The Origins of Anisotropy (includes associated papers 18394 and 18458 )
Authors: Lake Larry W. 
Published: December 2007

Lake, Larry W., SPE, U. of Texas

Introduction
A property is anisotropic if its value depends on the directionin which it is measured; otherwise, the property is said to beisotropic. Both terms are usually applied only to intrinsicproperties of permeable media. Only flow or transportproperties, which have a specific direction associated with theirmeasurement, can be anisotropic: permeability, relativepermeability, resistivity, thermal conductivity, and dispersivity, Static properties are intrinsically isotropic: density, porosity, and capillary pressure, Static properties apparently exhibitinganisotropy in homogeneous media are probably manifestingsize effects. Here we deal mainly with permeability anisotropy.A property is nonuniform if its values form a singledistribution with a nonzero variance-i.e., if there is a rangeof values about some average. Heterogeneity, is a descriptionthat refers to a property having values that form two or moredistributions. Both definitions are quite distinct fromanisotropy, but as we shall see below, heterogeneity inparticular is the major source of anisotropy.

The Effects of Anisotropy
Permeability anisotropy causes fluid to flow in a directiondifferent from that in which it is pushed. Fig. 1 illustrates thatthe flow and isopotential (isopressure if single-phase horizontalflow) lines in an isotropic medium are perpendicular to eachother. Because in a waterflood we normally impose theisopotential lines, the behavior illustrated means that we have agood deal of control over the direction in which fluids areflowing. In an anisotropic medium, flow is not perpendicularto isopotential lines. Here, there is much less control overwhere the fluid will go, with a consequent loss in sweepefficiency. Of course, the right panel of Fig. 1 is onlyschematic because, as the isopotential lines accommodatethemselves to the anisotropy, they will no longer be straightvertical lines.

Importance
The illustration in Fig. 1 is of single-phase flow through atwo-dimensional (2D) medium. The anisotropic nature ofpermeability can affect any process in which there is asubstantial density difference between fluids: primaryproduction below the bubblepoint, gas cycling, gas and/orwater coming, some waterfloods, some solvent floods, andmany steam and in-situ combustion processes. It also caninfluence fluid injection and production rates if the anisotropyis severe.

Representations
A medium property that causes the redirection of flow requiressix independent scalar values to represent it fully in threedimensions (three values in two dimensions). Mathematicianscall such representations tensors. Finding accurate values forthese quantities represents an immense challenge in datacollection. Many of our normal tools yield estimates that areusually (1) too integrated over a wide range of variation(single-well tests) and hence do not give good definition or (2)too local (core analysis) to be of much use by themselves. Moreover, the number of scalar quantities is so large in threedimensions, particularly when heterogeneity is included, thatcomplete representation is a considerable challenge even fornumerical simulation. For these reasons, nearly all numericalsimulators include only three principal components (horizontal, lateral, and vertical permeability) of anisotropy. Because therepresentation of anisotropy in simulators is coarse, we cannotfully appreciate the effects this phenomenon can have.

Causes
Sand grains with typical packing and shape can exhibitanisotropy ratios (largest/smallest permeability in twoperpendicular directions) of no more than 2 to 3. Yetmeasurements of anisotropy in the field and on whole coresindicate that horizontal permeability can be several factors of10 larger than vertical permeability. The origin of suchextreme values lies in the existence of heterogeneity on a scalesmaller than the measuring device. Two geologic features inparticular will account for this type of anisotropy: crossbeddingand shales. Crossbedding is the alternate layering of sands of differinggrain sizes and/or textures at an acute angle with majordepositional features. Crossbedding is an extremely commonfeature of sedimentary media, but unfortunately it is alsohighly irregular and usually intermittent. The right panel ofFig. 1 is a schematic of flow through a crossbedded sand. Thespacing between the layers usually is only a centimeter or two. There frequently is little difference between the mineralcomposition of the alternating layers. Shales, on the other hand, usually have a mineralcomposition distinct from the adjoining media, being mostclosely related to clays, which because of an extremely smallgrain size usually have a very low permeability. Dispersedshales reduce the permeability of most media, but do notimpart significant anisotropy. Continuous segregated shalesreduce or, under extreme conditions, eliminate flow throughthe shale. Fig. 2 shows a synthetic generation of discontinuousshales in a 2D cross section. In this figure, it is easy to seehow the horizontal permeability, which is an arithmeticaverage of the shale and sand permeabilities, is affected verylittle by the shales, while the vertical permeability, which iscloser to a harmonic average, can be greatly reduced. Thecalculated mean horizontal-to-vertical-permeability ratio in Fig.2 is between 103 and 105. Fig. 2 also illustrates three important points aboutanisotropy. First, severe anisotropy is the result of localheterogeneity. In the case of Fig. 2, the heterogeneity isseparate, uniform distributions of shale and sandpermeabilities.
JPT
P. 395^