Discretization methods are proposed for control-volume formulations on polygonal and triangular grid cells in two space dimensions. The methods are applicable for any system of conservation laws where the flow density is defined by a gradient law, like Darcy's law for porous-media flow. A strong feature of the methods is the ability to handle media inhomogeneities in combination with full-tensor anisotropy. This paper gives a derivation of the methods, and the relation to previously published methods is also discussed. A further discussion of the methods, including numerical examples, is given in the companion paper, Part II [SIAM J. Sci. Comput., pp. 1717--1736].

A companion paper, Part I [SIAM J. Sci. Comput., 19 (1998), pp. 1700--1716], presents discretization methods for control-volume formulations in two space dimensions. Properties of the developed methods are further discussed, including rotational invariance and stability under the influence of anisotropy. Strong anisotropy may have to be handled by use of stretched grid cells. Numerical examples that demonstrate strengths and weaknesses of the methods are presented. The examples focus on inhomogeneity, anisotropy, and rotational invariance.

Summary: The paper presents a class of control-volume discretization methods for general quadrilateral grids in three space dimensions. The methods allow full permeability tensor and arbitrary orientation of the grid. The paper discusses handling of inactive cells and flow barriers. Numerical examples are used to compare the new methods with conventional corner-point geometry methods.

Using numerical and analytical modeling, we assess the feasibility of marine controlled-source electromagnetic (CSEM) data for monitoring the flooding front during water flooding of an oil reservoir. We discuss the ability of time-lapse CSEM data to resolve changes in the electric conductivity small enough to be of interest for monitoring purposes. Measurement and modeling errors are discussed briefly, and analytical calculations comparing time-lapse signals with a certain type of time-lapse modeling errors are performed. Numerical calculations, performed with a volume integral-equation method, are used to study the effect on the electromagnetic (EM) fields of relevant conductivity changes. The numerical investigation includes robustness with respect to survey parameter values, discrimination between different front shapes, and ability to overcome time-lapse error. Simulated signals are found to be strong enough to overtake typical measurement errors and are fairly robust toward perturbations of survey parameters. It is found analytically and numerically that certain modeling errors experience a high degree of time-lapse cancellation.