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Flow of water in sub-surface geologic environments takes place through an array of interconnected pore spaces, the spatial distribution of which eludes us. The presence of complex variations in water pathways and geologic characteristics produces large spatio-temporal fluctuations in the values of the variables describing water flow and/or transport of contaminants. The situation is accentuated by the small number of measurements that are usually available in groundwater studies, the dependence of parameter values onto instruments' resolution, measurement and interpretation errors during sampling procedures, and model uncertainties in the description of a hydrologic system. Traditionally analyses of groundwater problems relied on solution of partial differential equations that required precise definition of parameters, water sources or sinks, and conditions at the boundaries of the hydrogeologic system. However, and in addition to the hydrogeologic parameters' variation, one typically has to estimate water inputs or outputs to aquifers from agricultural activities, leakage from lagoons, lakes, or other sources of surficial water in an area, and to a large extent hypothesize the hydraulic conditions at the boundaries of aquifers. This combination of spatio-temporal variation, measurement error, and uncertainty has led in the last 30 years to the development of stochastic sub-surface hydrology.