A Transformed Time Conformable-Type Slug Test Solution for Finite-Diameter Wells in Confined Aquifers: Verification, Identifiability, and Field Diagnostics

Slug test interpretation can fail when measured recovery follows a time scale that differs from the classical Cooper–Bredehoeft–Papadopulos (CBP) finite-diameter well solution. This study derives a conformable slug test formulation by showing that a local weighted derivative converts the governing problem into the classical solution evaluated in transformed time. The formulation therefore does not introduce a nonlocal memory kernel; instead, it provides a reproducible diagnostic with one fitted exponent for testing power law time scaling while retaining the finite-diameter wellbore storage boundary condition. The solution is evaluated using double-precision Stehfest numerical inversion with 12 terms and is verified by the exact classical limit and by sensitivity tests on the number of inversion terms. Type curves, Morris sensitivity indices, objective function slices, synthetic benchmarks, and measured slug test data from the Minnelusa and Madison aquifer system near Spearfish, South Dakota, are used to evaluate the added exponent. A benchmark with an exponent above one recovered fitted exponents of 1.397 without noise and 1.417 under Gaussian noise with a standard deviation of 0.01. Field fitting over exponents from 0.5 to 2.0 reduces root mean square error and information criteria relative to the classical model for the analyzed datasets, especially the LA-88B pressure tests. However, exponents above one are interpreted only as accelerated transformed time behavior, not as conventional fractional orders or unique physical mechanisms. Comparison with a published semi-analytical slug test model that represents near-well formation damage and non-Darcy flow for the same field dataset supports using the conformable exponent as a diagnostic indicator of time-scale mismatch alongside mechanistic slug test models.