Multicomponent induction log data can provide permeability readings more consistent with reservoir-scale data.

Permeability is a key parameter critical to the efficient production of hydrocarbons. A complete reservoir description including both vertical and horizontal permeability is needed to aid in completion and production designs. The ratio of vertical to horizontal permeability is a vital parameter in the efficient recovery of hydrocarbons as well as a required input to nodal analysis software programs used to optimize the recovery of hydrocarbons. All too often this ratio is an elusive value, and reservoir engineers use a field "rule of thumb" to come up with the best possible value.
In the early days of logging, the only logs that provided permeability information were micro-resistivity devices, but they provided only qualitative indications of permeability. This information, coupled with permeability measurements from sidewall or whole core samples, served as an attempt to provide a continuous permeability measurement. Today logs, well tests, formation tests and cores provide numerous ways to estimate permeability. It is now possible to estimate permeability from continuous formation evaluation measurements. However, it remains difficult to determine directional permeability. Nuclear magnetic resonance (NMR) logs are used routinely to estimate permeability, but NMR-derived permeability is based on scalar properties and is inherently a scalar property.
In order to add direction to permeability, a combination of directional and permeability measurements must be made. Dip and image logs provide bed thickness and layer dip information. However, only multicomponent induction instruments and crossed-dipole shear-wave acoustic tools provide direct measurements of macroscopic formation anisotropy.
Thinly bedded laminated reservoirs exhibit "macroscopic anisotropy" of physical properties, including electrical conductivity and permeability. Anisotropy is defined as the condition of having different properties in different directions, i.e. resistivity and permeability anisotropy. Macroscopic is defined to be the scale of the logging tool measurement. Macroscopic anisotropy is manifested in thinly layered sedimentary formations where geologic processes deposit sediments in layers much thinner than logging tools can resolve.
There are two types of layering:
Finely layered anisotropic sands (Bimodal sands). The fine layering can give rise to transverse anisotropic sands composed of layers of different grain sizes, porosity and/or sorting. When oil-saturated, the coarse layers exhibit a relatively high resistivity as result of low irreducible water saturation and a high permeability. The fine sand layers are characterized by lower resistivity values and lower permeabilities.
Laminated shaly sands. These formations consist of thinly bedded sand-shale sequences. The sand layers are characterized by high permeability and high resistivity (if oil-saturated); shale layers are characterized by low resistivity and extremely low permeabilities.
Laminated shaly sands normally exhibit higher contrasts in both resistivity and permeability than the finely layered anisotropic sands, especially for the permeability contrast.
All reservoir analyses, including the analysis for layered anisotropic sands, have the same goals: volumetric analyses and flow property determination. Volumetric reservoir descriptions include matrix volume fractions, porosity, permeability, fluid properties and volumes. Flow properties are characterized by permeability, but permeability is inherently a tensorial property that varies with direction. Only transversely isotropic (TI) media, which can be described by the horizontal and vertical permeability and resistivity of the constituent thin beds, is explored in this article.
New multicomponent induction-logging hardware makes possible the direct measurement of resistivity and resistivity anisotropy. In layered formations, the resistivity tensor data can be used to separate the macroscopic conductivities into the contributions from the two components of the layered formation. This data can be used to determine conventional volumetric properties (e.g., porosity, saturation) as well as to infer permeability anisotropy.
Background
Permeability is the relative ease with which a fluid flows through a formation, and consequently it is a critical reservoir property that controls hydrocarbon production. Permeability controls production rates, and the reservoir permeability controls the recovery efficiency. Since the first continuous wireline log data were collected, a goal of reservoir description has been to obtain permeability from log measurements. Generally, log-derived permeabilities have relied on the transformation of scalar properties to permeability. Most frequently, they are derived from core-based porosity-permeability correlations. Recently, log porosity data have been supplemented with NMR log-determined rock properties (e.g., mineralogy-independent porosity and bound water saturation). This has improved the accuracy of log-derived permeabilities, particularly in clastic reservoirs.
Log-derived permeabilities often are referred to as permeability indices with little regard to whether or not the computed permeabilities represent absolute or effective permeability, even though relative permeability effects are well understood by reservoir engineers and petrophysicists. Further, little regard is paid to whether the computed permeabilities represent horizontal or vertical permeability.
