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As conduit water carries pollutants introduced from a point recharge downgradient, a contaminant-free seepage from the surrounding limestone is added into the conduit water and actively dilutes the pollutants. In this study, the transport with advection and this active dilution but no dispersion, is solved using the method of characteristics. The complete solution considering initial condition, boundary condition, and production is presented. Scale analysis reveals that the model is applicable when injection of pollutants at sinkholes persists long enough but unsuitable for analysis of dye tracing experiments. An approach combining the model with dye tracing experiment is used to quickly estimate the geometry and flow of a major conduit from Ames Sink to Wakulla Spring, Northwest Florida.