Numerical Scale-Up of Two-Phase Flows in Strongly Heterogeneous Media.

California Institute Of Technology, Pasadena CA

Investigators

Abstract

Subject: NSF proposal: DMS-0073916 Principal Investigator: Thomas Y. Hou Abstract: We propose to develop an innovative numerical coarse grid model for two-phase flows in strongly heterogeneous media. The development of this coarse grid model consists of two steps. The first one is to develop a PDE-based adaptive mesh strategy to capture the dominating flow features using an adaptive coarse grid mesh. Using the adaptive mesh, the dominating flow features such as high velocity flow and strong shear flow can be accurately captured. The second step is to use the multiscale finite element method to model the effect of small scale components. The main idea of our multiple scale finite element method consists of the construction of finite element base functions which contain the small scale information within each element. In fact, we construct the base functions from the leading order differential operator of the governing equation. An important feature of the multiscale finite element method is that they can be used to reconstruct locally the small scale velocity within each coarse grid block. Since the dominating flow structures are already well captured by the adaptive coarse grid solution, the remaining small scale components are relatively small in amplitude. And we can effectively model the effect of these small scales by incorporating averages of high order moments. This gives rise to a robust coarse grid model for two-phase flows in strongly heterogeneous media. Generalization of this idea to the Navier-Stokes equations will also be considered. Many problems of fundamental and practical importance contain multiple scale solutions. The direct numerical solution of the above multiple scale problems is difficult even with the advent of modern super computers. The major difficulty of direct solutions is due to the scale of computation. To get an accurate solution, all scales contained in the problem need to be resolved. Therefore, tremendous amount of computer memory and CPU time are required. The requirement can easily exceed the limit of today's computing resources. On the other hand, from an engineering perspective, it is often sufficient to predict the macroscopic properties of the physical systems and to capture the averaged effect of small (and random) scales on the large ones. Therefore, it is desirable to develop a method that can capture the small scale effect on the large scales using a relatively coarse grid. If this can be done, this can lead to enormous economy saving. The factor of saving could be as high as ten thousands. This would enable us to perform many simulations very efficiently. The proposed research is to develop such a coarse grid model by incorporating the small scale features in the underlying physical problem to construct multiscale building blocks (bases). These multiscale building blocks (bases) capture the effect of small scales on the large scales locally. In order to increase the robustness of the multiscale modeling, we propose to introduce an automatic (PDE-based) adaptive mesh generator to generate the underlying coarse grid. The adaptivity is controlled by the local flow rate, using a finer mesh in high flow region. The idea of combining adaptivity with multiscale modeling may have a significant impact on a number of applications which go beyond the scope of this proposed study.