Examples of incorrect formulations

In this book, mathematical formulations are posed for well-defined physical problems and solved using rigorous solution methods. Assumptions are clearly stated and ad hoc analysis methods are avoided. In order to appreciate this philosophy, it is necessary to understand the proliferation of incorrect models and solutions in existing literature and software. A review of these formulations is imperative to teaching sound mathematics. But it is also important to petroleum engineers who purchase software responsible for field development and major financial commitments. For these reasons, we will explain common errors and their ramifications. To be entirely fair, only those problems that are actually solved in this book are listed and discussed here. 

Averaging methods. Equivalent resistance calculations in simple electric circuits is based on appropriate use of lumped or averaged properties. Similar results are desired in petroleum engineering, but in three widely used simulators we evaluated, averaging techniques are systematically abused. Formulas that are derived for linear (vs. cylindrical or spherical) flow under constant density, single-phase, identical-block-size assumptions are indiscriminately employed to process intermediate results in compressible, multiphase, variable grid block runs, leading to questionable results. 

Similar upscaling techniques, motivated by the need to reduce grid block number, are important in practice. But the equivalent permeabilities within any reservoir will change if the reservoir is produced by different arrangements or patterns of wells, because the parallel and serial nature of the flow has changed. Upscaled quantities are not properties of the formation but are also related to the production method. However, several simulators compute fixed upscaled properties and use them in contrasting production scenarios. 

Wells in layered media. Consider a layered reservoir produced by a general well, for simplicity, neglecting borehole friction and gravity. Production is controlled by one physical condition only the same constant pressure acts at the sandface along the entire length of the well, whether it is vertical, horizontal, deviated, or multilateral with arbitrary drainholes. Pressure itself may be 

Physical phenomena in science and engineering satisfy partial differential equations (PDEs), which relate changes in measurable quantities, like pressure or velocity, through partial derivatives taken in space and time. ETnlike ordinary differential equations (ODEs), whose integration constants are fixed by specifying values of the function and its derivatives at one or more points, PDEs require, in addition to functional information on curved boundaries, the specification of initial conditions for equations of evolution. Boundaries, we emphasize, may be external or internal, stationary or moving. The exact manner in which auxiliary conditions apply depends on the physical nature of the problem and is reflected in the type classification of the PDE studied. 

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