Over past decades, analytical solutions have been a popular mathematical tool for estimating hydrologic properties and predicting processes in the subsurface. Mathematical convenience, computational limitations, data scarcity, and practical concerns for field- and basin-scale problems are the main reasons for their popularity. The analytical approaches, however, adopt simplifying assumptions such as subsurface homogeneity and depth-averaged responses. Although geologic heterogeneity at various scales and 3-D nature of flow and transport are well-known realities, these approaches have been widely applied to many environmental management and engineering problems. As a result, estimation of effective properties for the simplified subsurface became a popular research topic, as did uncertainties associated with the predictions using the effective properties. Various stochastic and upscaling approaches thereby have emerged. Theories and formulas for field-scale effective properties were developed as well as for the uncertainty associated with the predictions using the effective properties. These theories and formulas relate the effective properties to the spatial statistics (mean, variance, and correlation structure) of the properties estimated at smaller scales (e.g., core sample, slug tests, single-hole tests, and flow meter tests).

After two decades, we have learned that effective hydraulic parameters are merely by-products of neglecting small-scale heterogeneity in our analyses. Resultant ensemble-average behaviors of a complex system are practical and useful in general application, but they rarely meet the resolution of our interests. Local-scale mixing processes (velocity variation) are too weak to make the ensemble-average behaviors equivalent to those observed at the scale of our interest. After all, high-resolution characterization, monitoring, and prediction are the key to advance our science and technology.