The global interaction between the spatial average of the autocatalytic variable in a reactive (catalytic, electrochemical, gel) medium and a surrounding mixed fluid in a continuously fed vessel may induce a rich structure of spatiotemporal patterns that would not exist otherwise. Patterns may form when the local kinetics are either excitable, oscillatory, or bistable and the reaction rate ascends with reactant concentration. Thus, a local change in the surface reaction rate may affect the reactant concentration in the surrounding well-mixed vessel, so that it arrests moving fronts on the surface. External control of the average temperature (or rate) of a catalytic ribbon by electrical heating is another form of interaction between a spatial average of a local oscillator and a space-independent variable that induces pattern formation. We study various patterns and bifurcations that can develop in a ring or a ribbon due to global interaction using a simple cubic kinetic expression. The predominant pattern on a catalytic ring is a rotating pulse. Other patterns, such as antiphase oscillations and crossing pulses, similar to those found previously on a controlled catalytic ribbon, may coexist for a sufficiently strong interaction. Several of the transitions between regions with qualitatively different patterns are via global bifurcation.

When a chemical reaction is carried out on a catalytic ribbon, the spatial average temperature of which is kept constant by electrical heating, spatiotemporal temperature patterns form when the uniform steady state is unstable at the set temperature. Numerical simulations reveal periodic and aperiodic patterns of moving pulses, ‘‘breathing’’ pulses, or stationary and oscillatory fronts. The transitions between some of these patterns are intricate and proceed via global bifurcations. Bifurcation maps of parameter regions leading to specific patterns are used to gain insight into pattern formation and organization of these parameter regions. The relations among the dynamics of the uncontrolled system, the ribbon length, and the selected pattern are discussed. Similar patterns are expected to evolve in other reaction–diffusion systems subject to control of space-averaged properties.

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Global interaction introduces a new mode of communication among all surface elements and may lead to the formation of a very rich variety of spatiotemporal patterns on a disk some of which cannot exist in its absence. While some of the motions on a disk are analogous to those observed in one-dimensional systems, some, such as targets and spirals, exist only in two-dimensional systems. The global interaction stabilizes motions such as target patterns, which are not stable in its absence. It may, however, destabilize the spiral motion and cause the spiral tip to meander in a rather complex fashion and in some cases to exit the disk. The global interaction also increases the sensitivity of the system to the initial conditions so that qualitatively different patterns may be obtained when different initial conditions are used.

Giant reservoirs of the Middle East are crucial for the supply of oil and gas to the world market. Proper simulation of these giant reservoirs with long history and large amount of static and dynamic data requires efficient parallel simulation technologies, powerful visualization and data processing capabilities. This paper describes GigaPOWERS, a new parallel reservoir simulator capable of simulating hundreds of millions of cells to a billion cells with long production history in practical times. The new simulator uses unstructured grids. A distributed unstructured grid infrastructure has been developed for models using unstructured or complex structured grids. Unconventional wells such as maximum reservoir contact wells and fish-bone wells, as well as faults and fractures are handled by the new gridding system. A new parallel linear solver has been developed to solve the resulting linear system of equations. Load balancing issues are also discussed. A unified compositional formulation has been implemented. The simulator is designed to handle n-porosity systems. An optimization-based well management system has been developed by using mixed integer nonlinear programming. In addition to the core computational algorithms, the paper will present the pre- and post-processing software system to handle large amount of data. Visualization techniques for billions of cells are also presented. Introduction For many oil and gas reservoirs, especially large reservoirs in the Middle East, availability of vast amount of seismic, geological and dynamic reservoir data result in high-resolution geological models. But despite the many benefits of parallel simulation technology for large reservoirs, average cell size still remains in the order of hundreds of meters for large reservoirs. In order to fully utilize the seismic data, smaller grid blocks such as 25 to 50 meters in length are required. This size of grid blocks results in billion (Giga) cell models for giant reservoirs. In order to simulate such models with reasonable turnaround time, new innovations in the main components of the simulator such as linear equation solvers and equation of state computations are essential. Also, next generation pre- and post-processing tools are needed in order to build and analyze giga-cell models in practical times.