Mathematical Methods for Surface and Subsurface Hydrosystems

The following sections are included:

• Preface

• Contents

The following sections are included:

• Introduction

• Resolution of the elliptic case with a scalar flux related parameter
  • The mixed finite element formulation
  • The finite volume formulation
  • The re-formulation of the mixed finite element

• General 2D formulation for the elliptic case
  • The mixed finite element formulation
  • The corresponding re-formulation

• The parabolic case
  • The standard mixte formulation
  • The re-formulation of the MFE

• Comments and conclusion

• References

The following sections are included:

• Introduction

• Finite elements for fully saturated steady flows
  • Darcy's equations
  • The pressure primal formulation
  • Mixed formulation in H_div × L²(Ω)
  • Mixed formulation in H_div × L²(Ω)
  • Mixed formulation in
  • Summary

• Finite elements for variably saturated flows
  • Richards' equation
  • Soil hydrodynamic functions
  • Space and time discretization
  • Water-table–ground interaction and the obstacle problem

• Overland flow
  • Model formulation
  • Coupling with Richards' equation
  • Numerical method

• Applications
  • One-dimensional infiltration
  • Two-dimensional hillslopes

• References

The following sections are included:

• Introduction

• Sharp front problem statement

• Classification of approaches

• Particle techniques

• Improving eulerian approaches
  • Improving a 1D Finite Difference Scheme
  • Improving a 1D Finite Element Scheme
  • Higher order schemes: spectral elements

• Lagrangian approaches

• Eulerian-lagrangian approaches

• Conclusion

• References

The following sections are included:

• General statements
  • How do biological processes differ from others (physical, chemical⋯)?
  • The Michaelis- Menten formalism (1911)
  • Principles of a model construction: calibration—validation—extrapolation
  • Numerical difficulties: non linear phenomena

• Aquatic biogeochemical modeling
  • Definitions and objects of the modeling
  • Modeling of phytoplankton growth
  • Usual modeling of organic matter biodegradation
  • Complex biogeochemical modeling of aquatic ecosystems

• Modeling of wastewater biodegradation
  • ASM1: a widely used wastewater treatment plant model
  • The estimation of the composition of wastewater samples through inverse modeling

• Strategies for a better modeling of biological processes in aquatic systems

• Cited References

• Some Recommended References

The following sections are included:

• Introduction

• The character of equilibrium point and its impact on the changing rate of phytoplankton concentration

• Red tide ecological dynamic model
  • Physical/Biological Coupling
  • Governing equations
  • General description of numerical methods

• Practical work
  • The governing equations
  • The numerical solution method
  • Results

• References

Submarine groundwater discharge (SGD) has been identified by the International Geosphere-Biosphere Programme as an important contamination source for coastal marine and estuarine environments. Inputs of land-derived contaminants associated with SGD cause serious threats to the near-shore ecosystem. Water exchange driven by oceanic oscillations such as tides and waves at the shore contributes to SGD significantly. Moreover, the oceanic oscillations modify the near-shore groundwater flow and induce mixing of fresh groundwater with seawater. These processes affect not only the transport but also the chemical reactions of the contaminants in the near-shore aquifer prior to the discharge. In this chapter, we first review studies of costal groundwater responses to tides and waves, focussing on analytical solutions of the groundwater table fluctuations. We then discuss the effects of the groundwater fluctuations on the fate of chemicals in the near-shore aquifer and chemical fluxes to coastal water. These discussions are based on several on-going studies aiming to improve the understanding and quantification of subsurface pathways and fluxes of chemicals to coastal environments.

In this paper, the variational adjoint method with Tikhonov regularization principle is applied to retrieving the initial concentration filed and the diffusion coefficient of a 3-D water pollutant model. The adjoint models are deduced for the complete and incomplete observation of water pollutant concentration respectively. Then a particular algorithmic scheme is designed for 1-D model as an example. Finally, a series of ideal numerical tests are performed to show the validity of the method. The numerical results show that the adoption of Tikhonov regularization principle can be used to eliminate oscillation in retrieval and the difficulty of short of information for the incomplete observations can be overcome efficiently by introduction of the background field of diffusion coefficient.

Tabu search method, a meta-heuristic approach, with which simulated annealing and genetic method are extensively used in combinational optimization. This paper successfully applied Tabu search method to the inverse analysis of hydrogeological parameters and validate its effectiveness in the simulation of the groundwater field affected by cutoff wall in Hanjiang dike. This method, which is based on human memory, has shown the predominance of random search.

The U.S. National Weather Service (NWS) FLDWAV model has capacity in simulating variety of hydraulic phenomena in river systems. However it is originally designed for simulating one-dimensional unsteady flow in dendritic channel systems and is not full suitable for river networks. The modified relaxation iterative method was proposed when simulating the unsteady flow and the water quality in river networks of Shanghai with more than 400 named rivers. The advantage of this improved model is that it can satisfy for any complex river networks. The convergence of Netwon iterative method using in this model for nonlinear systems was also theoretically proved and Netwon down-hill method was applied to the case of unconvergence situation in original numerical method. A combined Gauss elimination with partial pivoting method with compress-storage (GEPP-CS) technique was developed to solve the linear systems.

This study is focused on the equilibrium point, which is the value when phytoplankton concentration is unchanged with time, of phytoplankton concentration. The changing rate of phytoplankton population and the character of equilibrium point were studied using simple Nutrient-Phytoplankton mathematical model. The results indicated that under certain environment conditions phytoplankton concentration meets with the equilibrium point at the end and the differences between the equilibrium point and the concentration of phytoplankton determine the changing rate of phytoplankton concentration. The equilibrium point is essentially dependent on the environmental factors. In this paper, it is used to explain the phenomenon of red tide. The environmental factor, nutrient, was the only factor concerned in the work.

The two dimensional shallow water equations with heat convection and diffusion equation is proposed to simulate thermal pollution of Guangdong Nansha Thermal Power which is planed to be built. Finite volume method (FVM) and flux difference splitting(FDS) scheme is applied, and the result shows that this method can keep high precision when it is applied to simulate heat transferring in estuary.