Abstract:Heat is recognized as a natural tracer to identify the exchange of water between the groundwater and surface water compartment. One-dimensional (1D) heat transport models have the ability to obtain quantitative estimates of vertical fluxes through the sediment matrix. Input to these models can come from temperatures observed in the surface water and in the bed material of rivers and lakes. The upper thermal boundary condition at the groundwater-surface water interface is affected by seasonal and diurnal temperature variations. We hypothesize that effects of these transient influences are negligible at certain times of the year, such that the vertical temperature distribution can be approximated to be at steady state. Temperature time series observed over a year in the surface water and at several depths below a river in Belgium and in sediments of an acid mine lake in Eastern Germany were simulated with a heat balance model implemented in FEMME and the water and energy model VS2DH to obtain seepage fluxes. Temperature variations throughout the year at all depths could be adequately reproduced with the transient models. Vertical temperature profiles at several measuring times during the year were also fitted with an analytical, steady-state solution for 1D heat transport and the obtained fluxes compared to the results from transient simulations. Fluxes obtained from the much simpler steady-state solution were compared well with the flux rates from transient simulations for moments between mid and late summer, as well as during the winter. During transitional seasons (fall and spring), the fluxes from the steady-state solution deviated significantly from the transient estimates with a tendency to underestimate at the beginning and to overestimate towards the end of those seasons. We conclude that fitting a simple analytical solution for 1D vertical heat transport to temperature data observed at particular well-selected times of the year can provide an inexpensive, simple method to obtain accurate point estimates of groundwater-surface water exchange in rivers and lakes.

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The use of temperature-time series measured in streambed sediments as input to coupled water flow and heat transport models has become standard when quantifying vertical groundwater-surface water exchange fluxes. We develop a novel methodology, called LPML, to estimate the parameters for 1-D water flow and heat transport by combining a local polynomial (LP) signal processing technique with a maximum likelihood (ML) estimator. The LP method is used to estimate the frequency response functions (FRFs) and their uncertainties between the streambed top and several locations within the streambed from measured temperature-time series data. Additionally, we obtain the analytical expression of the FRFs assuming a pure sinusoidal input. The estimated and analytical FRFs are used in an ML estimator to deduce vertical groundwater-surface water exchange flux and its uncertainty as well as information regarding model quality. The LPML method is tested and verified with the heat transport models STRIVE and VFLUX. We demonstrate that the LPML method can correctly reproduce a priori known fluxes and thermal conductivities and also show that the LPML method can estimate averaged and time-variable fluxes from periodic and nonperiodic temperature records. The LPML method allows for a fast computation of exchange fluxes as well as model and parameter uncertainties from many temperature sensors. Moreover, it can utilize a broad frequency spectrum beyond the diel signal commonly used for flux calculations.

We introduce LPMLE3, a new 1-D approach to quantify vertical water flow components at streambeds using temperature data collected in different depths. LPMLE3 solves the partial differential equation for coupled water flow and heat transport in the frequency domain. Unlike other 1-D approaches it does not assume a semi-infinite halfspace with the location of the lower boundary condition approaching infinity. Instead, it uses local upper and lower boundary conditions. As such, the streambed can be divided into finite subdomains bound at the top and bottom by a temperature-time series. Information from a third temperature sensor within each subdomain is then used for parameter estimation. LPMLE3 applies a low order local polynomial to separate periodic and transient parts (including the noise contributions) of a temperature-time series and calculates the frequency response of each subdomain to a known temperature input at the streambed top. A maximum-likelihood estimator is used to estimate the vertical component of water flow, thermal diffusivity, and their uncertainties for each streambed subdomain and provides information regarding model quality. We tested the method on synthetic temperature data generated with the numerical model STRIVE and demonstrate how the vertical flow component can be quantified for field data collected in a Belgian stream. We show that by using the results in additional analyses, nonvertical flow components could be identified and by making certain assumptions they could be quantified for each subdomain. LPMLE3 performed well on both simulated and field data and can be considered a valuable addition to the existing 1-D methods.

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