Transient Water Age Distributions in Environmental Flow Systems: The Time-Marching Laplace Transform Solution Technique

TL;DR

This paper introduces a novel time-marching Laplace transform method for efficiently modeling transient water age distributions in complex environmental flow systems, addressing challenges in simulating time-dependent age transport.

Contribution

The paper presents a new algorithm combining Laplace Transform and time-marching schemes to solve transient age distribution problems in 3D flow systems, extending to exposure time analysis.

Findings

Validated with analytical solutions in 1D, 2D, and 3D domains.

Effective for large time periods and complex flow regimes.

Applicable to both steady-state and time-varying conditions.

Abstract

Environmental fluid circulations are very often characterized by analyzing the fate and behavior of natural and anthropogenic tracers. Among these tracers, age is taken as an ideal tracer which can yield interesting diagnoses, as for example the characterization of the mixing and renewal of water masses, of the fate and mixing of contaminants, or the calibration of hydro-dispersive parameters used by numerical models. Such diagnoses are of great interest in atmospheric and ocean circulation sciences, as well in surface and subsurface hydrology. The temporal evolution of groundwater age and its frequency distributions can display important changes as flow regimes vary due to natural change in climate and hydrologic conditions and/or human induced pressures on the resource to satisfy the water demand. Steady-state age frequency distributions can be modelled using standard numerical techniques, since the general balance equation describing age transport under steady-state flow conditions is exactly equivalent to a standard advection-dispersion equation. The time-dependent problem is however described by an extended transport operator that incorporates an additional coordinate for water age. The consequence is that numerical solutions can hardly be achieved, especially for real 3-D applications over large time periods of interest. A novel algorithm for solving the age distribution problem under time-varying flow regimes is presented and, for some specific configurations, extended to the problem of generalized component exposure time. The algorithm combines the Laplace Transform technique applied to the age (or exposure time) coordinate with standard time-marching schemes. The method is validated and illustrated using analytical and numerical solutions considering 1-D, 2-D and 3-D theoretical groundwater flow domains.