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FEFLOW is an acronym of Finite Element subsurface FLOW simulation system and solves the governing flow, mass and heat transport equations in porous and fractured media by a multidimensional finite element method for complex geometric and parametric situations including variable fluid density, variable saturation, free surface(s), multispecies reaction kinetics, non-isothermal flow and multidiffusive effects. FEFLOW comprises theoretical work, modeling experiences and simulation practice from a period of about 40 years. In this light, the main objective of the present book is to share this achieved level of modeling with all required details of the physical and numerical background with the reader. The book is intended to put advanced theoretical and numerical methods into the hands of modeling practitioners and scientists. It starts with a more general theory for all relevant flow and transport phenomena on the basis of the continuum approach, systematically develops the basic framework for important classes of problems (e.g., multiphase/multispecies non-isothermal flow and transport phenomena, discrete features, aquifer-averaged equations, geothermal processes), introduces finite-element techniques for solving the basic balance equations, in detail discusses advanced numerical algorithms for the resulting nonlinear and linear problems and completes with a number of benchmarks, applications and exercises to illustrate the different types of problems and ways to tackle them successfully (e.g., flow and seepage problems, unsaturated-saturated flow, advective-diffusion transport, saltwater intrusion, geothermal and thermohaline flow).

Inhaltsverzeichnis

Chapter 1. Introduction

Abstract
Flow, mass and heat transport through porous and fractured media occurs in many branches of engineering and science. Of particular concern are those processes in the subsurface occurring beneath the surface of the earth’s ground, that means flow and transport in geologic media with their complexity and uncertainty.
Hans-Jörg G. Diersch

Chapter 2. Preliminaries

Abstract
The purpose of this chapter is to introduce basic definitions, mathematical conventions, expressions and fundamental principles.
Hans-Jörg G. Diersch

Chapter 3. Porous Medium

Abstract
The processes of flow, mass and heat refer to extensive quantities (such as mass, momentum, energy and entropy), cf. Sect. 2.2.2, which are transported through a spatial domain of interest. This spatial domain is said to behave as a continuum which is occupied by matter for which a continuous distribution can be postulated. The matter may take a number of M aggregate forms or phases α, particularly: solid s, liquid l and gaseous g. It retains their continuity regardless how small volume elements the matter is subdivided in and interior material interfaces or surfaces exist. Any mathematical point we select can be assigned to matter as a physical point of given finite size.
Hans-Jörg G. Diersch

Chapter 4. Discrete Feature

Abstract
Essential flow and transport equations are derived for discrete features, which are separated from the porous-medium approach. Discrete features are very useful to model flow and transport processes in fractures, conduits, channels, faults, boreholes and many other macroscopic geometric representations. Typically, diffusion-type flow conditions are assumed in those discrete features. The developments are summarized in Tables 4.5, 4.6 and 4.7 for flow, mass and heat transport, respectively, in discrete features.
Hans-Jörg G. Diersch

Chapter 5. Chemical Reaction

Abstract
The quantities r k α , R k α , R k , $$\tilde{R}_{k}$$, $$\bar{R}_{k}$$ or $$\bar{\tilde{R}}_{k}$$ that appear in the species mass transport equations (3.​50), (3.​51), (3.​248) and (4.​71) and those of Tables 3.5, 3.7, 3.9 3.11 and 4.6 represent rates of production of mass of chemical species k due to chemical reactions occurring within a phase α, termed as homogeneous reactions, or between two or more phases, termed as heterogeneous reactions.
Hans-Jörg G. Diersch

Chapter 6. Initial, Boundary and Constraint Conditions

Abstract
Initial, boundary and constraint conditions are thoroughly discussed for flow, mass and heat transport. Required special formulations of boundary conditions refer to free-surface, seepage-face, surface ponding, integral, gradient-type, multi-layer well and outflow conditions. It is shown that a Neumann-type boundary condition of the divergence form of a transport equation is equivalent to a Cauchy-type boundary condition of its convective form, which easily allows to impose load conditions for mass and heat.
Hans-Jörg G. Diersch

Chapter 7. Anisotropy

Abstract
Anisotropy is described in full three dimensions and two dimensions. Important special cases are developed for the shape-derived 3D anisotropy and axis-parallel anisotropy.
Hans-Jörg G. Diersch

Chapter 8. Fundamental Concepts of Finite Element Method (FEM)

Abstract
In this chapter the basic principles of FEM are systematically developed and reviewed for prototypical advection-dispersion equations (ADE’s). The different spatial and temporal discretization techniques are addressed. The important approximate solutions for the divergence and convective forms of ADE are carefully developed. Emphasis is given on adaptive solution strategies. Implicit and explicit time integration methods are reviewed and compared. It clearly shows the superiority of implicit strategies for the present problem classes, in particular automatic error-controlled predictor-corrector schemes are favorited. Upwind methods are thoroughly discussed and examined in comparison to the standard Galerkin-based FEM (GFEM). Stability and error analyses for the favorite schemes are presented in some detail. Techniques for solving the resulting matrix equations are discussed. Of important interest is the solution of the nonlinear equations by using Picard and Newton iteration techniques, which are embedded in adaptive time stepping strategies for solving transient problems. A particular focus is given on derived quantities, i.e., the evaluation of fluxes and balance quantities. It is shown that the FEM is locally conservative.
Hans-Jörg G. Diersch

