Anomalous diffusion in simulations of pumping tests on fractal lattices

Sellers, Shaun and Barker, John A. (2005) Anomalous diffusion in simulations of pumping tests on fractal lattices. In, Shaw, R.P. (ed.) Understanding the micro to macro behaviour of rock-fluid systems. (Geological Society, London, Special Publications, 249) London, UK. The Geological Society of London, pp. 79-90. (doi:10.1144/GSL.SP.2005.249.01.07).

Abstract

The recent interest in fractals in the geosciences literature has led to several proposed theoretical models for the hydraulic testing of fractured-rock systems that exhibit a fractal-like geometric structure. There is, however, no agreement on the correct form of the resulting model equations. In order to gain some insight into the range of possible behaviours to be expected from pumping tests on such systems, as well as the type of theoretical models needed, extensive simulations of pressure diffusion for transient groundwater flow, modelled by random walks on both deterministic and random fractal lattices were performed. For simplicity, the focus was on measurements of the random-walk dimension for generalized Sierpinski carpets, a proposed model for porous and fractured media. In addition to the expected anomalous slow down in diffusion in fractals as measured by the random-walk dimension, the simulations show further novel and unexpected anomalous behaviour due to the presence of internal boundaries at all scales. None of the proposed theoretical models for pumping tests on fractals appears consistent with all of the observed anomalous behaviours. The simulations suggest that interpretation of experimental pumping tests in terms of well-defined non-integer dimensions can be difficult, even when finite-size effects are negligible

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