### THE VELOCITY ORIENTED APPROACH REVISITED

#### Abstract

Abstract. A great deal of hydrogeologtcal situations requires an extremely accurate calculation of the 3-dimensional groundwater discharge rates in the subsoil. Examples are: hydrology of wetlands, water balances of aquatic ecosystems depending on groundwater recharge, river-groundwater interaction, adveetive transport of pollution underneath waste disposal sites, particle trajectories in aquifer-aquitard systems with contrasting heterogeneities and many others. Numerical determination of the *vertical* groundwater velocity is a notoriously difficult problem. In nature this component may be two or three orders of magnitude smaller than the horizontal velocity components. In such cases application of Darcy's law to the numerically calculated hydraulic heads obtained from a finite difference or finite element model may lead to relatively inaccurate vertical velocities. More specifically, when estimating vertical velocity components in cases where the Dupuit approximation - negligible vertical head gradient - holds, numerical differentiation of hydraulic heads yields zero vertical velocity. In the 1980s of the last century Zijl and Nawalany proposed to invert the order of calculating the velocity field by eliminating the head from Darcy's law and to consider the Darcy velocity as the primary variable. For 2-dimensional flow this was already common practice and the challenge was a 3-dimensional extension, which was called the Velocity Oriented Approach (VOA). In two dimensions such methods were conventionally based on a stream function as primary variable. However, at that time application of a 3D stream function was not feasible and, therefore, the Darcy velocity itself was considered as the primary variable. This approach has been proven to yield a high accuracy for all three components of the specific discharge, including the relatively small vertical component, especially in cases where the subsoil is smoothly heterogeneous in the horizontal directions. In the 1990s the mixed-hybrid finite element method was developed. The physical interpretation of this method shows the way how to liberate the VOA from its smoothness requirement by introduction of a practical applicable 3D stream function. In conclusion, the velocity oriented approach indicates a change in paradigm regarding the accurate calculation of specific discharge tit groundwater flow.

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