Field-scale permeability distribution does not have an unique description but it includes normal-distribution, log-normal distribution, multi-modal distribution and Levy stable distribution.
Despite having such a complex background, we generally approximate the expression of permeability variation as a simple function of square of mean grain size. Complex physics with such a simple solution.
Where are we heading towards?
What could be the possible fundamental problem?
Is it because - given a complicated lithology and geologic structures of a given geological formation - deducing a meaningful Representative Elementary Volume (REV) almost becomes impossible as the nature and intensity of heterogeneity keeps increasing with the scale of the problem - without attaining the asymptoticity?
In fact, the problem becomes extremely complex for multi-phase fluid flow - where - permeability becomes a function of fluid property as well on top of reservoir property - and thereby fundamentally violating from Darcy's approach.
Is it fair to follow Darcy's approach while neglecting his fundamental assumuptions?
Well,
Hydraulic conductivity= K= k(rg/v)
Fluidity= rg/v
Permeability=k=Cd2 , where C; porous media characteristic, d; grain size
r; liquid density [M L-3]
g; acceleration of gravity [L t-2]
v; viscosity coefficient of the liquid in [Pas] [M L-1 t-1]
Fluidity at 20oC is: 9.81x106 [m s]-1. Note ; permeability has dimensions of L2.
While in nature the heterogeneity is the dominants, here the distribution of field scale permeability differs from place to another due to fluidity and heterogeneity.
Sincerely
Natural fractures and bedding planes are the most prominent features that interfere with scaling up small (core size) measurements to blocks that are used in reservoir simulation. Weather one is using dual permeability models or simple matrix models, there are ways to incorporate those effects into up-scaled permeability vectors. Although at the field scale level, it would be ideal to represent permeability as a tensor, the most common finite difference formulations require a simple XYZ vector representation. Given the large uncertainties involved in the measurement of permeability, the stratigraphic complexities of laminations and natural fractures, in the case of very tight formations, we might need to reconsider our approaches in solving reservoir engineering problems. The idea of addressing these uncertainties seperately and producing a combined probabilistic output constrained is probably the reasonable thing to do. However, as the flow behavior of natural fractures is influenced by changes in stresses (and pore pressure), field scale permeability is a dynamic property.
An answer from operational perspective. The conventional process to handle field scale permeability rely upon discrete non sortable variables (aka facies, rock types) and handling petrophysical properties conditioned to these facies including permeability and relative permeability. Such approach is reasonably efficent in most cases (but quite cumbersome). It is not perfect for the reasons mentionned in the question. Some improvement might be gained by stopping the ridiculous practice of handling relative permeability in tabular manner but rather treat it as a 3 phase parameteric functions whose control parameters (e.g. Corey exponent and endpoints in very simplistic 2 phase approach) vary in 3D space and by streamlining the cross correlations we develop in different facies to make sure they tie in petrophysical multispace with other facies. To conclude, I dont think there is enough evidence of on-shelf solutions being inappropriate to revisit altogether the Darcy structure even thought I agree with the observations made in the question. So yest it is fair.
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