# Methods model flows

and assess uncertainty

*Posted February 9, 2009*

No one can be quite sure what goes on underground, and that’s largely why brothers Daniel and Alexandre Tartakovsky find the subterranean so fascinating.

Daniel is a professor at the University of California, San Diego, and Alexandre is a scientist at Pacific Northwest National Laboratory. Working with other researchers, they are developing mathematical methods to model subsurface flow and transport and to quantify predictive uncertainty.

Computer models of subsurface flow and transport are important to DOE – and to the United States. Protecting the environment, remediating contaminated sites, enhancing oil and gas recovery and geological carbon dioxide sequestration to slow climate change are just some of the applications requiring such models. The methods the researchers develop also could be used in a range of fields, including nuclear engineering, material science and cellular biology.

Subsurface flow and transport also are good subjects for developing and testing tools that quantify uncertainty in models’ predictions. “Many of the parameters are, in principle, not knowable,” Daniel says. The dearth of data and questions about the validity of mathematical conceptualizations of physical and biochemical subsurface processes contribute to uncertainty.

Uncertainty quantification is one of Daniel’s main interests, whereas Alexandre works more on multiscale methods to describe fluid flow and biochemical processes. The brothers followed the same path, however, to specialize in hydrology and applied mathematics and they collaborate frequently. See sidebar.

Tracking and predicting the migration of contaminants like strontium, neptunium or other contaminants as water carries them in the subsurface is a tricky proposition. “You have your subsurface environment and you are interested in some biological or chemical processes there,” Daniel says. “To model things with certainty one would, in principle, need to know the shape of every pore in a heterogeneous environment that consists of billions upon billions of pores of various sizes and shapes.”

Such information is never available, and even if it were, modeling processes in those pores would demand tremendous computer power and processor time. To make the problem tractable, researchers commonly average the pore-scale governing equations – including the advection-diffusion equation describing solute transport and the Navier-Stokes equation describing the flow of incompressible fluids – over a large volume of a porous medium.

This gives rise to continuum or macroscopic models that “work fine in most cases, except when they don’t,” Alexandre says. A case in point is reactive transport, which is described by a coupled system of nonlinear differential equations.