Townley, L.R., and Turner, J.V. (2024, submitted), Consistent solutions for simultaneous dynamic water, solute, stable isotope and radon balances to estimate steady or average exchange fluxes between surface water bodies and underlying groundwater, Journal of Hydrology. Submitted on 31 March 2024 for publication in a special issue on surface water - groundwater interaction.

Many papers show that water, conservative solute, stable isotope and radon balances can be applied to the study of surface water – groundwater interaction. When undertaken simultaneously, such analysis provides an even more powerful method for estimating groundwater inflows and outflows, i.e. the exchange fluxes that for most people are hardest to estimate. The overarching objective of this paper is to provide practitioners with a consistent conceptual and simulation modelling framework with which to achieve a rapid initial understanding of surface water – groundwater interaction, as well as to guide the efficient and targeted acquisition of field data. This paper makes several important contributions. First, after an introduction that explains the complexities of groundwater flow patterns near surface water bodies, a set of general balance equations is developed using consistent terminology and notation. Second, after focusing on a simple conceptual model with steady groundwater inflow I, groundwater outflow O, precipitation P and ever-present evaporation E, analytical solutions are developed for transient and steady state conditions: one water balance solution is found to apply in all situations; solutions for the solute balance equation are developed for 23 different subcases; a transient balance equation for stable isotopes written terms of 1+\delta is found to apply in all situations; and a new transient solution for radon is found in terms of the upper incomplete gamma function. On the path towards developing the isotope balance equation, we present a rigorous approach, based on coupled nonlinear balance equations for abundant and rare isotope pairs (1H and 2H, then 16O and 18O), using isotope ratios and the fractions of each isotope. Third, we describe a Lake Water Balance Calculator (LWBC) written using GoldSim simulation software, a tool to help others to conduct the simultaneous analyses that we promote; we use GoldSim’s numerical methods to solve general balance equations, we implement the analytical solutions to demonstrate that both GoldSim’s methods and the analytical solutions are correct, and we also show that the 1+\delta approximation is equivalent to the rigorous coupled equations, at least for heavy stable isotopes with very low abundance. Fourth, we describe field studies: we provide three examples of applications of LWBC to short-term studies of Perry Lakes East, Lake Jasper and Georgetown Billabong in Australia; we then describe WSIBal, a simulation model based on the same balance equations and describe its application to Lake Jasper over a 19-year period; finally, we describe R-WSIBal, which evolved from WSIBal, and explain its application to 2500 km of the Murray River in Australia.