Computational Methods in Environmental Fluid Mechanics

The basic idea of continuum mechanics is that the evolution of a physical system is completely determined by conservation laws, i. e. basic properties such as mass, momentum, and energy are conserved during the considered process at all times. Any physical system can be completely determined by this conservation properties. In contrast, other quantities such as pressure or entropy do not obey conservation laws. The only additional information concerns the consistence of the material (e. g. fluids, solids, porous medium) in form of constitutive laws.

The Reynolds number of flow gives a measure of the importance of inertial related to viscous forces. Experiments show that all flows become unstable above a certain Reynolds number. Below values of the so-called critical Reynolds number Re _crit the flow is smooth and adjacent layers of fluid slide past each other in an orderly regime. This regime is called laminar flow. At Reynolds numbers larger than the critical value a complicated series of physical events takes place which eventually result in a radical change of the flow behavior. Finally, the flow becomes turbulent, i. e. velocity and other flow properties become chaotic and random. The flow is then unsteady even with constant boundary conditions. Turbulence is a kind of a chaotic and random state of motion. Nevertheless, velocity and pressure vary continuously with time within substantial regions of flow. Velocity fluctuations associated with turbulence give rise to additional stresses on the fluid — so-called Reynolds stresses. Examples of turbulent flows are: free turbulent flows (jet flow), turbulent boundary layer flows.

Soil or rock can be considered as a multiphase medium consisting of a solid phase (solid matrix) and of one or more fluid phases (gas and liquids), which occupy the void space (Fig. 3.1). Fluids are immiscible, if a sharp interface is maintained between them. In general, a phase is defined as a part of a continuum, which is characterized by distinct material properties and by a well-defined set of thermodynamic state variables. State variables describe the physical behaviour at all points of the phase. They must vary continuously within the considered phase of a continuum. Phases are separated from each other by surfaces referred to as interphase boundaries. Transport of components may occur within a phase and/or across interphase boundaries. Those interphasic exchange processes between adjacent phases can result from diffusive and/or advective mechanisms.

The governing equations for fluid flow and related transport processes are partial differential equations (PDE) containing first and second order derivatives in spatial coordinates and first order derivatives in time. Whereas time derivatives appear linearly, spatial derivatives often have a non-linear form. Frequently, systems of partial differential equations occur rather than a single equation.

There are many alternative methods to solve initial-boundary-value problems arising from flow and transport processes in subsurface systems. In general these methods can be classified into analytical and numerical ones. Analytical solutions can be obtained for a number of problems involving linear or quasi-linear equations and calculation domains of simple geometry. For non-linear equations or problems with complex geometry or boundary conditions, exact solutions usually do not exist, and approximate solutions must be obtained. For such problems the use of numerical methods is advantageous. In this chapter we use the Finite Difference Method to approximate time derivatives. The Finite Element Method as well as the Finite Volume Method are employed for spatial discretization of the region. The Galerkin weighted residual approach is used to provide a weak formulation of the PDEs. This methodology is more general in application than variational methods. The Galerkin approach works also for problems which cannot be casted in variational form.

The basic steps in order to set up a finite difference scheme are:

The Finite Volume Method (FVM) was introduced into the field of computational fluid dynamics in the beginning of the seventies (McDonald 1971, Mac-Cormack and Paullay 1972). From the physical point of view the FVM is based on balancing fluxes through control volumes, i. e. the Eulerian concept is used (see section 1.1.4). The integral formulation of conservative laws are discretized directly in space. From the numerical point of view the FVM is a generalization of the FDM in a geometric and topological sense, i. e. simple finite volume schemes can be reduced to finite difference schemes. The FDM is based on nodal relations for differential equations, whereas the FVM is a discretization of the governing equations in integral form. The Finite Volume Method can be considered as specific subdomain method as well. FVM has two major advantages: First, it enforces conservation of quantities at discretized level, i. e. mass, momentum, energy remain conserved also at a local scale. Fluxes between adjacent control volumes are directly balanced. Second, finite volume schemes takes full advantage of arbitrary meshes to approximate complex geometries. Experience shows that non-conservative schemes are generally less accurate than conservative ones, particularly in the presence of strong gradients.

Object-oriented (OO) methods are a necessary tool in responding to many of the challenges in scientific computation, in particular, in managing modeling of complex systems such as multicomponent/multiphase processes in porous / fractured media. Use of 00 methods can significantly reduce the effort to maintain and extend codes as requirements change. In addition, OO techniques offer means to increase code reuseability.

