\)
\(k = \dfrac{q \ln(R/r_w)}{\pi\left(H^2 - H_w^2\right)}\)
\(h^2 = H^2 - \dfrac{1}{\pi k}\sum_{i=1}^{n} q_i \ln(R/r_i)\)
Unit Conversions:
- 1 m = 3.28 ft
- 1 ft = 12 in
- 1 in = 2.54 cm
- 1 lb = 2.2 kg
- 1 mi = 5280 ft
- 1 acre = 43,560 ft²
- 1 acre-ft = 43,560 ft³