Ground water flows makes up an interesting category from the standpoint of modelling. Such flows essentially involve incompressible fluids flowing through porous media. The equation governing such flows is dominated by diffusive and source terms and includes the time derivative. It is usually possible to derive a single pressure equation reminiscent of the inhomogeneous Laplace equation to characterize and solve the problem. Problems involving ground water flows are analogous to classical heat conduction problems in multi-metallic complex objects.
The pressure equation is conventionally expressed as one with total head as the dependent variable. It is numerically well behaved and the solution readily converges with modern robust solution algorithms. However, when considering variably-saturated conditions, the conductance parameter can vary over tens of orders of magnitude for relatively small changes in pressure, a feature that can lead to convergence problems. To a lesser extent, density effects introduced by high concentrations of dissolved constituents has been known to slow down convergence. In general, however, the equation is well behaved and mathematically interesting solutions are usually the result of complex source terms.
Principia's experience with ground water modelling is extensive, covering over two decades and over one hundred projects. It has varied from models that cover an area of a few hundred metres on a side, to basin models that cover an area hundreds of kilometers on a side. Models representing complex flow patterns as a result of large variations in material properties and complex sources together with transport of multiple, reacting compounds have been constructed and successfully applied by Principia.
A significant difficulty in representing a natural system such as the earth is that the parameters which describe the sub-surface material are not well known and can vary by many orders of magnitude. Therefore a large part of constructing a ground water model is to infer many of these parameters through a computational process known as calibration. By performing many model simulations and comparing model predictions with observed quantities such as water levels or concentrations, parameter values are estimated. The uniqueness of such a calibration requires great care and often becomes an issue.
Modelling river and stream systems presents a different set of challenges. In some instances such flows can be modeled using a pipe network approach, which reduces the mathematical problem to a set of algebraic equations. Water rights introduce an additional layer of complexity to such a problem by imposing a set of mathematically-arbitrary rules of human origin on the water distribution system.
A particularly interesting class of problems involve conjunctive surface water and ground water flows. In many instances only the ground water portion or only the surface water portion is actually represented and the counterpart is considered to be a prescribed quantity. In some problems, however, the coupling between surface and groundwater becomes dominant and it is necessary to represent both systems. A difficulty to overcome in such a coupled system is that the time scale for ground water flows is orders of magnitude different from the time scale of surface water flows. Care must be taken to appropriately couple the two systems.
Principia's experience covers all of these types of surface water modelling.
For regulatory purposes, the U.S. EPA mandates the use of certain simple gaussian plume models. These models can also be used in other modelling contexts when the circumstances are appropriate and sufficient data are available. Gaussian plume approaches assume that temporally-discrete steady-state plumes exist with the height of the plume centerline and its horizontal and vertical dimensions are algebraic prescriptions that are functions of parameters such as the ambient temperature, wind speed, plume exit velocity, diameter and temperature. A variation of the gaussian plume models are the gaussian puff models. In these models an algebraic prescription is still used for the size and height of the puff, but by tracking discrete puffs more complex variations in source strength and ambient conditions can be modelled.
When more complex physical phenomena such as condensation are to be represented, the simplest type of modelling represents a plume in terms of a set of coupled ordinary differential equations (ODEs). Each of these equations tracks the conservation of a quantity such as mass, momentum, energy or moisture. The coupling between the equations comes from effects such as the release of latent head during condensation and endothermic or exothermic chemical reactions. The mathematical problem usually becomes initial valued and standard coupled ODE solvers such as an adaptive fourth order Runge-Kutta method works well in solving the problem. It is necessary to use numerical methods to, for example, determine the distribution of moisture between the liquid and vapor phases, but standard fixed point methods can be used for this purpose.
To account for more complex phenomena such as the turbulent motions of air around solid objects like buildings or aircraft wings or through objects like power plants, heat exchangers or jet engines, solution of the complete Navier-Stokes equations are required. This often requires equations for turbulence quantities to be included. In some instances, the equations can be simplified. For example, the Euler equations can be used in situations where diffusive effects are negligible, or a boundary layer equation can be used when only skin effects are of interest. In general parameters for the problem such as the properties of air and the geometry of the objects are well known. However, numerical convergence of the equations typically cause difficulties, especially if the initial conditions are not well chosen. In some instances, phenomena such as shock waves poses significant challenges to the solution procedure, and the solution algorithms have to be carefully chosen to satisfy the constraints of the problem.
Principia's experience varies from development and applications of simple gaussian plume models used to represent atmospheric transport and deposition over several decades, to the analysis of complex plumes involving condensation and chemical reaction, to the solution of the complete Navier-Stokes equations including turbulence and combustion associated with involving air flow around power stations, heat exchangers, jet engines, etc.
Contact Us |
Directions | Our Logo | Software Links | Data Links | Download | Upload | Restricted
Page updated on Mon Aug 2 11:38:40 MDT 2004
Copyright © 1997-2004, Principia Mathematica, Inc. - All Rights Reserved