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from the datasets conventionally used in reser-
voir modelling.
Improved treatment of variance in reservoir
modelling is clearly needed and presents us with
a significant challenge. The statistical basis for
treating population variance as a function of
sample support volume is well established with
the concept of Dispersion Variance (Isaaks and
Srivastava 1989 ), where:
needed to handle and upscale the data in order to
derive an appropriate average. Assuming that we
have datasets which can be related to the REV's in
the rock system, we can then use the same multi-
scale framework to guide the modelling length
scales. Reservoir model grid-cell dimensions
should ideally be determined by the REV
lengthscales. Explicit spatial variations in the
model (at scales larger than the grid cell) are
then focussed on representing property variations
that cannot be captured by averages. To put this
concept in its simplest form consider the follow-
ing modelling steps and assumptions:
1. From pore scale to lithofacies scale: Pore-scale
models (or measurements) are made at the
pore-scale REV and then spatial variation
at the lithofacies scale is modelled (using
deterministic/probabilistic methods) to estimate
rock properties at the lithofacies-scale REV.
2. From lithofacies scale to geomodel scale.
Lithofacies-scale models (or measurements)
are made at the lithofacies-scale REV and then
spatial variation at the geological architecture
scale is modelled (using deterministic/probabi-
listic methods) to estimate reservoir properties
at the scale of the geological-unit REV (equiva-
lent to geological model elements).
3. From geomodel to full-field reservoir simula-
tor. Representative geological model
elements are modelled at the full-field reser-
voir simulator scale to estimate dynamic flow
behaviour based on reservoir properties that
have been correctly upscaled and are (arguably)
There is no doubt that multi-scale modelling
within a multi-scale REV framework is a chal-
lenging process, but it is nevertheless much pre-
ferred to 'throwing in' some weakly-correlated
random noise into an arbitrary reservoir grid and
hoping for a reasonable outcome. The essence of
good reservoir model design is that it is based on
some sound geological concepts, an appreciation
of flow physics, and a multi-scale approach to
determining statistically representative properties.
Every reservoir system is somewhat unique,
so the best way to apply this approach method is
try it out on real cases. Some of these are
illustrated in the following sections, but consider
trying Exercise 4.2 for your own case study.
2 b
ðÞ ¼ ˃
within blocks
between blocks
where a, b and c represent different sample
supports (in this case, a
point values, b
block values and c
total model domain).
The variance adjustment factor, f , is defined as
the ratio of block variance to point variance and
can be used to estimate the correct variance to be
applied to a blocked dataset. For the example
dataset (Table 4.2 , Fig. 4.26 ) the variance adjust-
ment factor is around 0.8 for both scale adjust-
ment steps.
With additive properties, such as porosity,
treatment of variance in multi-scale datasets is
relatively straightforward. However, it is much
more of a challenge with permeability data as
flow boundary conditions are an essential aspect
of estimating an upscaled permeability value
(see Chap. 3 ) . Multi-scale geological modelling
is an attempt to represent smaller scale structure
and variability as an upscaled block permeability
value. In this process, the principles guiding
appropriate flow upscaling are essential. How-
ever, improved treatment of variance is also crit-
ical. There is, for example, little point rigorously
upscaling a core plug sample dataset if it is
known that that dataset is a poor representation
of the true population variance.
The best approach to this rather complex prob-
lem, is to review the available data within a multi-
scale REV framework (Fig. 4.24 ). If the dataset is
sampled at a scale close to the corresponding
REV, then it can be considered as fairly reliable
and representative data. If however, the dataset is
clearly not sampled at the REV (and is in fact
recording a highly variable property) then care is
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