Thiem equation for a confined aquifer (steady radial flow) In a pumping test on a confined aquifer of saturated thickness b, a steady discharge Q is pumped from a well. Two observation wells at radii r1 and r2 from the pumped well show piezometric heads h1 and h2 respectively. According to Thiem, the coefficient of permeability K is obtained from which equation (use natural logarithm)?

Introduction / Context:
The Thiem equation is a classical steady-state solution for radial flow toward a pumping well. It allows hydrogeologists to estimate the hydraulic conductivity (permeability K) of an aquifer from discharge and drawdown observations in two observation wells.

Given Data / Assumptions:

• Confined aquifer of uniform thickness b and homogeneous/isotropic properties.
• Steady pumping at constant discharge Q.
• Two observation wells at radii r1 and r2 with observed piezometric heads h1 and h2.
• Dupuit–Forchheimer assumptions: predominantly horizontal flow, negligible vertical gradients near observation points.

Concept / Approach:
For a confined aquifer under steady radial flow, the head distribution satisfies a logarithmic decline with radius. Integrating Darcy’s law in cylindrical coordinates between r1 and r2 gives a direct relationship among Q, K, b, and the measured heads.

Step-by-Step Solution:

Verification / Alternative check:
Dimension check: Q has L^3/T; denominator 2 * π * b has L; the remaining ratio ln(r2/r1)/(h2 - h1) has 1/L; overall gives L/T as expected for K.

Why Other Options Are Wrong:

• Option B is for unconfined Thiem (uses h^2), not confined.
• Option C is the algebraic inverse (K in denominator of Q); it gives 1/K form.
• Option D omits thickness b and misplaces terms.
• “All the above” is false because only one equation is correct for the confined case posed.

Common Pitfalls:
Mixing the confined and unconfined forms (h vs h^2), using base-10 log instead of natural log without adjusting constants, and selecting observation wells too close to the pumped well (well losses).

Final Answer:
K = (Q / (2 * π * b)) * ln(r2 / r1) / (h2 - h1)