Research works on the coupling of incompressible surface flow with subsurface porous media flow arouse increasing interest recently. The coupled problem is a typical multi-domain problem with multi-physics. In-depth understanding of this problem requires both modeling process and numerical study. In this work, some existing surface flow models and subsurface flow models are reviewed; the interaction mechanisms of surface flow with subsurface porous media flow are discussed; numerical algorithms for solving coupled surface/subsurface flow models are proposed; in particular, preconditioning techniques and two grid algorithms are mathematically and numerically investigated. In Chapter 1, we present some existing models for describing surface fluid flow motion as well as those for subsurface porous media flow motion. Moreover, we study some coupled models including both surface fluid flow and subsurface porous media flow. The interactions of fluid flow and porous media flow are reflected by transmission conditions at the interface between surface and subsurface. Other related multimodels for incompressible flows will also be introduced. In Chapter 2, we focus on preconditioning techniques for the coupled Stokes/Darcy model, a linear model for the surface/subsurface flows coupling. Several decoupled preconditioners are proposed and analyzed. Especially, the convergence rate of GMRES method with these preconditioners is shown to be independent of meshsize. For improving the robustness with respect to physical parameters, coupled preconditioners are also theoretically and numerically investigated. In Chapter 3, a two grid algorithm for decoupling the coupled Stokes/Darcy model is studied. The two grid algorithm consists of solving a coupled coarse grid problem, then solving two sub-problems in parallel. We use coarse grid solution to supplement boundary conditions at the interface for fine grid subproblems. Theoretical analysis shows that the two grid algorithm retains opt.

Subsurface flow problems are inherently multiscale in space due to the large variability of material properties and in time due to the coupling of many different physical processes, such as advection, diffusion, reaction and phase exchange. Subsurface flow models still need considerable development. For example, nonequilibrium effects, entrapped air, anomalous dispersion and hysteresis effects can still not be adequately described. Moreover, parameters of the models are diffcult to access and often uncertain. Computational issues in subsurface flows include the treatment of strong heterogeneities and anisotropies in the models, the effcient solution of transport-reaction problems with many species, treatment of multiphase-multicomponent flows and the coupling of subsurface flow models to surface flow models given by shallow water or Stokes equations. With respect to energy and the environment, in particular the modelling and simulation of radioactive waste management and sequestration of CO2 underground have gained high interest in the community in recent years. Both applications provide unique challenges ranging from modelling of clay materials to treating very large scale models with high-performance computing. This book brings together key numerical mathematicians whose interest is in the analysis and computation of multiscale subsurface flow and practitioners from engineering and industry whose interest is in the applications of these core problems.

The primary interest of this work is in the area of flow through permeable media in the context of the subsurface. This area receives growing interest in the scientific community due to the wide range of applications that rely on an in-depth understanding of subsurface systems. The multi-scale and multi-physics nature of flow through permeable media pose significant challenges for modeling and simulation as scientific tools to understand such processes. The aim is to propose predictive mathematical models motivated by physical observations or high-fidelity simulations, and to develop reliable and robust numerical framework to predict the dynamics of flow across different scales. The first part targets discretization schemes for simulation of fractured reservoirs. A discrete fracture model is implemented using an unstructured Voronoi mesh that can capture complex fracture geometries. The mesh generation algorithm is extended to enable modeling complex fractures in fully three-dimensional space. The proposed approach is validated against results obtained from a commercial simulator. The second part develops a high-resolution numerical framework for immiscible two-phase flow, which helps to exploit the interplay of nonlinearity and heterogeneity, as well as nonequilibrium models on the behavior of fluid mixing. A generalized analytical scaling relation is derived for two-phase flow subject to self-similar heterogeneity. Moreover, we proposed a novel, self-consistent, and physics-based mathematical formulation for multiphase flow. The proposed model generates evolving fronts that match experimental observations. The third part investigates rarefied gas flow using the lattice Boltzmann method. New optimal values for slip coefficients are proposed to capture slip velocities. Results are validated against molecular dynamics simulation reported in the literature. To capture the confined phase behavior of methane, we developed a modified extension of Peng-Robinson equation of state. The proposed equation is coupled with lattice Boltzmann method to investigate transport of methane in slit nanopores. Result indicates that pressure plays an important role in determining the transport characteristics in confined systems.

