Dynamical systems describe the time evolution of moving spatial points under the influence of a smooth, bounded vector field. The theory of these systems is focused on the global properties of their flow paths, not on their integration, and so gives general, qualitative information that does not depend on the details of the spatial variability of the vector field. This approach was applied to describe the global consequences of the Darcy law for steady groundwater flows in isotropic, heterogeneous aquifers. No particular model of the spatial variability of the hydraulic conductivity (K) was assumed. Vorticity in the flow paths was shown to exist wherever isoconductivity and equipotential surfaces intersect transversely, and the importance of the Lamb vector (the vector product of vorticity and specific discharge) for the geometry of flow paths was established. Because steady groundwater flows governed by the Darcy law have zero helicity, they cannot exhibit tangled vorticity lines or become chaotic. The absence of chaos is related closely to the impossibility of closed flow paths, the asymptotic stability of isolated minima of the hydraulic head (H), and the existence of a function ℋ(K, H) on whose level surfaces all flow paths are confined. This last function also permits groundwater flows to be represented by moving points in the K, H plane, with motion there generated by a form of Hamilton's equations. The results obtained are not related to any stochastic approach to aquifer spatial variability, but instead may be applied to constrain stochastic models on purely dynamical grounds.
Water Resources Research – Wiley
Published: Aug 1, 1994
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