Core-derived permeability anisotropy. Permeability of cores is routinely determined with small-diameter core plugs (typically a few inches in length and about an inch in diameter). These small plugs are usually cut either parallel to bedding or perpendicular to bedding. This makes it possible to estimate vertical and horizontal bedding (assuming that the bedding is horizontal).
With core permeability data from adjacent plugs cut at right angles to one another, it is possible to estimate the ratio of the vertical to horizontal permeability, kv:kh.
Reservoir engineers often refer to this ratio, which is a key required input for nodal analysis and reservoir simulation software. This ratio generally is less than 1 and typically is taken to be much smaller in reservoir simulation. The permeability anisotropy ratio is defined as the ratio of kv:kh and is a number generally greater than one.
Well test-derived permeability anisotropy. When a well test exhibits partial penetration effects and early- and late-time data can be clearly distinguished, it is possible to extract both spherical and radial permeabilities. The late-time radial permeabilities can be interpreted as horizontal permeability and the early-time data as a spherical permeability.
From the combined early- and late-time derived-permeability data, one can extract the horizontal and vertical permeabilities and the permeability anisotropy ratio. The anisotropy ratios based on published figures can be very large (~100 to more than 1,000).
On core plugs scale permeability anisotropy is generally small. However, on the reservoir scale kv:kh is much smaller than 1.0, and permeability anisotropy can be greater than 100. When the horizontal and vertical core data permeabilities are averaged, the resulting kv:kh ratios are similar to those obtained from well test analysis and consistent with those used in reservoir simulation.
Modeling permeability anisotropy. For isotropic porous reservoirs, permeability can be described in terms of porosity, pore radius, tortuosity and properties expressing the pore channel geometry.
Permeability determination in macroscopic anisotropic reservoirs is a two-step process. Since pore geometry is anisotropic or directionally dependent, a second-step directional dependence of permeability must be introduced in the macroscopic scale (tensor of permeability). For macroscopic anisotropic sediments, additional information about the anisotropic rock properties is necessary. The directional information contained in the multicomponent resistivity data can be taken advantage of to derive directional permeabilities.
The difference between core-based and fluid flow-based (e.g., well tests and production test/reservoir simulation) estimates of kv:kh is related to the scale of the investigation. Core plugs are small, consisting of a few tens of cubic centimeters of material. Well tests involve larger volumes of the reservoir, at minimum cubic meters. Measured horizontal and vertical resistivities involve depths of investigation of
1 m or so and involve 1 or 2 cu m of reservoir material. Resistivity data samples a scale between cores and well tests, and if the data can be used to compute permeability and impart directionality, it should simplify up-scaling.
Log analysis relates log-measured properties (i.e., resistivities, gamma radiation, NMR echoes, etc.) and reservoir properties (i.e., porosity, shale content, saturation, permeability). For isotropic materials all related properties are uniform with respect to direction. For anisotropic materials, reservoir properties and logged properties must be classified into scalar (directional independent) and tensorial (directional dependent) properties.
Scalar properties (porosity, saturation, nuclear cross sections, etc.) depend only on volume fractions of the rock components (e.g., minerals, pore fluids). Tensorial properties are controlled also by the spatial distribution of the rock-building components. In the particular case of "macroscopic anisotropy," the anisotropy is originated by the different (microscopic) properties of the individual laminae and the integrating effect of the tools with insufficient resolution to characterize the individual layers.
Laminated sands can be described by "microscopic" properties related to two or more distinct individual components, while "macroscopic" properties are related to the response and resolution of the tools.
The easiest way to describe anisotropy in layered sediments is to distinguish between the horizontal direction (h) or parallel direction (plane of layering) and the vertical direction (v) or perpendicular direction (normal direction to plane of layering). Figure 2 summarizes the parameters and the classification of rock properties.
The relationships between microscopic and macroscopic tensorial properties are based on the laws for series and parallel electrical and hydraulic circuits with the volume fractions as weight functions and the material balance equations for the scalar properties.
Anisotropy calculation
Conventional induction logging tools are limited to measurements in one dimension because their sensors are aligned along the tool. Such measurements are satisfactory only when evaluating formations at least as thick as the tool's vertical resolution, which is generally several feet. Newer multicomponent tools use three orthogonally mounted transmitter-receiver coil arrays in the X,Y and Z planes relative to the tool axis. This configuration, along with specially developed software, provides the information necessary to determine vertical and horizontal resistivity.
