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The Simplification of the Material Balance Formulas by the Laplace Transformation
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1958
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Numerical AnalysisEngineeringMechanical EngineeringPetroleum Production EngineeringContinuum MechanicComputational MechanicsReservoir EngineeringMechanicsNumerical SimulationPetroleum ProductionWater Influx EquationsMaterial NonlinearitiesBoundary Element MethodReservoir CharacterizationMultiphase FlowReservoir SimulationReservoir ModelingReservoir PressureMaterial ModelingContinuum ModelingMaterial Balance FormulasReservoir GeologyReservoir ManagementLaplace TransformationPetroleum TransactionsPetroleum EngineeringMultiscale Modeling
Published in Petroleum Transactions, AIME, Volume 213, 1958, pages 292–303. Abstract For many years the author has been cognizant of the difficulty encountered by some in treating with the water influx formulas for unsteady-state fluid flow as pertain to the material balance equation. This has particularly applied in establishing reservoir performance and identifying reservoir pressure, which to the practicing engineer has entailed a trial-and-error procedure, and for others has necessitated resorting to computing devices and reiteration processes. In retrospect this difficulty stems from the fact that reservoir pressure in the material balance formulas, as well as associated with the water influx equations, is an inexplicit term, and the work reported in the past is irrefutable. However, what will be presented in this paper is another approach to the problem, whereby the entire material balance equation will be treated by the Laplace transformation, and reservoir pressure which hereto has been inexplicit, can now be isolated by mathematical procedure to relate that parameter with all the factors contributing to its change. This is the simplification entailed, that treats first with an undersaturated oil reservoir as an integrated effect from the inception of production. The second phase pertains to saturated oil reservoirs that encompass a survey traverse. Although both methods of approach are necessarily different in aspect, the most interesting fact is that the mathematics so deduced are identical. Both the linear and radial water-drive systems are incorporated, for which an illustrated factual example is offered for the latter, treating with a saturated oil reservoir. Introduction What is performed in this work is the simplification of an involved computation by advanced analysis. Although such may be construed as a contradiction when one treats with higher mathematics; nevertheless, when direction is given to such an undertaking the results can be most revealing. Likewise, it is to be mentioned that the bases for these mathematics have been developed on the expediency of the occasion. This is not to be inferred as a qualification of this work, but rather the demands frequently placed upon the author in his private practice in meeting a time limit. A situation, instead of being fraught with hazards, often has given emphasis to creative thought.