A Note on the Role Played by Sediment Bedding in Causing Permeability Anisotropy
Authors: Rose Walter 
Published: December 2007

Rose, Walter; EG and G Continental Laboratories
Recently data have shown unexpectedly that the ratio of maximum to minimum permeability was of the order of only 1.5 as measured on four sandstone permeability was of the order of only 1.5 as measured on four sandstone samples where bedding and crossbeddings were quite marked. The definitive work of Bear showing how bedding gives rise to anisotropy makes it interesting to ask the simplistic and obvious (but somewhat ignored) question about how the thickness and permeability contrasts of the adjacent layers exactly affect the directional trends. The motivation for the analysis presented here is twofold. On one side, Muskat showed in 1949 that, when the ratio of the maximum to minimum permeability is less than 10, ignoring it would not introduce serious error permeability is less than 10, ignoring it would not introduce serious error when accounting for the production performance, even of a partially penetrating well. On the other side, both laboratory and field penetrating well. On the other side, both laboratory and field state-of-the-art methods of measurement are quite complicated, and serious ambiguities are encountered frequently in the interpretation of the data. It is as though sediment heterogeneity, resulting from spatial variations in permeability and porosity, rather than anisotropy is the principal factor to take into consideration when simulations are undertaken as a step toward history matching (past) and predicting (future) well and reservoir (field) performance. Greenkorn already has touched on these considerations. Simplistic mathematical models lead more often than not to predictions of limited utility; on occasion, however, results can be obtained that are useful and surprising. A case is illustrated here. The main assumption is that two-dimensional anisotropy cases occur as a consequence of the bedding and cross bedding of layered sediments. The secondary assumption is that there are alternating layers of contrasting permeabilities and thicknesses made up of otherwise uniform strata. If, within any given representative region, all bedding planes are more or less parallel with each other, it is clear that the maximum and minimum permeabilities will be in the parallel and perpendicular directions with respect to the bedding planes. Since only two types of repeating beds are being considered here (having permeabilities and thicknesses of, and, respectively), it is easy permeabilities and thicknesses of, and, respectively), it is easy to show that the maximum and minimum permeabilities will be given by
and
............................(1)
where A=( / ) and B=( / ), and where the 's and 's are the permeabilities and thicknesses of the alternating layers. Eq. 1 is valid permeabilities and thicknesses of the alternating layers. Eq. 1 is valid only for the case where the two kinds of layered strata have the geometry of right cylinders alternately stacked on top of one another. From Eq. 1 it follows that the ratio of maximum to minimum permeability will be given by
............................(2)
Taking she derivative of Eq. 2 with respect to B for constant A, we expose the surprising result that the / ratio is maximum when B . Fig. 1 is a graph of Eq. 2, which indeed shows that curves of constant / ratio are orthogonal to the B abscissa line. We also show that, for all values of A, the / ratio is the same whether / equals B . This means that the anisotropy effects will be greatest when, and that when, the anisotropy effects depend more on the absolute value of B than on whether the most permeable layer is the thickest (or vice versa). Other limiting relationships of interest are referenced in Table 1. Here it becomes clear that, unless A is large with respect to B, the / ratio will have values not much greater than two, as when A and B are of the same order, nor much greater than one if A is considerably smaller than B. In conclusion, it is interesting to consider the cases of A=B=10 3 and A=B=10 3, which can be thought of as describing reservoir situations where more or less homogeneous blanket sands are broken by thin shale laminae of meager thicknesses. For example, suppose that these laminae have a permeability such as =; then / is of the order of only permeability such as =; then / is of the order of only two (1.996 and 1.9996, respectively), while itself will range from 0.1 to 1 md for the two cases under discussion. If the shale partings (laminae) themselves are of the order of 1 (1 micron) thick, this implies that they are separated by 0.1 to 1 cm (0.04 to 0.4 in.) for the same two cases. On the other hand, if we imagine that B=10 5 to 10 6, so that the 1= (1-micron) laminae are separated by 10 to 100 cm (4 to 40 in.) of permeable sand and with A still in the range of 10 3 to 10 4, the / ratios are only slightly greater than unity (1.010). In other words, the conclusion can be drawn that permeability anisotropy is of negligible importance in many cases. This is clearly shown in Fig. 1, which indicates that very large values for A will be required (say ranging from 40 to 10 3) for cases where B ranges from 1 to 10 2, if Muskat's criterion for / to exceed 10 is to be met. Some of the practical implications are that shale laminae may not affect the drainage of tight sands, that water coning may not be much different for anisotropic and heterogeneous reservoirs and that the deliverability of partially penetrating wells may not be affected seriously by anisotropy.
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