MEAN (Ave) – SD (Sigma) – Reservoir Heterogeneity:
Given the fact that the probability density function for permeability remains log-normal, leaving aside arithmetic-mean, can a simple geometric-mean or harmonic-mean be used - in order to deduce - the mean value of permeability, while characterizing a heterogeneous petroleum reservoir?
If yes, what will be the value of permeability "at the interface" between two different beds/layers with significantly differing permeability?
If no, should we need to consider the ranges of standard-deviation as well, apart from considering the mean-value of reservoir-permeability, while characterizing the degree of reservoir heterogeneity?
A simple answer to your first question is no, not in a general sense. Using geometric means correspond to the assumption of random spatial distribution which is generally speaking untrue. Simple answer to your second question can be that permeability is in essence a property at a given scale. At zero scale length (interface), permeability is meaningless.
Geometric mean is sensitive to the lower values. Therefore, any outliners in your data will yield the results far more pessimist. It is ideal for the estimation of horizontal permeability but some errors are there for horizontal permeability as well.
Simple arithmetic mean is ideal for additive properties like porosity, saturation and Net-to-gross thickness ratio but not for permeability as you indicated. While harmonic mean is for the layered systems.
Generally, the situation for the estimation variation from optimist to pessimist is as follows
Arithmetic Mean > Geometric Mean > Harmonic Mean
However, when a heterogeneous system is considered the best estimates can be made using the combination of Arithmetic and Harmonic Mean. Directional Permeabilities can be estimated as follows
Minimum Permeability: Arithmetic Mean of the Harmonic Mean of Rows which is usually zero
Maximum Permeability: Harmonic Mean of the Arithmetic Mean of Columns which yields the maximum averaged permeability (while upscaling).
Then arithmetic mean of maximum permeability and minimum permeability can be used to estimate the upscaled value of permeability for that particular strata.
Regarding the agreement between the derived averaged value and other values of permeability, one can use Pearson Coefficient of Covariance.
These are all my learning from class of geostatistics. Hope it may be useful to you. If there is some confusion regarding the estimation of maximum and minimum permeabilities, I can share an example with you
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