Ying-Fan Lin

Groundwater models estimated during constant-rate pumping can reproduce observed head yet yield different predictions after the pumping rate changes. This study determines when distinct adjustment times for groundwater flux and hydraulic gradient can be estimated and improve prediction of subsequent drawdown and recovery. The Lagging Darcy law (LDL) yields impulse and unit step responses for a leaky aquifer that can be used in continuous-time groundwater models. Four dimensionless groups relate the two adjustment times to leakage time, pumping duration, and observation distance. The analysis spans 2601 combinations and locates conditions where the two times provide independent information and Hantush prediction error exceeds 5% of the initial drawdown range. Separate observation periods estimate parameters, select the pumping response, and evaluate later predictions. The selection rule chose LDL for 39 of 1296 synthetic Hantush hydrographs; the exact 95% binomial interval was 2.15–4.09%. Field applications comprise a constant-rate pumping test at Dewey and a Louisville well field affected by precipitation and Ohio River stage. The selected LDL response reduced recovery RMSE by 81.0% at Dewey and prediction RMSE by 57.5% at Louisville C26, with a moving block bootstrap interval of 45.5–67.3%. Both predictions also had lower RMSE than persistence. Pumping duration and separation of the constitutive times determine when the classical response is adequate and when delayed adjustment materially affects a later groundwater prediction.

A disturbed annulus around a well can transmit and store water differently from the surrounding aquifer. A finite two-zone model describes this annulus directly, whereas the traditional skin factor replaces it by a steady head difference at the well. This study derives and compares five skin-factor boundary conditions that successively include radial resistance, skin storage, vertical flow within the skin, and first and second finite-thickness corrections. A semi-analytical solution is obtained in the Laplace domain and is used for the drawdown calculations, sensitivity analysis, and Bayesian parameter estimation. An independently assembled finite-volume model verifies the semi-analytical calculations. Pumping drawdown, water-level recovery, and non-oscillatory slug displacement in an unconfined aquifer are compared with the finite two-zone solution. The skin-factor condition with second finite-thickness correction gives the smallest difference from the finite two-zone solution and is the only tested condition whose differences for all three tests are below one tenth of the observation standard deviation. Close agreement in drawdown or water level does not ensure unique parameter estimates. Well-level data alone cannot separate the three skin properties, whereas depth-distributed heads make them identifiable in the common synthetic case. Three monotonic slug-out tests from the Mississippi River Valley alluvial aquifer further show that inferred aquifer conductivity depends on the selected skin condition. The second finite-thickness correction is recommended when the finite annulus is not solved explicitly, with uncertainty from observations and the skin approximation quantified.

Groundwater models often treat hydraulic conductivity estimates from aquifer tests as interchangeable inputs, but slug tests and pumping tests sample different support volumes and transform water-level responses through different interpretation equations. This mismatch can turn method-dependent interpretation error into apparent hydraulic-conductivity heterogeneity. This study develops TRUST-K, short for Transformation-Residual Uncertainty and Support-scale Treatment of hydraulic conductivity, a transformation-uncertainty framework for integrating slug-test and pumping-test hydraulic conductivity estimates as method-dependent soft data. TRUST-K estimates interpretation bias and residual scatter from dimensionless test controls before field use. Verified synthetic experiments simulate paired slug and pumping tests on shared heterogeneous aquifers with known support-scale conductivity, screen response-fit quality, and estimate conditional transformation priors. Hold-out validation and controlled spatial estimation test calibration and field-scale consequences. A U.S. Geological Survey data set from Lovelock, Nevada provides a field-data predictive check with co-located slug tests and multi-well pumping and recovery responses. Co-located well pairs show slug-test hydraulic-conductivity contrasts of 3.0 to 185.7 times. Direct transfer of slug-test conductivity into pumping-response prediction performs no better than a zero-drawdown predictor, whereas TRUST-K preserves method-specific uncertainty before combining estimates. These results show that aquifer-test interpretation should be treated as a transformation step before hydraulic conductivity estimates are used in groundwater modeling, stochastic hydrogeology, or site-scale parameter mapping.

