Many of the natural materials that make up the Earth contain small pockets of space called pores, which make them permeable enough to hold and transport fluids such as water or petroleum. The pore size distribution, permeability and other characteristics or properties of these natural porous media are often not uniform, but rather can vary immensely throughout the area of the medium being studied. Fluid flow and contaminant transport in soils and aquifers are influenced significantly by these variations in the properties of natural porous media across a range of scales. The theory developed by subsurface hydrologists to account for these variations and its applications is the subject of a series of short articles in the December 2016 issue of Water Resources Research.
Broadly speaking, the practical consequences of the heterogeneity of porous media are two-fold. First, porous media properties inferred from laboratory-scale experiments on core samples 0.1-1m in size cannot be applied for predicting behavior at field scales. Second, field-scale model predictions that are not based on a complete characterization of medium property variations at an approximately 0.1-1m scale of resolution over the entire region of interest are bound to be uncertain. It is practically infeasible to characterize subsurface property variations at the required resolution.
In response to these challenges, stochastic approaches for predicting fluid flow and transport in aquifers and soils became relevant, and the field of stochastic subsurface hydrology became established. A stochastic process or random field, for those who are not familiar with the term, is a mathematical model for describing seemingly random variations of a variable of interest by designating a probability distribution to the values that the variable can take at any point and correlations between nearby values. In a stochastic approach, the coefficients in the partial differential equations describing groundwater flow and contaminant transport are treated as stochastic processes, which turns them into stochastic partial differential equations. When solved, these equations help to quantify the mean and probabilistic structure of dependent variables such as groundwater pressures and contaminant concentrations. There are at least five authoritative textbooks and over 3500 journal publications in the field of stochastic subsurface hydrology. However, despite 30 years of development, there has been very limited transfer of research findings to the practical enterprise of groundwater flow and transport modeling, which focuses on groundwater resource evaluation and management and on the prevention, containment, and cleanup of contamination.
A series of debate papers in Water Resources Research (“Debates–Stochastic subsurface hydrology from theory to practice”) addresses the reasons for the gap between research and practice in this field from various perspectives. Fiori et al.  and Sanchez-Vila and Fernandez Garcia  explore and synthesize key developments in this field and evaluate their successes and limitations. Cirpka and Valocchi  address the relevance and limitations of of these developments in the context of contaminant transport and remediation. Fogg and Zhang  provide a geologic perspective on the origins of heterogeneity and associated implications for the high degree of heterogeneity that may be expected in typical groundwater aquifers. Together, the papers highlight the important need for improved, geologically realistic descriptions of heterogeneity, and the need to acknowledge the high degree of heterogeneity that may be expected in typical groundwater aquifers at scales relevant to practical problems. Many of the early theoretical efforts in stochastic subsurface hydrology assumed a relatively low degree of heterogeneity (to facilitate simple and elegant theoretical results), greatly limiting their application to practical problems.
The authors suggest that flexible and user-friendly stochastic subsurface hydrology toolkits that incorporate site-specific measurements of porous medium properties, groundwater pressure, and contaminant concentrations (in other words, toolkits that use stochastic models conditioned on site-specific measurements) may hold the key to closing the gap between theory and practice. Community efforts like those that have been successful in other earth sciences, which leverage the tremendous advances in subsurface characterization and computational technologies over the last decade, will help to realize the vision emerging from this Debate series.
Groundwater is estimated to supply drinking water to more than 2 billion people and over 40% of irrigation water demand worldwide. It is by far the largest easily-accessible freshwater resource and accounts for 99% of the rural water supply in the United States. As this precious resource becomes more and more stressed, there will be an increased need for improved and more precise approaches to its management and protection. Flexible, practically-applicable and user-friendly toolkits for dealing with heterogeneity and uncertainty are highly relevant in this context.
—Harihar Rajaram, University of Colorado, Boulder; email: [email protected]