The Influence of Oriented Arrays of Thin Impermeable Shale Lenses or of Highly Conductive Natural Fractures on Apparent Permeability Anisotropy
Authors: Prats Michael 
Published: December 2007

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The apparent permeability of a reservoir normal to its gross depositional surfaces (considered here to be horizontal for purposes of discussion) may be extremely important in determining the flow behavior within a reservoir. When there is no vertical communication, there can be no crossflow, and each element in the depositional sequence will respond independently to driving forces. For the other extreme, when the vertical and horizontal permeabilities are equal, density differences between fluids alone can cause great deviations from horizontal flow. Intermediate values of net vertical-to-horizontal permeability ratios also occur. Although these permeability ratios also occur. Although these intermediate ratios may arise from several causes, we are going to consider here the case where the apparent anisotropy arises from a large number of very thin horizontal shale streaks distributed throughout the reservoir. These shale streaks, or laminae, are considered to be impermeable and are idealized as having zero thickness. Their areal shape, extent, and distribution are presumed to be determined primarily by depositional environment, but these data are generally not known every in a statistical sense. The distribution of these impermeable barriers is assumed to have the repetitive pattern illustrated in the top of Fig. 1. The impermeable barriers are considered to be infinite strips running normal to the plane of the page, so that the problem becomes two-dimensional. Idealizing the shale barriers by these infinite strips, which is done solely to help obtain a quantitative analysis, is by far the most significant assumption made in the development.
To determine the vertical flow resistance of the array of impermeable barriers and to compare it to that in the absence of the barriers, it is sufficient to consider the flow of only a single incompressible fluid. Accordingly, the pressure distribution is harmonic. Because of the symmetry resulting from bulk vertical flow around the barriers only the elements shown in the bottom of Fig. 1 need be considered. Them is no How across the boundaries ADF and CBE, and the open intervals EA and FC are lines of equal pressure. pressure. The method used to obtain results is discussed in the Appendix. Calculated values of R, the ratio of vertical permeability to horizontal permeability, are plotted in permeability to horizontal permeability, are plotted in Fig. 2 vs twice the width:height ratio of the elemental rectangle (2W/H) and for several values of the fractional aperture, a. When no barriers are present, the aperture is complete (alpha = 1), and the ratio of vertical permeability to horizontal permeability is unity. When the impermeable barriers are continuous, the aperture is zero (alpha = 0), and there is no vertical permeability. Results are intermediate for other values of the aperture. There are two interesting cases for very large W/H ratios. When the apertures from the opposite sides of the elemental rectangle do not overlap (a less than 0.5), the vertical permeability vanishes as W/H increases. When the apertures from the opposite sides of the elemental rectangle overlap (a &gt;0.5), the ratio of vertical permeability to horizontal permeability approaches 2 - 1 asymptotically.
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