Abstract
Accurate estimation of three-dimensional (3D) groundwater velocity vectors is critical for interpreting subsurface transport, groundwater-surface water exchange, and hyporheic residence times. Heat-pulse methods provide a promising approach to estimating groundwater flow because advective transport distorts the thermal plume around a known heat source, allowing flow magnitude and direction to be inferred from surrounding temperature responses. Existing field applications of heat-pulse flowmeters have commonly relied on temperature-difference or sensor-axis approaches to infer 2D flow direction, whereas temperature-ratio methods have been developed primarily for one-dimensional soil-water flux or sap-flow applications. Their extension to 3D groundwater velocity estimation remains limited. Here, we extend heat-pulse temperature-ratio theory from one-dimensional flux estimation to three-dimensional groundwater velocity-vector estimation. We derive a decoupled linear/ratio-based inversion for a proposed 16-thermistor geometry and test its performance against synthetic breakthrough curves generated from a finite-duration point-source heat-transport solution. For the tested geometry and thermal-property assumptions, the method recovered imposed 3D velocity vectors accurately under low-flow conditions, with reliable performance up to approximately 5 m/day. Above this range, errors increased because advective plume distortion and early-time ratio instability produced underestimation of velocity magnitude and increasing errors in the vertical component. A realistic vertical thermistor misalignment of 1.59 mm introduced additional geometric contributions to the log-ratio response, highlighting the sensitivity of ratio-based inversions to sensor geometry. A multistart nonlinear optimization solver substantially reduced errors under high-velocity and misaligned conditions, but at much greater computational cost. These results define the validity range of a simplified 3D heat-pulse ratio method and provide a practical decision framework for analyzing heat-pulse data from probes deployed in monitoring wells or directly within sediments.