Uncertainty Estimation and Modeling of Porous Media FlowsJune 10, 2005
Figure 1. Viscous fingering in a porous medium. Low-viscosity gas (at left) is injected into higher-viscosity oil (right). The displacement is unstable, and gas fingers through the oil, reducing the efficiency of recovery.
By Michael A. Christie
Large-scale computer-based simulations are used increasingly to predict the behavior of complex systems. Prime examples include weather forecasting, global climate modeling, the performance of nuclear weapons, and fluid flow through an oil reservoir. Simulations invariably involve theory, experimental data, and numerical modeling, all with their attendant errors. It is thus natural to ask, Are the simulations believable? How are the accuracy and reliability of the results to be assessed?
In the oil industry, lack of accuracy in predictions can be traced to a variety of sources. Firstly, the properties of the rocks through which the oil flows are generally unknown and have to be inferred from local measurements at wells and from measurements of analogous reservoir systems. Secondly, the equations are usually solved at a relatively coarse scale, with significant amounts of subgrid detail ignored, leading to uncertainties in predictions. Finally, the equations solved contain a number of empirical relationships, leading to uncertainties arising from the lack of a complete physically based model.
Figure 1 shows an image of viscous fingering in an oil reservoir. A hydrodynamic instability analogous to the Rayleigh–Taylor instability, viscous fingering occurs as low-viscosity gas channels through the higher-viscosity oil, resulting in lower oil production rates. Overall oil recovery is affected by both the small-scale behaviour of the instability and the larger-scale variations in the rock permeability throughout the reservoir. In most operational simulations, the details of the gas fingering are below the grid resolution and, as such, are dealt with in a relatively ad hoc manner.
The talk featured details of a statistical approach for capturing the errors introduced by ignoring subgrid flow phenomena. The approach relies on a model of the changes in mean solution error and variance with parameters specifying the evolution of large-scale flow features. This statistical error model is then linked to a Bayesian framework for quantifying uncertainty in estimated parameters. In this case, use of this approach resulted in a robust and accurate estimation of physical parameters via coarse-grid or effective models, and the ability to predict the uncertainties arising from incomplete knowledge at a level of accuracy close to that obtainable with a fine grid. A goal of current work is to extend the ideas to more realistic cases with a larger number of parameters describing the physical system.
The ideas presented in the talk seem to have much in common with other areas of science. A recent article that provides more detail and describes the relationship of the ideas presented here to similar ideas in gas dynamics can be found in .
 Michael A. Christie, James Glimm, John W. Grove, David M. Higdon, David H. Sharp, and Merri M. Wood-Schultz, Error analysis and simulations of complex phenomena, Los Alamos Sci., 29 (2005), 6–25; available online from http://library.lanl.gov/cgi-bin/getfile?01057075.pdf.
Michael A. Christie, a professor of reservoir engineering at Heriot–Watt University, was an invited speaker at the conference ("Accurate Estimation of Uncertainty Bounds for Porous Media Flows").