Chapter 9. Flow in Saturated Porous Media: Groundwater Flow

Abstract
This chapter deals with the finite element solutions for saturated porous media (groundwater). Groundwater modeling is concerned with the motion of subsurface water in aquifers and aquifer systems, which can be unconfined or confined. Solutions are required for fully 3D, vertical or essentially horizontal 2D and axisymmetric isothermal flow of homogeneous flow in saturated porous media with and without free surface(s), including fully anisotropic flow situations and the incorporation of multi-layer well condition.
Hans-Jörg G. Diersch

Chapter 10. Flow in Variably Saturated Porous Media

Abstract
This chapter deals with the finite element solutions for variably saturated porous media (unsaturated-saturated flow). The different formulations of Richards equations with the favorite solution strategies, including the computation of hysterestic effects and time-varying porosity, are discussed. Typical examples and benchmark tests are described to illustrate the usefulness, efficiency and accuracy of the proposed numerical techniques.
Hans-Jörg G. Diersch

Chapter 11. Variable-Density Flow, Mass and Heat Transport in Porous Media

Abstract
In Chap.​ 3 the continuum approach of the porous medium has been described. A fluid (or better a phase) appears there as an effectively continuous medium with a mass density ρ (fluid mass per unit volume of fluid) as a fundamental bulk property. The density of a fluid is often not uniform. In general, the fluid is composed of N miscible chemical species with a partial density ρ k (mass of the constituent k per unit volume of fluid), so that for the mixture $$\rho =\sum _{ k}^{N}\rho _{k}$$ (density increases when dissolved mass of constituents increases). Moreover, the density of a fluid can be influenced by the temperature T (density decreases when temperature increases) and by the pressure p (density increases when pressure increases due to compressibility). In a formal manner, the density is to be regarded as a dependent thermodynamic variable for which an equation of state (EOS) ρ =ρ(p, ρ k , T) holds, cf. Sect. 3.​8.​6.​1.
Hans-Jörg G. Diersch

Chapter 12. Mass Transport in Porous Media with and Without Chemical Reactions

Abstract
In this chapter the computation of multispecies (including single-species) mass transport in porous media with chemical reaction in particular is examined. The complexity of those reactive transport processes arising in natural and engineered porous media requires some specific treatment due to their nonlinearity and the occurrence of multiple unknowns. In the preceding Chap.​ 5 the constitutive relations in form of reversible reaction and irreversible chemical kinetics have been developed. It ends up with a set of mass transport equations for each chemical species k = 1, , N of an arbitrary number, nonlinearly coupled by the rate expressions of chemical reaction in form of degradation type, Arrhenius type, Monod type or freely editable kinetics. A given species k can be either mobile associated with a liquid (aqueous) phase l or immobile associated with a solid phase s, so that N = N l + N s . Chemicals in the liquid phase are subject to advection and dispersion, while in a solid phase there is no advection and dispersion. We solve the reactive multispecies mass transport processes in multi-dimensional porous media under variably saturated, variable-density and nonisothermal conditions. The focus of this chapter is on the treatment of the species mass transport PDE system, while for the flow computations we refer to Chap.​ 9 for saturated porous media, to Chap.​ 10 for variably saturated porous media and to Chap.​ 11 for density-coupled problems. Nonisothermal aspects are subject of Chaps.​ 11 and 13.
Hans-Jörg G. Diersch

Chapter 13. Heat Transport in Porous Media

Abstract
In this chapter we discuss the finite-element computation of heat (thermal energy) transport in porous media. Nonisothermal porous-medium processes can be found in many areas of application to natural and engineered systems, for instance exploitation of geothermal reservoirs as a viable and renewable source of energy, underground energy storage and recovery for heating and cooling purposes, waste disposal of heat-generating materials, chemical reactor engineering, insulation of buildings, material technology and many others. Modern industrial developments have expanded significantly the fields, where numerical simulation is required as a powerful tool to aid the design and operation of equipments.
Hans-Jörg G. Diersch

Chapter 14. Discrete Feature Modeling of Flow, Mass and Heat Transport Processes

Abstract
The discrete feature approach provides the crucial link between the complex geometries for subsurface and surface, porous and fractured continua as well as to incorporate engineered structures in modeling flow, mass and heat transport processes. In such a holistic approach a 3D geometry of the subsurface domain (aquifer system, rock masses) in describing a porous-medium structure can be combined by interconnected 1D and/or 2D discrete features as shown in Fig. 14.1. In the finite element context the 3D mesh for the porous medium can be enriched by discrete line (channel, borehole, pipe network, tunnel, mine stope) and/or areal (overland, fault, fracture) elements.
Hans-Jörg G. Diersch

Chapter 15. Specific Topics

Abstract
The finite element solution of the governing flow, mass and heat transport equations as described in the preceding chapters requires the discretization of the equations to replace the continuous PDE’s with a system of simultaneous algebraic equations (cf. Chap.​ 8). The spatial discretization is accomplished by subdividing the study domain with its boundary into a number of nonoverlapping finite elements of different shapes, such as triangles, tetrahedra, bricks (see Fig. 8.6), forming the finite element mesh associated with a set of nodes and interpolation functions.
Hans-Jörg G. Diersch

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