In Figure 3 we show a histogram of resistivity anisotropy for typical data. For most of the reservoir the ratio is less than 1.5. The higher resistivity anisotropies are, in general, less than 2.2.
For a bimodal sand composed of a coarse and a fine fraction, Figure 4 shows the anisotropy ratio for resistivity and for permeability as a function of the fractional sand composition. For both examples the permeability anisotropy is distinctly higher than the resistivity anisotropy.
Comparison
How does the scalar NMR permeability respond to the macroscopic, directional permeability? The Coates equation is applied with averaged input properties.
To investigate the difference between the NMR permeability and the true directional permeabilities, the ratios of the parallel direction permeability and the NMR permeability: kh/kNMR are formed. Similarly, the ratio for the perpendicular direction, kv/kNMR is formed.
Based on the model for bimodal laminated sand, an algorithm was developed to derive the BVI (bulk volume irreducible water), porosity and microscopic permeability for the coarse and fine sand components. The algorithm also provides the permeability parallel and perpendicular to the sand laminations as part of the single, consistent solution.
Input for the calculation are the "macroscopic properties" resulting from measurements: resistivity values parallel and perpendicular to the lamination of the sand; NMR bulk volume irreducible water saturation; and/or the spherical permeability obtained from formation test tools porosity data.
The following general assumptions are made:
• the sand is clean;
• the bimodal laminated sand consists of a coarse and a fine component;
• the individual sand layers are isotropic; and
• the sand layers are hydrocarbon-bearing and at irreducible water saturation.
"Tensorial information" is used by resistivity measurements made in orthogonal directions, and the "averaged" NMR or formation test data is used as constraints to find the consistent solution from the equivalent solutions; it carries the "dimensional information."
The combined iterative interpretation gives the volume fraction, the BVI values, the permeability of the coarse and the fine fraction, and the horizontal and vertical permeability.
Conclusions
The relationship between resistivity and permeability anisotropy was investigated. Generally the observed resistivity anisotropy, though greater than 1, is less than 5. For permeability based on reservoir simulation practices and well test results, permeability anisotropy is more extreme.
The Archie and Coates equations provide a simple petrophysical connection between resistivity anisotropy and permeability anisotropy in case of bimodal sand. Generally, based on these equations, the permeability anisotropy is greater than the resistivity anisotropy. Only in a very special case are the two anisotropies equal.
Resistivity and permeability anisotropy are scale-dependent. Core scale permeability data can be upscaled to reservoir scales by simply mathematically averaging the kh data and harmonically averaging the kv data. The resulting permeability anisotropy is consistent with well test data. This suggests that permeability anisotropy computed from multicomponent induction data will be more consistent with reservoir scale permeability anisotropy.
Finally, the NMR-derived permeability data can be converted to vertical and horizontal permeabilities for bimodal sands. It is necessary to determine the fine and coarse volume fractions and can partition the porosity and BVI between the two components. This can be done with the aid of the T2-distribution data using the consistent solution methodology put forth by Schoen, et al.
References
Hagiwara, T., "Macroscopic Anisotropy Approach to Analysis of Thinly Laminated Sand/Shale Sequences: Sensitivity Analysis of Sand Resistivity Estimate and Environmental Corrections," Paper SPE 38669, 1997.
Klein, J.D., P.R. Martin, P.R Allen, D. F., "The Petrophysics of Electrically Anisotropic Reservoirs," SPWLA Paper HH.
Schoen, J. H., Liming Yu, and Georgi, D. T., "Aspects of Multicomponent Resistivity Data and Macroscopic Resistivity Anisotropy," Paper SPE 62909, 2000.
Schoen, J. H., Mollison, R. A., and Georgi, D. T., "Macroscopic Electrical Anisotropy of Laminated Reservoirs: A Tensor Resistivity Saturation Model," Paper SPE 56509.
Georgi, D. T., Harville, D. G., Phillips, C., and Ostroff, G. M., "Extrapolation of
Core Permeability Data with Wireline Logs to Uncored Intervals," SPWLA Paper KK, 2000.
Kriegshäuser, B., Fanini, O., Forgang, S., Itskovich, G., Rabinovich, M, Tabarovsky, L., Yu, L., Epov, M., and v. d. Horst, J., "A New Multicomponent Induction Logging Tool to Resolve Anisotropic Formations," SPWLA Paper D, 2000.

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