# How to Upscale Permeability

Permeability is a particular difficult property to upscale for different reasons. For one, the property has a logarithmic scale making the extreme high or low permeability’s have a much larger impact depending on the averaging method. The method used for averaging should ideally be based on the reservoirs character. Permeability is a vector and is typically very different for flow in different directions; during upscaling the flow direction thus needs to be taken into account. Permeability is, more often than not, anisotropic. Lastly, permeability can not be accurately measured from logs and is typically only available from core plugs or well test. Both techniques measure permeability at different scales and do not measure the exact same permeability.

The data you’re using to upscale (the log) is already based on many assumptions and is very error prone and uncertain; the only way to validate your results are via well test (or dynamic simulation) which actually are based on

effective permeability. In this article you will be taken through the methods and assumptions for upscaling permeability and the most important QC steps to perform.

What is Permeability?As permeability is a measure of the rock’s ability to flow fluid it controls the rate of flow (together with a fluids viscosity). It can thus be considered a vector and therefore the direction of flow becomes important when averaging. Usually only horizontal and vertical flow directions are considered and any anisotropic behaviour in the horizontal plane is neglected. Therefore vertical permeability (Kv) and horizontal permeability (Kh) are the most common.

Different averaging techniquesThere are several methods available for averaging (as shown below), but for permeability upscaling the most typical are: Arithmetic, Harmonic and Geometric.

**Arithmetic average**As a general rule Arithmetic averaging is used when flow is parallel to the main permeability changes. For horizontal permeability this can thus be used for most sedimentary rocks due to the horizontal bedding and general laminar layering. If permeability has a log linear relationship with porosity at log scale, the upscaled values will lie slightly above this line, as the large values have larger impact on flow as the low permeability values.

**Harmonic average**The harmonic average is more representative when flow is perpendicular to the main permeability changes. In most cases this is thus used to upscale vertical permeability. For horizontal permeability the harmonic average is hardly ever used, but may be applicable in very steep dipping layers or fractured systems. If permeability has a log linear relationship with porosity at log scale, the upscaled values will lie slightly below this line, as the low permeability values control the flowrates.

**Geometric Average**The geometric average is used when there is no apparent preference for vertical or horizontal flow, the rock has no significant anisotropy for flow. This is often used in carbonates, ut could also be true in well sorted sandstones, with hardly any vertical baffles. Also fractured systems might have these characteristics. If permeability has a log linear relationship with porosity at log scale, the upscaled values will lie approximately on this line, as this method will try to retain both high and low values.

Below you find a image depicted these different averaging techniques.

Upscaling Permeability from Core to LogPermeability can not be measured by logging (some Magnetic resonance techniques are used, but rarely offer a good result) and can only be accurately measured on core plugs. The measurements from these core plugs are usually linked to a drillers depth and to start off with this drillers depth needs to be adjusted with a depth shift to match the core to logging depth.

The Petrophysicist then usually checks if the log porosity matches the core porosity, while taking into account the differences between core calculated (intermediate)

total and effective porosity. If this relationship is clear a relationship between the core porosity and permeability is sought. Usually a log-linear relationship is sough (either split by facies or zones), but as logs sample the formation on a lower resolution (~30cm) the core data needs to be averaged/upscaled to be able to compare. Usually the same assumptions as described above are used for this averaging either by using an arithmetic, harmonic or geometric regression technique or by binning the different porosity classes and applying a arithmetic, harmonic or geometric average to all the permeabilities in this bin. A regression is then run through the averages of each bin. Once the log linear relationship is defined this can be applied to the porosity log and a permeability log is generated. The binning technique is shown below. The orange points are the Arithmetic average, the light blue points the Harmonic average.

Other techniques that are sometimes used are the FZI method (Flow Zone Indicator) or by using Neural networks so multiple logs can be used to predict permeability. If sturdy relationships can not be found, different facies might have to be used.

After a permeability log has been generated it can be QC-ed by crossplotting it against saturation logs and investigating if it confirms capillary behavior (via saturation height functions) and confirms the cut-off (eg. permeabilities above 0.5mD are indeed Net Reservoir).

Upscaling Permeability from Log to static ModelWhen upscaling from log to the cells of the static model usually the same averaging technique is used as applied by the petrophysicist to upscale from core to log. However, when the reservoirs character at log resolution is very different to that at cell resolution this could be reconsidered.

To make sure that only the permeability of the Net Reservoir is upscaled the permeability log should be set to undefined values below the

Net-to-gross cut off. In this case also a NtG log should be created with a value of 0 for non-net reservoir and 1 for net-reservoir. This log should also be upscaled arithmetically and can theoretically result in 0 or low NtG in cells that have high permeability’s. At first this may seem strange, but in this case the cell represent a very thin sand with a high permeability, which when looking back at the log is exactly what has been upscaled.

To ensure cells with certain facies do actually get the permeabiliteis associated with those facies you should use bias during upscaling. In this case only the length of log corresponding to the facies in the upscaled cell is used in the averaging. An example is shown below.

To check your upscaling result first visually inspect your logs and check if important heterogeneities have been preserved. If not, apply a finer layering scheme. You should also create a crossplot of upscaled permeability versus upscaled porosity and compare it to the crossplot at the log resolution. For arithmetic averaging your upscaled points hould be higher than the log points, for harmonic averaging lower and for geometric averaging approximately the same.

If you have

DST data you can now sum the horizontal permeability over the perforated interval height and compare it with the KH from the well test. A reservoir engineer should be involved to understand the differences that will most likely be present.

Some geologists also compare the log scale Variogram to the upscaled Variogram. Another technique is using Lorenz plots. Both techniques require experience and understanding of the effects of permeability acting at different scales.

Upscaling Permeability from Static to Dynamic Model When upscaling permeability from a fine scale geological model to a coarse scale dynamic model to speed up run times several considerations need to be made. First of all the cell sizes need to be such that they still capture heterogeneities if they are important for the model. If there are very thin thief zones or very thin but extensive vertical baffles than these will most likely need to be modeled. In those case one can also consider using a larger lateral cell size to speed up run times.

When upscaling from fine to coarse scale cells, there are two basic techniques that can be used. The most accurate is using flow based upscaling in which the flow through the larger cell is matched to the simulated flow through the fine scale cells by adjusting the permeability. A more simple technique is by using the same averaging techniques again as explained above (Harmonic, Arithmetic or Geometric), but if you have also models NtG you will have to volume weigh the permeability to this NtG property. This is something what most modeling packages can do.

Permeabiltiy-porosity Crossplots and visual inspections form the most used QC steps, but after this upscaling step the dynamic behavior becomes more important and calibration to well test KH or well behaviour and productivity are paramount.