Auxiliary material for this article contains a data set of the analytic formulae for the partial derivatives of the piezometric head with respect to the logarithm (base 10) of the resistance c and the hydraulic conductivity k. Additional file information is provided in the README.txt.
 Analytic relations between input parameters, output variables, and statistics for confidence are derived for semiconfined groundwater flow in a circular domain around a well. The statistics of the two calibrated parameters are derived analytically using a least squares target function. The statistics of the piezometric head are approximated using linear variance analysis (LVA, also known as first-order second moment method (FOSM)) and quadratic variance analysis (QVA). LVA values for the capture zone radius have been calculated based on numerical derivatives. The results are used to verify TrCalCon, the calibration and confidence module of the groundwater flow simulation package TRIWACO. The LVA and Monte Carlo (MC) results of TrCalCon compare well with the analytic LVA and QVA values, respectively. The problem is only weakly nonlinear in the heads, so that the LVA is accurate enough for practical purposes, and QVA or MC analysis would not be needed. The test can be applied as well to other groundwater simulation packages with a module for automatic parameter optimization and reliability analysis. As presented, the problem is quite simple, and the parameter optimization is relatively easy. The analysis can be expanded easily to make the problem more intricate by adding parameters, changing parameter values, and including other measurements.