## Deterministic models of groundwater age, life expectancy and transit time distributions in advective-dispersive systems

### Cornaton, Fabien ; Perrochet, Pierre (Dir.)

### Thèse de doctorat : Université de Neuchâtel : 2003 ; 1731.

The main objective of this dissertation consisted in the elaboration of a methodology to determine reservoir groundwater age, life expectancy, and transit time probability distributions in a deterministic manner, considering advective-dispersive transport in steady velocity fields. In the first section, it is shown that by modelling the statistical distribution of groundwater age at aquifer scale... More

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- The main objective of this dissertation consisted in the elaboration of a methodology to determine reservoir groundwater age, life expectancy, and transit time probability distributions in a deterministic manner, considering advective-dispersive transport in steady velocity fields. In the first section, it is shown that by modelling the statistical distribution of groundwater age at aquifer scale by means of the classical advection-dispersion equation (ADE) for a conservative and non-reactive tracer, associated to proper boundary conditions, the obtained function corresponds to the density of probability of the random variable age, defined as the time elapsed since the water particles entered the aquifer. In a second step, the evaluation of the life expectancy, being the time remaining before a water particle leaves the aquifer was derived from an adjoint backward model, yielding the life expectancy distribution. The convolution of these two distributions (age and life expectancy) is then shown to correspond to the groundwater total transit time distribution, from inlet to outlet, and is fully defined for the entire aquifer domain. From the ADEs simulating the full distributions of age and life expectancy, moment averaged equations are defined, like e.g. the well-known mean age equation. The mathematical models developed in the first section are illustrated by two-dimensional numerical experiments based on a scaled groundwater simulator model. In the second section, the focus is directed towards the reservoir theory (RT). An accurate and efficient method is presented to simulate the transit time distribution at discharge zones. It was shown that for systems with a known internal age probability distribution, the application of the RT to advective-dispersive aquifer systems allows full definition of the discharge zone transit time distribution. The RT can also be applied to internal life expectancy probabilities, yielding the recharge zone life expectancy distribution. One-, two-, and three-dimensional theoretical examples are presented to illustrate the application of the RT in advective-dispersive systems, and make inferences on the effect of boundary conditions, aquifer structure, and macro-dispersion on age, life expectancy and transit time distributions. Also, the particular case of vertically averaged forward and backward ADEs is developed. In the last section, the RT is extended to arbitrary aquifer configurations by subdividing the entire flow system into subsystems, treating each of them as a compartment. Transfer of water fluxes within these compartments from recharge zones to a particular discharge zone could then be considered isolated from any other subsystem. Nevertheless, the effects of mixing and interaction with other compartments and dispersion processes are considered in this approach. In this way, the RT was made applicable to any sub-drainage basin of an aquifer of arbitrary complexity. It was then found that the backward transport of the life expectancy to a specific outlet could predict the forward transport of a contaminant introduced anywhere in space. In other words, the concentration breakthrough curve at any particular outlet, which would result from the transport of a unit mass release at any point can be predicted with only one single realization of the life expectancy field. The usefulness of the elaborated method to deal with environmental settings such as the well-head vulnerability and protection problem, or also the problem of underground storage of high-level nuclear waste, is illustrated on twodimensional synthetic examples. Finally the work is concluded with a brief summary and with a critical view on the obtained results, as well as possible directions for future investigations