Multiscale Analysis and Computation for Flows in Heterogeneous Media  
                           (supported by DOE grant DE-FG02-06ER25727)

                                                     
    Thomas Y. Hou (Caltech), PI                                               Louis J. Durlofsky (Stanford), Co-PI

  Yalchin Efendiev (Texas A&M), Co-PI                                    Hamdi A. Tchelepi (Stanford), Co-PI

Abstract: The project is concerned with making fundamental advances in the development of multiscale methods for flow and transport in highly heterogeneous porous media. Numerical simulation of dynamic flow and transport processes in natural geologic formations is the primary computational tool for the management of natural resources, including oil reservoirs and water aquifers. However, the development of algorithms for modeling these dynamic processes in large-scale highly heterogeneous formations is quite challenging  because the properties of natural geologic porous formations (e.g., permeability) display high variability levels and complex spatial correlation structures, which span a rich hierarchy of length scales. Thus, it is usually necessary to resolve a wide range of length and time scales in order to obtain accurate predictions of the flow process under investigation.  In practice, however, some type of coarsening (or upscaling) of the detailed geologic model is usually performed before the model can be used to simulate complex displacement processes. Many approaches have been developed and applied successfully when the spatial variability of the subsurface properties (e.g., permeability) has scale separation. In this case, solutions of the local problems in a representative elementary volume can be used to represent the effects of the small scales on large-scale descriptions. The quality of these approaches deteriorates, however, if there is no apparent scale separation. The challenging problem is to develop a rigorous methodology that can capture the nonlocal effects accurately for heterogeneous systems with multiple scales where a scale separation assumption is not appropriate. The main thrust of this research is to develop a systematic multiscale analysis and efficient coarse-scale models that can capture the global effects. The emphasis is on problems without an apparent scale separation.

Reports:
   Report for Year 1

Report for Year 2



Publications:

1. An adaptive local-global multiscale finite volume element method for two-phase flow simulations, L.J. Durlofsky, Y. Efendiev, V. Ginting, Advances in Water Resources, 30 (2007), pp. 576-588

2. Multiscale simulations of porous media flows in flow-based coordinate system, Y. Efendiev, T. Hou, T. Strinopoulos, to appear in Computational Geoscience

3. Operator Based Multiscale Method for Compressible Flow, H. Zhou and  H. A. Tchelepi, SPE 106254, 2007 SPE Reservoir Simulation Symposium held in Houston, Texas, U.S.A., 26-28 February 2007. To appear in SPE Journal

4. Flow based oversampling technique for multiscale finite element methods, C.C. Chu, Y. Efendiev, V. Ginting, T. Hou, to appear in Advances in Water Resources

5. Multiscale finite element methods for stochastic porous media flow equations and application to uncertainty quantification, P. Dostert, Y. Efendiev, and T.Y. Hou, to appear in Computer Methods in Applied Mechanics and Engineering.

6. Adaptive multiscale computation of the nonlinear transport equation, H.A. Tchelepi, S.H. Lee and H.Zhou, in preparation, will be posted here soon (see the report for the description).

7. The accuracy of multiscale finite element methods for high-contrast elliptic interface problems by C.C. Chu, I. Graham and T.Y. Hou, in preparation, will be posted here soon (see the report for the description).