Established closed-form analytical solutions for using heat as a tracer of vertical groundwater fluxes typically rely on assumptions of steady hydraulic conditions. We introduce a novel analytical approach and associated tool, PyTFLUX, to account for transient changes in vertical groundwater fluxes. The analytical solution uses a Fourier series to represent diurnal surface temperature variability and a differential method to represent vertical flux changes. Optimization techniques are employed to achieve faster convergence and prevent the estimation of unreasonable vertical fluxes. The PyTFLUX script, presented in a Python Jupyter notebook, enables the easy adoption of the new analytical framework. To test the new approach, illustrative transient vertical flux time series were developed for three time-varying groundwater flux scenarios: a step-change, a single sine-wave, and a mixed sine-wave. These profiles were analyzed to infer vertical groundwater flux time series using PyTFLUX and previously published methods implemented in VFLUX2. Results show that PyTFLUX can reproduce temporal variability in groundwater fluxes not typically captured by existing methods. Finally, previously published high-resolution sediment temperature data from the Quashnet River in Massachusetts, USA, were analyzed to demonstrate the efficacy of PyTFLUX in analyzing complex field data. The analysis of field data yielded a vertical flux time series with mean values that agreed with fluxes yielded from other approaches, but the new approach also revealed pronounced temporal flux variability that was obscured by other methods.

Lagging theory has emerged to construct mathematical models to describe groundwater flow since 2017 due to the addition of two lagging parameters to simply represent a complex physical model; however, the original theory, called dual-phase lag theory, has been widely applied to heat transfer problems since 1995. As yet, lagging theory has already been applied to develop the mathematical model related to well hydraulic in confined or unconfined aquifers and stream depletion prediction problems. For example, the effects of water inertia, dead-end or small-pore storage, capillary fringe exceeding storage, capillary suction, and streambed storage on the hydraulic response can all be simply represented by two lagging parameters, whereas the physical-based model may necessitate more in situ measures as inputs to the model. Although it has some benefits for data interpretation, there are only a few studies (merely five published papers) that specifically focus on the application of lagging theory to the problem of groundwater flow because the physical meaning of lagging parameters remains somewhat unclear. This study aims to present a brief review of studies on groundwater flow problems and to discuss the physical insights behind the concept of lagging theory. The threshold value analysis is used to investigate the lagging effect on the drawdown. In addition, we introduce several candidate models regarding the hydrology or well hydraulic for future research directions.

Mathematical models for stream depletion typically use the constant-head Dirichlet boundary condition or the general Robin boundary condition at the stream. Both approaches fix stream stage as constant during pumping. Fixed the stream stage implies the stream acts as an infinite water source with depletion affecting stream discharge but having no impact on stream stage. We refer to this depletion without drawdown as the “depletion paradox.” It is a glaring model limitation, ignoring the most observable adverse effect of long-term groundwater abstraction near a stream – dry streambeds. Our field data demonstrate that stream stage responds to pumping near the stream. This motivates the development of a model considering transient stream drawdown using the concepts of finite stream storage and mass continuity at the stream-aquifer interface. The models include the cases for fully- and non-penetrating the stream. First-order mass transfer is also assumed across the streambed. The proposed model reduces to the fixed-stage model as stream storage becomes infinitely large and the confined flow case with a no-flow boundary at the streambed when stream storage vanishes. Sensitivity analysis for hydraulic properties of the stream-aquifer system is also included. Our results suggest that fixed-stage models (a) underestimate late-time aquifer drawdown to pumping adjacent to a stream and (b) overestimate the available groundwater supply from streams to pumping wells because of the infinite stream storage assumption. This can have significant implications for the sustainable management of water resources in interacting stream-aquifer systems with heavy groundwater abstraction.