A closed-form solution for pre-asymptotic dispersion with transient velocity

Advective-dispersive transport in natural systems is often modeled assuming steady one-dimensional flow and complete cross-sectional mixing, the latter allowing the use of a constant dispersion coefficient. In reality, however, many environments exhibit pre-asymptotic dispersion (PAD), where the effective dispersion coefficient increases as solutes progressively sample velocity heterogeneity. Time- and/or distance-dependent dispersion coefficients have historically been used to address PAD but can produce inconsistencies when solute pulses interact, particularly in nonlinear systems where superposition fails. Here we generalize our recent closed-form solution for one-dimensional advective-dispersive transport with PAD to the case of transient velocity. This new solution involves expressing dispersion as a function of both solute age in the flow (residence time), and physical time. These solutions apply to both initial and boundary value problems over an infinite domain, including cases with first-order reactions. We validate our boundary value solution by reproducing breakthrough curves from turbulent, accelerating pipe flow experiments. Our results show that a solute’s “memory” of reduced early-time dispersion continues to influence its transport even after complete mixing is achieved. More broadly, this framework fills a critical modeling gap by enabling physically consistent analysis of solute transport under PAD and transient flow—conditions common in natural systems but rarely studied together. • Closed-form solution for transport with transient flow and pre-asymptotic dispersion. • Solutions apply for both initial and boundary value transport problems. • Solutions account for the residence time of solute in the flow. • We demonstrate solution applicability to published data in accelerating pipe flow.