Object-oriented programming (OOP) has become exceedingly popular in the past few years. OOP is more than rewriting programs in modern languages, OOP is a new way of thinking about designing and realizing software projects. This requires a complete re-evaluation of existing programs (Budd 1996).

In this chapter we discuss the implementation of finite elements within the object-oriented framework presented in chapter 10. We consider all required steps for introduction of element types into the code. As an example we present the implementation of triangular elements. Theory of triangular elements is described in section 7.4.3. Based on this element template, new ones can be introduced very easily (see Problems).

This chapter deals with theory and computation of fluid flow in fractured rock. Non-Darcian flow behavior was observed in pumping tests at the geothermal research site at Soultz-sous-Forêts (France). Examples are examined to demonstrate the influence of fracture roughness and pressure-gradient dependent permeability on pressure build-up. A number of test examples based on classical models by Darcy (1856), Blasius (1913), Nikuradse (1930), Lomize (1951) and Louis (1967) are investigated, which may be suited as benchmarks for nonlinear flow. This is a prelude of application of the non-linear flow model to real pumping test data. Frequently, conceptual models based on simplified geometric approaches are used. Here, a realistic fracture network model based on borehole data is applied for the numerical simulations. The obtained data fit of the pumping test shows the capability of fracture network models to explain observed hydraulic behavior of fractured rock systems.

In this chapter we examine heat transfer during forced water circulation through fractured crystalline rock using a fully 3-D finite-element model. We propose an alternative to strongly simplified single or multiple parallel fracture models or porous media equivalents on the one hand, and to structurally complex stochastic fracture network models on the other hand. The applicability of this “deterministic fracture network approach” is demonstrated in an analysis of the 900-day circulation test for the Hot Dry Rock (HDR) site at Rosemanowes (UK). The model design in respect to structure, hydraulic and thermal behavior is strictly conditioned by measured data such as fracture network geometry, hydraulic and thermal boundary and initial conditions, hydraulic reservoir impedance, and thermal drawdown. Another novel feature of this model is that flow and heat transport in the fractured medium are simulated in a truly 3-D system of fully coupled discrete fractures and porous matrix. While an optimum model fit is not the prime target of this study, this approach permits to make realistic long-term predictions of the thermal performance of HDR systems.

In this chapter we examine variable-density flow and corresponding solute transport in groundwater systems. Fluid dynamics of salty solutions with significant density variations are of increasing interest in many problems of subsurface hydrology. The mathematical model comprises a set of non-linear coupled partial differential equations to be solved for pressure/hydraulic head and mass fraction/concentration of the solute component. The governing equations and underlying assumptions are developed and discussed. The equation of solute mass conservation is formulated in terms of mass fraction and mass concentration. Different levels of the approximation of density variations in the mass balance equations are used for convection problems (e. g. the Boussinesq approximation and its extension, full density approximation). The impact of these simplifications is studied by use of numerical modeling. Numerical models for non-linear problems, such as density-driven convection, must be carefully verified in a particular series of tests. Standard benchmarks for proving variable-density flow models are the Henry, the Elder, and the salt dome problems. We studied these benchmarks using two finite element simulators — ROCKFLOW, which was developed at the Institute of Fluid Mechanics and Computer Applications in Civil Engineering, and FEFLOW, which was developed at the Institute for Water Resources Planning and Systems Research Ltd. Although both simulators are based on the Galerkin finite element method, they differ in many approximation details such as temporal discretization (Crank-Nicolson versus predictor-corrector schemes), spatial discretization (triangular and quadrilateral elements), finite element basis functions (linear, bilinear, biquadratic), iteration schemes (Newton, Picard), and solvers (direct, iterative). The numerical analysis illustrates discretization effects and defects arising from the different levels of the density approximation. We present results for the salt dome problem, for which inconsistent findings exist in literature (Kolditz et al. 1998).

We consider both cases: multiphase flow in deformable as well as non-deformable (static) porous media. In addition to flow of two fluid phases (compressible and incompressible fluids) we also apply the Richards approximation, which is valid for most cases of infiltration in soils. We assume isothermal conditions. For additional information the reader should refer e. g. to Bear & Bachmat (1990) and Lewis & Schrefler (1998).