Scaling Fluid Flow Through Heterogeneous Permeable Media

Li, Dachang (U. of Texas) | Lake, L.W. (U. of Texas)

OnePetro 

Abstract Despite several decades of study, there is no rigorous method for scaling flow through heterogeneous permeable media. Existing methods neglect heterogeneity and spatial correlation, which we now know to be of fundamental importance in many processes. On the basis of statistical techniques and inspectional analysis, we present a general method to scale flow through heterogeneous permeable media for an immiscible displacement of oil by water in a two-dimensional, anisotropic, heterogeneous cross section with statistically stationary properties. Our results illustrate the interplay among local heterogeneity (the variance at a lower cutoff in a power-law variogram), global heterogeneity (the variance at an upper cutoff) and zonal heterogeneity (the power-law upper cutoff and exponent). The effects of the scaling groups obtained from the method have been examined in detail, and some important flow characteristics, which were unknown or poorly understood before, have been revealed. Introduction Scaling is the translating of results from one system to another. The first system is usually small in scale and the second of a much larger size. The classical approach to scaling is to formulate the results of all responses of geometrically similar systems in terms of dimensionless variables and (scaling) groups. All systems with the same dimensionless groups will have the same dimensionless response. The dimensionless groups also can point to various limiting cases where one factor is clearly dominant over another and, consequently, simpler descriptions can be employed. Our previous work demonstrated an exhaustive procedure, using inspectional analysis (IA), to determine the scaling groups of immiscible displacements through homogeneous permeable media. We extend this to heterogeneous media here. The always-present heterogeneity of a permeable medium is a major roadblock that hinders the study of scaling. During the last three decades, many researchers have attempted to find ways to scale flow through heterogeneous permeable media. Carpenter et al. published the first paper in this field that extended the Rapport scaling relationships from homogeneous media to heterogeneous media of communicating strata of different permeability. Van Daalon and Van Domselear later extended this work to a medium with either a continuously or a discontinuously varying permeability distribution. The basic difficulty with these works is the simplified representation of reservoir heterogeneity. More complicated representations have appeared in works that do not deal with scaling. Spivak and Giordano et al. included random heterogeneity generated by using a normally distributed permeability field without spatial correlation. Moissis et al., Araktingi et al., and Waggoner et al. used geostatistical techniques to generate permeability fields based on the coefficient of permeability variation, or the Dykstra-Parson coefficient, and autocorrelation. Sorbie et al. extended Waggoner et al.'s work to define flow regimes and their significance for small autocorrelation. They also demonstrated that the importance of a "shape" scaling group. None of these have indicated a scaling method for flow through heterogeneous permeable media. Therefore, the major objective of this work is to illustrate an exhaustive method to provide such scaling. Another objective of the work is to examine effects of the scaling groups obtained, and to reveal some important flow characteristics that were previously unknown or poorly understood. General concepts A. Heterogeneity and scale Permeable media are heterogeneous if they display spatial variability of their properties. Reservoir heterogeneity has long been recognized as an important factor in determining reservoir performance and describing heterogeneity is a crucial step toward to understanding it.

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