Groundwater hydraulics: For discharge of a confined (artesian) tube well under steady conditions, which classical formula is used?

Introduction / Context:
Estimating the discharge of tube wells requires groundwater flow theory. For steady, radial flow to a well in a confined aquifer, the classical Thiem equation relates drawdown and discharge. Recognizing which expression applies to which aquifer type is a core hydrogeology skill.

Given Data / Assumptions:

• Confined aquifer of thickness b with transmissivity T = K * b.
• Steady-state pumping with two observation wells at radii r1 and r2 giving piezometric heads h1 and h2.
• Radial flow, Dupuit–Thiem assumptions (horizontal flow in aquifer, negligible vertical gradients).

Concept / Approach:
For confined conditions, Thiem’s formula for discharge is commonly written as Q = 2 * π * K * b * (h1 − h2) / ln(r2 / r1) = 2 * π * T * (Δh) / ln(r2 / r1). It stems from integrating Darcy’s law in cylindrical coordinates under steady conditions.

Step-by-Step Solution:

Verification / Alternative check:
Plot drawdown versus ln(r); the slope gives Q / (2 * π * T), a standard field method for confined aquifers.

Why Other Options Are Wrong:
Darcy’s law is the basis but not, by itself, the discharge formula; Tolman is not the standard reference here; Dupuit’s unconfined form uses heads squared (h^2) rather than linear heads.

Common Pitfalls:
Mixing unconfined and confined forms; using base-10 logarithms without converting constants; ignoring well losses and partial penetration effects.

Final Answer:
Thiem’s formula