WHY MODEL SUBSURFACE SOLUTE TRANSPORT STOCHASTICALLY?

Subsurface solute transport has been traditionally described by a deterministic advection-dispersion equation based on analogy to Fick's laws of diffusion. According to this analogy, the spread of a nonreactive tracer in a hydrogeologic environment is controlled by a constant directional medium property called dispersivity which, when multiplied by absolute velocity, yields a directional dispersion coefficient. Field tracer experiments raise questions about the validity of the traditional deterministic transport model by suggesting that dispersivity varies with tracer mean travel distance and time. To date, the only successful way to explain and reproduce such anomalous non-Fickian behavior theoretically has been to treat the medium as being randomly heterogeneous and the transport equation as being stochastic. This motivates an examination of the traditional deterministic transport model from a stochastic viewpoint. We ask: What is the meaning of velocities and dispersion coefficients entering into the traditional model when the latter is applied to a randomly heterogeneous medium of the kind ubiquitously encountered in nature? What is obtained upon calibrating such a model against randomly fluctuating tracer data as is commonly done in practice? What is the utility of traditional tracer tests in light of our answers to these questions? We propose that applying the traditional model to heterogeneous media is valid at best as an approximation, provided further that the model is interpreted in a nontraditional manner and its variables (velocity, dispersion coefficient, solute concentration and mass flux) are recognized to be inherently nonunique by depending on information, scale and the prevailing flow regime. Not only does the stochastic approach explain field observations that might otherwise remain puzzling, but it also provides a way to quantify predictive uncertainty in a way that a deterministic model necessarily cannot.