An Introduction to Permeability Averaging and the Effects of Scale on the Permeability of Heterogeneous Rocks
James W. Jennings
Shell International Exploration and Production, Houston, Texas, USA
Most naturally occurring porous media exhibit some degree of spatial permeability variation, usually referred to as heterogeneity. Few rocks are homogeneous, although some are more variable than others. One of the consequences of heterogeneity is scale dependence. That is, the permeability of a large volume of rock, often called the “effective permeability,” will in general be different than the permeabilities of smaller volumes within it. Given the way that effective permeability is typically used in numerical fluid-flow simulation it is tempting to think of it as a “fudge factor” that makes a simulator give the correct answer. However, physical experiments conducted on different-sized sub volumes of a limestone block demonstrate that the scale dependence of permeability is a real property of heterogeneous rocks and not just a numerical artifact (Henriette and others, 1989).
Effective permeability estimates are required at various stages of reservoir modeling. A common application is the scaleup of a fine-grid static model to a coarser grid for use in fluid-flow simulation. The various methods of effective permeability estimation fall into two broad classes: analytical calculations and numerical approximations. The numerical methods, otherwise known as “flow-based” upscaling are well established, but the analytical calculations have important applications as well and should not be neglected. One obvious application of an analytical method is to test a numerical approximation, but there are other perhaps less obvious uses in reservoir modeling that are especially important in carbonates.
This paper is concerned with analytical methods of matrix permeability averaging for single phase flow. Multiphase flow and fracture flow present a different set of issues that are not directly addressed, although the insights presented here provide an important conceptual foundation for modeling more complicated settings.
The special importance of analytical effective permeability methods in carbonates is a consequence of the fact that carbonates typically exhibit large permeability variances concentrated at relatively small scales (Jennings and others 2000). It is not unusual for more than half of the total variance of plug-scale permeability measurements in a carbonate reservoir to exist at scales smaller than the cells of the so-called “fine-grid” static models. In such cases one should not rely exclusively on routine upscaling of the static model to the dynamic model, no matter how accurate those methods may be, because most of the scaleup effects on effective permeability have already occurred in other scaleup steps used to construct the static model itself.
One common example of these other scaleup steps is the vertical averaging of well-log samples into the cells of the static model, known as “log-upscaling” in some static modeling workflows. Flow-based upscaling would be very cumbersome at this point in model construction and is not generally available; a selection of statistical averages are usually offered to the modeler instead. However, the choice of averaging method can have a significant effect on the final result.
Other implied scaleup steps common in static modeling are: (1) the scaleup that occurs when a porosity-permeability transform calibrated with plug-scale data is applied to well-log porosity to predict effective permeability in the well-log volume of investigation, and (2) the scaleup that occurs when variograms computed from well-log-scale samples are used to generate stochastic simulations of permeability in the laterally larger cells of a static model. In general, the relative importance of each scaleup step is controlled by the amount of variance that is absorbed in the calculation; scaleup steps that account for more variance are more important. In carbonate reservoirs the most important scaleup steps will often occur at the smallest scales where the analytical scaleup methods are usually the only methods available.
The most well known permeability averaging methods are the arithmetic and harmonic averages for flow along and across the layers of stratified media (see for example Amyx and others, 1960). A less familiar theoretical result is the 1/3 power average for isotropic random fields (Noetinger, 1994; Hristopulos and Christakos, 1999). The 1/3 power average is especially useful in carbonates because their small-scale heterogeneities are often well-approximated by isotropic random fields. A common misconception derived from a misunderstanding of a well known paper by Warren and Price (1961) states that the effective permeability of an isotropic random field is a geometric average. However, the effective permeability of an isotropic random field depends on the dimensionality of the flow: the geometric average applies to two dimensional flow, the 1/3 power average applies to three dimensional flow (Delhomme and de Marsily, 2005).
All of these permeability averages are special cases of a simple generalized formula, the power average. Harmonic, geometric, and arithmetic averages are power averages with exponents of -1, 0, and 1 respectively. Power averages are more general and may be computed for any real-valued exponent, but an important result of permeability averaging theory is that all effective permeabilities must correspond to a power average with an exponent in the range of -1 to 1. That is, all effective permeabilities must be between harmonic and arithmetic averages regardless of the spatial geometry. Thus, the problem of estimating an effective permeability can be restated as the problem of finding the correct averaging exponent between -1 and 1 for a given geometry.
The conjecture of Ababou (1990) provides a convenient method of estimating the averaging exponents in cases where the spatial correlation lengths are intermediate between the isotropic case and the maximum-anisotropy layered case. The combination of power averaging, exponents estimated via the Ababou conjecture, and a stepwise application of these methods to composite random fields containing different anisotropies at different scales produces a simple generalized approach to permeability averaging spanning a surprisingly broad spectrum of spatial variability that includes many cases of practical importance in carbonates.
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AAPG Search and Discovery Article #120034©2012 AAPG Hedberg Conference Fundamental Controls on Flow in Carbonates, Saint-Cyr Sur Mer, Provence, France, July 8-13, 2012