Computational Free-Surface Flows Computational Techniques for Nonlinear Seepage Flow in Porous Media with Free and Moving Boundaries

Doctoral thesis, 2002

Motion of a viscous fluid interface is encountered in many physical processes in engineering applications. The accurate representation and efficient numerical solution of the fluid interface are crucial problems in simulations of the physical processes in these applications. To this end, the present study is focused on modeling two-phase flow problems in porous media. Fixed-mesh techniques are used in the numerical analysis of the motion of the two-fluid interfaces in porous media. The numerical solutions of the interface for qualitative two-dimensional (2D) problems are compared with available analytical solutions of the linear and nonlinear flow equations. As the interface motion can be computed by techniques based on either moving-mesh or fixed-mesh perspectives, the level set method (LSM) in an Eulerian view establishes the interface-capturing technique (ICT) on a fixed mesh domain. The ICT is integrated in the finite element (FE) approximation for the (2D) and three-dimensional (3D) Navier-Stokes (N-S) equations. However, sharp physical interfaces that separate different fluids often introduce singularities to the simulation of these problems. The effect of the singularities arises in the form of discontinuous density and viscosity functions across the interface. These quantities are more naturally represented numerically in the FE field where a weak form of the equations is used. The integrals in the weak formulation include the discontinuous density and viscosity functions. Accordingly, the study has also focused on the numerical instabilities of the LSM [initial value problem, IVP] in a FE approximation. The presentation includes a brief analysis for the integration of the IVP for stationary motion of two-fluid interfaces in porous-media flow. A similar analysis is conducted for transient motion of the interfaces in porous media in 2D and 3D. The approximations have also included the implementations of different time-integration methods where the stability and the time of the computation have been considered in the analysis. The numerical analysis has yielded results that include test problems and comparison of different solver techniques for various free and moving boundaries problems in porous media flow. Simulations of stationary phreatic-surface flows in dams are presented, using the initial value problem of the LSM and a stationary modified version of the LSM. Comparison with the analytical solutions for a qualitative and quantitative flow problem has shown that the numerical solutions have better representations of the position and the shape of the phreatic surface. Simulations of transient motion of the phreatic surfaces in real-time applications are also discussed, using the regularized IVP. The implementations in an object-oriented programming (OOP) are examined as well. The implementations show that the extension from lower-dimension to higher-dimension problems is straightforward. Furthermore, the ICT has proved a robust and simple tool for computing problems with two-fluid interfaces. The thesis shows that this technique is also applicable to various two-phase flow problems in Computational Fluid Dynamics (CFD).