A fully penetrating well pumps a confined, homogeneous, isotropic aquifer of constant thickness at a constant rate Q. Water comes only from elastic storage, so the cone of depression keeps growing outward. The page integrates the radial diffusion equation numerically (conservative finite volumes on a logarithmic radial grid, Crank-Nicolson in time) and compares it with the exact Theis solution.
S ds/dt = T (1/r) d/dr ( r ds/dr ) s(r,0) = 0, s(inf,t) = 0, 2 pi r T ds/dr -> Q at the well
s(r,t) = Q / (4 pi T) * W(u), u = r^2 S / (4 T t), W(u) = E1(u) = integral_u^inf e^-x / x dx
Cooper-Jacob (u < 0.05): s = Q / (4 pi T) ln( 2.2458 T t / (r^2 S) ) slope per log10 cycle = 2.303 Q / (4 pi T)
Recovery: s = Q / (4 pi T) [ W(u) - W(u2) ], u2 = r^2 S / (4 T (t - t_off))
- Q and T - drawdown scales with Q / T everywhere; doubling T halves the cone depth at every radius.
- Storativity S - sets how fast the cone spreads: its reach grows like the square root of 4 T t / S. A small S (confined) gives a wide, shallow cone.
- Pump-off - after the chosen day the same field recovers; the solver and the superposition formula agree.
- Checks - the well function against its integral, the solver against Theis, the water balance, the Cooper-Jacob slope, recovery and time-step convergence.
- Honest simplifications - no leakage from adjacent layers, no boundaries, no well-bore storage or skin, no partial penetration, no unconfined dewatering (S would then be the specific yield and T would fall as the water table drops). The vertical scale of the 3D cone is exaggerated. A teaching model, not a design tool.