Analysis of Non-Linear Flows in Heterogeneous Porous Media and Applications

Texas Tech University, Lubbock TX

Investigators

Abstract

Ibragimov DMS-0908177 Almost all off-the-shelf industrial simulators of the process of filtration in porous media utilize the linear Darcy's approximation of the momentum equation. On the other hand, deviations from the assumed linear relationship between the pressure gradient and fluid velocity occur even at low Reynolds numbers, and in porous media containing fractures. Though these deviations are commonly attributed to inertial forces, the actual nature of this phenomenon is not adequately understood. The objective of this project is to arrive at a model that transforms current understanding and prediction of fluid flow in heterogeneous porous media. Approaches and methodologies from partial differential equations, computational fluid dynamics, and differential geometry are deployed to attain this goal. The models and methods involved include the following: a family of generalized nonlinear flow models, which spans existing models and forecasts new ones; a mathematical theory of the nonlinear Darcy equation where the permeability tensor depends on the gradient of pressure; a thorough study of the dynamics and robustness of the equations; analysis of solutions to Navier-Stokes-Forchheimer equations; applications of the methods of differential geometry to nonlinear hydrodynamic flows; a mathematical model for coupled flows in discontinuous fractured media; computational methods to investigate the stability and convergence of the multigrid algorithm for coupling nonlinear multiphase flows in highly irregular porous media. This project aims to better understand the complex phenomena of fluid filtration observed in nature or arising in modern technology and industry. The work provides a novel theoretical and numerical framework to explore a wide class of nonlinear flows in porous media. The methodology is based on modern research in nonlinear partial differential equations, real analysis and geometry, as well as new numerical methods in computational fluid dynamics. The mathematical models and methods are novel, and their applications are not limited to nonlinear fluid filtration in porous media. The research is integrated in educational programs spanning various special topics. The results of this project are disseminated not only through professional meetings and journal publications, but also through a network of collaborations and partnerships with industrial and reservoir engineering companies interested in employing the tools developed in the course of the project. On a larger scale, results of the project bear on the prediction and evaluation of complex bio-hydrodynamic processes in the lumen and arterial walls, transport of microorganisms in soil, new techniques for image processing, and management of the nation's groundwater and energy resources.