Introduction One can argue that the nature of heterogeneities in an underground formation is the most influential factor limiting the success of mathematical models of flow or transport in the aquifer. Lack of adequate knowledge of aquifer heterogeneity and the attendant difficulty in assessing the realism of a model's predictions make the modeler's job a frustrating one.
The importance of heterogeneity elicits discomfort among many of us whose research concerns new numerical techniques for groundwater modeling. Part of the uneasiness over heterogeneity arises from a widely shared view of its importance. The valid premise of this view is that difficulties in accurately quantifying underground heterogeneity impose constraints on the accuracy of mathematical models, owing to limitations in the quality of the input data. The argument then proceeds as follows: Since poorly quantified heterogeneity is the dominant source of prediction error in most groundwater models, there is little point in focusing research on improved numerical techniques. After all, even if we use more accurate numerics, the deleterious effects of inadequate input data will still be present, swamping any improvements to be gained through mathematical refinements. The natural conclusion is that research into methods for detecting and characterizing underground heterogeneity have much more potential for improving mathematical models than does research into the numerical techniques themselves.
We offer a different perspective. No one would deny that improved methods for quantifying heterogeneity are crucial to advances in the realism and utility of groundwater models, in accordance with the popular maxim, "garbage in, garbage out." However, we contend that the most commonly used mathematical methods are inadequate to model heterogeneous aquifers. As we review in Section 3, even in the ideal case when the heterogeneities are "perfectly" known, standard methods can perform poorly, suggesting a new adage: "heterogeneity in, garbage out."
In a more realistic scenario, where one relies on detailed statistical characterizations of heterogeneous aquifers, existing mathematical techniques are largely inadequate at answering the hydrologist's questions. Here, quantifiable control over the uncertainties in aquifer parameters can fail to yield reasonable control over the reliability of the numerical solution. We illustrate this problem in Section 4. This observation suggests the even more distressing adage: "statistics in, garbage out."
In what follows, we examine these notions and briefly indicate promising avenues for overcoming the difficulties. We hope to affirm the importance of continued research into mathematical techniques used in numerical models of groundwater flow and transport, thereby rebutting the conclusions of the conventional wisdom.
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