Delayed recovery in pressurized slug tests can carry information about near-well hydraulic memory, but the same short records also reflect wellbore storage, pressure amplitude, support scale, and model non-uniqueness. This study develops an observation-to-memory transformation framework that extracts apparent-memory diagnostics from pressurized slug tests with explicit uncertainty checks. A new radial analytical solution embeds Lagging Theory within the classical state-factor form and introduces separate response times for flux memory and hydraulic-gradient memory. The solution reduces to the classical instantaneous-flow response when lag effects vanish. We evaluated the framework with tabulated Minnelusa aquifer slug-test records from three wells near Spearfish, South Dakota. The analysis combined classical and lagging-theory pathways, deterministic fitting, Bayesian posterior sampling, response-level residual diagnostics, load-block validation, posterior principal-component analysis, and a hierarchical pressure-dependent model for repeated tests in one well. The lagging-theory pathway reduced structured residuals, but the inferred delay times did not behave as universally identifiable aquifer constants. Repeated partial-release tests at one well supported a comparatively stable shared-memory interpretation, whereas repeated pressure-amplitude tests at another well produced pressure-dependent apparent memory. In the multi-pressure well, the median flux-memory response time increased from 2.50 s at 5 psi to 61.7 s at 30 psi, and the apparent memory length increased almost linearly with pressure amplitude. Load-block validation improved average withheld-pressure prediction, but three pressure levels were insufficient to define a general pressure-lag law. Pressurized slug tests can therefore screen for delayed hydraulic response and classify memory behavior; unique material interpretation still requires independent constraints on wellbore response, support scale, and pressure-dependent near-well effects.

Constant-head aquifer tests prescribe the well head and measure the transient well flowrate, making them the Dirichlet-boundary counterpart of constant-rate pumping tests. This study derives and verifies an analytical solution for that response under a first-order lagging Darcy law. The derivation demonstrates that the well-wall flowrate must use a frequency-dependent flow transmissivity derived from the same constitutive relation that controls radial propagation. Infinite-domain and finite-outer-boundary solutions recover the classical zero-lag solution and the expected steady radial-flow limit. Stehfest, de Hoog, and Talbot inversions provide numerical cross-checks over early, intermediate, and late times. Temporal sensitivity analysis demonstrates that flowrate mainly resolves transmissivity, whereas multi-well specific-drawdown responses add time-dependent information on storage and response times. A multi-response field demonstration evaluates four inverse strategies: discharge-based inversion, joint discharge-drawdown calibration, specific-drawdown fitting, and a hybrid multi-response objective. Specific-drawdown fitting closely reproduces observation-well specific-drawdown responses, whereas information criteria and absolute-flow prediction favor the hybrid objective. The hybrid objective combines pumping-well discharge with multi-well specific drawdown, favors the lagging-Darcy response function by information criteria, and improves local identifiability. Remaining raw-drawdown bias and well-specific response-time estimates identify structural ambiguity in the field response. Fitted lag times therefore function as effective response-time parameters in the absence of independent physical constraints.

Quantifying uncertainty across many observation wells complicates lagging-theory inversion in pumping tests. To resolve this, this study introduces LAGWISE (Lagging Theory Workflow for Inference, Screening, and Evaluation) to advance the framework from single-curve fitting to full-network inversion. LAGWISE systematically evaluates lag-term support, rapidly generates posterior ranges, anchors selected cases with Bayesian sampling, and diagnoses time-window dependencies. The workflow couples Flow Matching as an amortized posterior engine with Markov chain Monte Carlo (MCMC) as the Bayesian anchor and time attribution as the diagnostic layer. This study applies LAGWISE to synthetic twins and a fractured crystalline-bedrock pumping test with 12 reproducible observation wells. Field results show the lagging model reduced the root mean square error by a median factor of 5.42; furthermore, the Bayesian information criterion supported the lagging model across all 12 wells. For the synthetic twins, a delay-scale and delay-ratio parameterization successfully improved the identifiability of delay times. Mapping posterior summaries to field coordinates demonstrates how parameter uncertainty informs monitoring-network design. Crucially, amortized inference breaks the linear computational scaling of traditional methods. After a single 3.21-second training step, Flow Matching sampled 12 curves in 0.164 seconds, compared to 78.8 seconds for repeated MCMC. LAGWISE therefore positions Flow Matching as a highly scalable posterior engine within a rigorous hydrological verification framework, ensuring explicit model selection, Bayesian checking, and field-design translation.