This volume certainly is a Conference Proceedings, the Proceedings of the NATO Advanced Research Workshop (ARW) on "Unsaturated Flow in Hydrologic Modeling" held at "Les Villages du Soleil" near ArIes, France from June 13 to 17, 1988. Let me therefore acknowledge properly, at the very beginning, the gratitude of all the participants to the NATO Science Committee for its generous support and worthwhile goal of bringing together scientists of many countries to communicate and share their experiences. Particular thanks are extended to the director of the program, Dr. Luis Vega da Cunha for his interest and understanding. On the other hand this volume is also, and probably more so, a Textbook that fills a gap in the field of unsaturated flow. Many treatises on the subject present the theory in its different aspects. Hardly any explain in details how the different pieces can be put together to address realistic problems at the basin scale. The various invited contributions to the ARW were structured in a subject progression much as chapters are organized in a book. The intent of the ARW was to assess the current state of knowledge in "Unsaturated Flow" and its use in "Hydrologic Modeling Practice". In a sense the interest in fundamentals of unsaturated flow in this ARW was not just for the sake of knowledge but also and primarily for the sake of action. Can such fundamental knowledge be utilized for better management of the water resource? was the basic question.

Free-Surface Flow: Computational Methods presents a detailed analysis of numerical schemes for shallow-water waves. It includes practical applications for the numerical simulation of flow and transport in rivers and estuaries, the dam-break problem and overland flow. Closure models for turbulence, such as Reynolds-Averaged Navier-Stokes and Large Eddy Simulation are presented, coupling the aforementioned surface tracking techniques with environmental fluid dynamics. While many computer programs can solve the partial differential equations describing the dynamics of fluids, many are not capable of including free surfaces in their simulations.

Computational Methods in Subsurface Flow explores the application of all of the commonly encountered computational methods to subsurface problems. Among the problems considered in this book are groundwater flow and contaminant transport; moisture movement in variably saturated soils; land subsidence and similar flow and deformation processes in soil and rock mechanics; and oil and geothermal reservoir engineering. This book is organized into 10 chapters and begins with an introduction to partial differential and various solution approaches used in subsurface flow. The discussion then shifts to the fundamental theory of the finite element method, with emphasis on the Galerkin finite element method and how it can be used to solve a wide range of subsurface problems. The subjects treated range from simple problems of saturated groundwater flow to more complex ones of moisture movement and multiphase flow in petroleum reservoirs. The chapters that follow focus on fluid flow and mechanical deformation of conventional and fractured porous media; point and subdomain collocation techniques and the boundary element technique; and the applications of finite difference techniques to single- and multiphase flow and solute transport. The final chapter is devoted to other alternative numerical methods that are based on combinations of the standard finite difference approach and classical mathematics. This book is intended for senior undergraduate and graduate students in geoscience and engineering, as well as for professional groundwater hydrologists, engineers, and research scientists who want to solve or model subsurface problems using numerical techniques.

This tutorial provides the application of the coupling interface OGS#IPhreeqc (open-source scientific software) to model reactive mass transport processes in environmental subsurface systems. It contains general information regarding reactive transport modeling and step-by-step model set-up with OGS#IPhreeqc and related components such as GINA and ParaView. Benchmark examples (1D to 2D) are presented in detail. The book is intended primarily for graduate students and applied scientists who deal with reactive transport modeling. It also gives valuable information to the professional geoscientists wishing to advance their knowledge in numerical simulation, with the focus on the fate and transport of nitrate. It is the third volume in a series that represents the further application of computational modeling in hydrological science.