Some current methods to represent the heterogeneity of natural media in hydrogeology

We have known for a long time that the material properties of the subsurface are highly variable in space. We have learned that this variability is due to the extreme complexity and variation with time of processes responsible for the formation of the earth's crust, from plate tectonics to erosion, sediment transport, and deposition, as well as to mechanical, climatic, and diagenetic effects. As geologists, we learned how to "read" this complex history in the rocks and how to try to extrapolate in space what we have understood. As physicists, we then learned that to study flow processes in such media we must apply the laws of continuum mechanics. As mathematicians using analytical methods, we learned that we must simplify by dividing this complex continuum into a small number of units, such as aquifers and aquitards, and describe their properties by (constant) equivalent values. In recent years, as numerical modelers, we learned that we now have the freedom to "discretize" this complex reality and describe it as an ensemble of small homogeneous boxes of continuous media, each of which can have different properties. How do we use this freedom? Is there a need for it? If the answer is "yes," how can we assign different rock-property values to thousands or even millions of such little boxes in our models, to best represent reality, and include confidence levels for each selected rock property? As a tribute to Professor Eugene S. Simpson, with whom the first author of this paper often discussed these questions, we present an overview of three techniques that focus on one property, the rock permeability. We explain the motivation for describing spatial variability and illustrate how to do so by the geostatistical method, the Boolean method, and the genetic method. We discuss their advantages and disadvantages and indicate their present state of development. This is an active field of research and space is limited, so the review is certain to be incomplete, but we hope that it will encourage the development of new ideas and approaches.