Many real life problems are modelled by differential equations, for which analytical solutions are not always easy to find. One of the most difficult problems is how to solve these differential equations efficiently. Several researchers have tried to do this in various different ways (e.g. via Finite Element Methods, Standard Finite Difference Methods, Spline Approximation Methods, etc.). In recent years, to get reliable results with less effort, researchers have applied nonstandard finite difference methods (NSFDMs) and obtained competitive results to those obtained with other methods. In this survey article, the author tries to provide as much stimulating information as available regarding these NSFDMs to the researchers, which will be helpful for them as the research proceeds in this direction. While the author made the utmost efforts to include whatever he could, he would like to apologize if there are any omissions which are totally unintentional.

On the use of nonstandard finite difference methods

Transport and Relaxation Phenomena in Porous Media

The usual macroscopic equations of motion for two-phase immiscible displacement in porous media are known to be physically incomplete because they do not contain the surface tension and surface areas governing capillary phenomena. Therefore, a more general system of macroscopic equations is derived here that incorporates the spatiotemporal variation of interfacial energies. These equations are based on the theory of mixtures in macroscopic continuum mechanics. They include wetting phenomena through surface tensions instead of the traditional use of capillary pressure functions. Relative permeabilities can be identified in this approach that exhibit a complex dependence on the state variables. A capillary pressure function can be identified in equilibrium that shows the qualitative saturation dependence known from experiment. In addition, the proposed equations include a description of the spatiotemporal changes of residual saturations during immiscible displacement.

Macroscopic equations of motion for two-phase flow in porous media

Non-standard numerical methods applied to subsurface biobarrier formation models in porous media.

Publisher Summary This chapter describes the development of a compositional reservoir simulator capable of modeling multiphase transport of carbon dioxide (CO2) in brine aquifers. As an example, it considers the radial injection of supercritical CO2 in a brine aquifer, and results are provided that illustrate the evolution of a 2-phase fluid system in which the injected CO2 resides in a dense supercritical phase and also dissolves into the liquid phase. Salt precipitation into a solid phase is found to occur close to the injection well where the gas phase dominates. To determine the feasibility of deep aquifer sequestration it is necessary to have a numerical model that can accurately predict the fate of CO2 under the conditions of interest. Conditions that support the existence of supercritical CO2 should be present at depths greater than about 800 m where pressure and temperature would be above the critical point of CO2.

Numerical modeling of carbon dioxide injection and transport in Deep Saline Aquifers

A petroleum reservoir is a complicated mixture of porous rock, brine, and hydrocarbon fluids, usually residing underground at depths that prohibit extensive measurement and characterization. Petroleum reservoir engineers face the difficult task of using their understanding of reservoir mechanics to design schemes for recovering hydrocarbons efficiently. In the past decade or so the reservoir engineer’s reliance on mathematical models has grown at a remarkable rate, and, in view of the evolution of many other successful branches of engineering, it seems fair to expect this reliance to continue increasing. This monograph examines several aspects of the development of mathematical reservoir models. Before embarking on technical discussions, though, we shall give a brief, relatively nontechnical survey of how an oil reservoir works.

Myron B. AllenIII

Papers

Basic Mechanics of Oil Reservoir Flows