Scaling Laws for Use in Design and Operation of Water-Oil Flow Models

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Abstract
Published in Petroleum Transactions, Volume 204, 1955, pages 143–150. This paper is intended as an aid in the performance and interpretation of experimental studies of multi-phase flow in porous media. The mathematical formulation of incompressible, two-phase flow phenomena in a threedimensional porous medium is presented. The resulting equations which account for the effects of gravitational and capillary forces are used to derive general scaling laws permitting accurate representation of water-oil displacement processes by means of experimental flow models. The significance of the scaling laws is examined in the light of the physical concepts of micro-behavior of fluids in porous media. Formulations of scaling conditions obtained by application of dimensional analysis are reviewed. The principles of application and the limitations of flow model studies are discussed. Introduction: Most of the problems facing a reservoir engineer involve three-dimensional, two-phase flow systems, influenced by the effects of gravitational and capillary forces. Present computational procedures generally do not permit the evaluation of such systems without introducing greatly simplifying assumptions. The actual reservoir configuration is usually replaced by one or a combination of several "equivalent" linear systems, and the effects of capillarity and gravity are considered to be negligible. While these assumptions may in many instances be justified, there are a number of important practical problems for which the results obtained on the basis of such simplifications remain open to question. An alternate approach to the problems that are not amenable to analytical treatment consists of seeking their solution by experimental laboratory investigations. The main difficulty of such investigations lies in the fact that multiphase flow phenomena subjected to gravity and capillary forces proceed differently at different rates and in systems of different dimensions. Therefore, unless great discrimination is exercised in selecting the conditions and interpreting the results of laboratory tests, such tests may be entirely misleading in regard to field applications. The purpose of this paper is to present a theoretical background for the performance of laboratory tests that are more nearly representative of field behavior. It will be shown that simulation of field behavior can be achieved by conducting laboratory tests under conditions selected in accordance with definite scaling laws.

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