Flow of underground water is a classical example of fluid flow in porous media. It has been studied by hydrologists and soil scientists in connection with applications to civil and agricultural engineering. The reservoir modeling of multi-phase and multicomponent flows has been used in the petroleum industry for production and recovery of hydrocarbon. Transport of radionuclides into the aquifer has been studied by nuclear engineers in connection to possible leaks from tanks with radioactive materials. Finally, various problems of flows in porous media are related to the design and evaluation of remediation technologies and water quality control.

In the last several decades hydrology and petroleum engineers have become increasingly involved in modeling and computer simulation of flows and transport in underground reservoirs. These efforts have led to the development of a wide range of mathematical models for saturated single phase flow, saturated/unsaturated two-phase flow and multi-phase, multicomponent flow and transport. In general, these are systems of nonlinear partial differential equations of convection-diffusion-reaction type. The formulation of the differential model is usually based on the mass conservation principle enhanced with constitutive relations such as the Darcy's and Henry's laws.

In many practical situations the system of equations can be simplified substantially. For example, incompressible fluid flow in fully saturated reservoir is adequately described by a single elliptic equation for the pressure and transport equation(s) for the concentration(s) of the pollutant(s). This model has been successfully used in underground hydrology in the past century.

Driven by the needs for design of technologies for production and recovery of oil and gas the petroleum industry has developed and implemented a variety of compositional models of multi-phase multicomponent flows. In environmental protection considerations the problems are similar to those of the petroleum industry, but differ in many respects. Here the pressures are much lower, the variety of species is much larger, the topography of the reservoir plays an important role, and the needed accuracy is often very high (especially for the concentration of the pollutants). This requires adequate modeling and accurate numerical techniques. An example of such complex phenomenon is the transport of radionuclides in multi-phase groundwater flow in combination with sorption, desorption and radioactive decay.

My research in this area has been concentrated on the implementation and numerical investigation of a variety of models in groundwater hydrology that have been used in computer simulation for design of remediation and clean-up technologies. One of the most important question one tries to answer definitely is the question of the choice of the approximation method for the corresponding mathematical problem. In fluid reservoirs (aquifer and petroleum reservoirs) there are two imperative practical requirements: the method should conserve the mass locally and should produce accurate velocities (fluxes) even for highly nonhomogeneous media with large jumps in the physical properties. This is the reason that the finite volume method with harmonic averaging of the coefficients has been very popular and successful in computer simulation of flows in porous media. However, when the problem requires accurate description of the topography and the hydrological structure, a more general technique based on the finite element approximation is needed. The mixed finite element method has these properties. Since its introduction by Raviart and Thomas and its implementation by Ewing and Wheeler for flow problems, it has become a standard way of deriving high-order conservative approximations. It should be noted that the lowest-order mixed method realized on rectangles (or parallelepipeds) with certain numerical integration produces cell-centered finite differences with harmonic averaging.

A comprehensive overview of this subject can be found in one of my recent papers Mathematical Modeling, Numerical Techniques, and Computer Simulation of Flows and Transport in Porous Media. Technical Report ISC-95-09-MATH. (to appear in CTAC-95). Some more information concerning the computer implementation of this model and results from simulations are also available.