Geoscience Reference

In-Depth Information

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)

representative.

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

2

2
b

˃

ðÞ
¼
˃

a

;

c

ðÞþ

a

;

b

˃

ðÞ

;

c

Total

Variance

Variance

variance

within blocks

between blocks

ð

4

:

14

Þ

where a, b and c represent different sample

supports (in this case, a

¼

point values, b

¼

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

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

¼