Benchmark study using a multi-scale, multi-methodological approach for the petrophysical characterization of reservoir sandstones

Haruzi, Peleg; Katsman, Regina; Halisch, Matthias; Waldmann, Nicolas; Spiro, Baruch

doi:https://doi.org/10.5194/se-12-665-2021

Articles | Volume 12, issue 3

https://doi.org/10.5194/se-12-665-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/se-12-665-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 12, issue 3

Method article

|

19 Mar 2021

Method article |

| 19 Mar 2021

Benchmark study using a multi-scale, multi-methodological approach for the petrophysical characterization of reservoir sandstones

Peleg Haruzi, Regina Katsman, Matthias Halisch, Nicolas Waldmann, and Baruch Spiro

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Final revised paper (published on 19 Mar 2021)
Supplement to the final revised paper
Preprint (discussion started on 04 May 2020)
Supplement to the preprint

Interactive discussion

Status: closed

AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment

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- Supplement

RC1: 'Manuscript review', Anonymous Referee #1, 02 Aug 2020
- AC1: 'Response to Reviewer #1 (attached as a pdf file)', Regina Katsman, 27 Aug 2020
RC2: 'Benchmark study using a multi-scale, multi-methodological approach for the petrophysical characterization of reservoir sandstones', Anonymous Referee #2, 11 Aug 2020
- AC2: 'Response to Reviewer #2 (attached as a pdf file)', Regina Katsman, 27 Aug 2020
EC1: 'Decision on manuscript se-2020-47', Joachim Gottsmann, 12 Aug 2020

Peer-review completion

AR: Author's response | RR: Referee report | ED: Editor decision

AR by Regina Katsman on behalf of the Authors (24 Sep 2020)

ED: Referee Nomination & Report Request started (01 Oct 2020) by Joachim Gottsmann

RR by Anonymous Referee #1 (01 Dec 2020)

Suggestions for revision or reasons for rejection

While the variographic analysis of the three samples has been
re-performed, partially accepting the suggestions of my first review,
there still are quite relevant problems with how it was done and the
resulting interpretation by the authors which can't be published as is
in my opinion.

- In the variograms of figure 12(a) and (b) and all of Fig. 14 I can
only recognize pure nugget effect combined with high-frequency
fluctuations. The computation of the experimental variograms in
terms of retained "bins" and "lags" must be probably adjusted (here
I mean: increased to supports larger than these high frequencies
fluctuations, and not following an incremental pixel by pixel lags.
In other words: the original image should probably be "regolarized"
in order to apply variographic analysis) in order to smooth them
out. Furthermore, it's not clear which are the calculated
experimental values near the origin, since they are covered by the
model line. In any case, relying on the fitted range of the gaussian
models as measure of correlation length for the porosity in these
samples is in my opinion non supported by the experimental
evidences. First of all, the gaussian model is always a suspect one
to the geostatistician, since realizations of the corresponding
random function are infinitely often differentiable (they are
analytic functions): this is contradictory with their randomness
(see Matheron 1972, C-53, p73-74; available at
http://cg.ensmp.fr/bibliotheque/public ). Secondly, a practical
argument for rejecting this variogram model in the presented cases
is the fact that at the origin it is supposed to have
near-horizontal behavior (roughly, an S-shaped function). The
calculated experimental variograms do not display such behavior,
with possible exception of figure 18(b).

- For the same graphs, it would help to have the experimental
variograms plotted as points at the centres of each actually
computed bin instead of (what it seems) regularly spaced points,
which I can't tell if they correspond to the actual computed bins or
if it's a graphical artifact.

- The z-direction in fig. 12(c) shows a clear nugget effect, I don't
see why the authors don't acknowledge that. The same applies to fig.
18(b).

- The de-trending of the sample S1 if I understand correctly, has been
done just subtracting the average of the porosity, and not using the
"external drift" represented by the cement vol. percent, which is
actually provided by figure 20 and is an excellent observation. So,
when doing a linear detrend the authors should use a linear model
based on the cement volume-% in direction z. Even better however,
since it's clear from figure 11(c) that we are in presence of a
clear stratification with a period of little less than 4 mm in z
direction (corresponding indeed to the minimum value of the
variogram for distances at little less than 4 mm), it's clear from
the beginning that a linear de-trend would not be sufficient in this
case. A solution would be to apply here a higher order de-trend
(polynomial but also smoothing splines or loess...), so to focus on
the fluctuations around the middle and eliminate the hole effect
alltogether.

- Concerning figure 19 and the whole study about it, it seems to me
that it just refers to subsets of the original sample, in which the
variograms were still computed slice by slice, whereas I initially
suggested to use 3D "supports" for the computation of the
experimental variograms, and not restrictions of the domain. I am
not convinced that this part adds valuable informations to the
analysis at the moment.

Given the flaws of this variographic analysis it's not surprising to
me that the "apparent", interpreted range is ten times smaller than
the classical REV determination. The authors should consider either
doing it properly, possibly involving some experienced geostatistician
in the analysis, or else leave it alltogether.

I salute however the new simulations of permeability within the whole
sample S3. In my opinion this part should not be in appendix but
discussed in more detail in the main text.

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ED: Reconsider after major revisions (02 Dec 2020) by Joachim Gottsmann

AR by Regina Katsman on behalf of the Authors (15 Jan 2021) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (24 Jan 2021) by Joachim Gottsmann

RR by Anonymous Referee #1 (04 Feb 2021)

Suggestions for revision or reasons for rejection

To me, this revision is satisfactory and can be now published as is. I
read also with interest the replies of the authors to my concerns in
matter experimental variogram determination and I decided to respond
to some of them here.

1) Experimental semivariogram values near the origin: we now added to
the semivariogram of samples S1-S3 (Figures 12_new, 14_new, 18_new,
below) insets that focus on the short lag distances (subplots d-f
in the Figures 12_new, 14_new, 18_new below). Now, it can be seen
that near the origin a nugget effect is not observed and the
experimental semivariograms are characterized by a parabolic
behavior (red dots), which implies a continuity (due to the
smoothing occurred when averaging over a slice) of the investigated
property and with a continuous slope. We remind that the data
investigated in the semivariograms are an averaged porosity along
incremental slices (whereas the averaging is applied on binarized
voxel value: ‘1’ – pore or ‘0’ – grain) with a single voxel width
along (x-,y-,z-) directions (Figure 11,13,17 in the manuscript).
The parabolic behavior near the origin is fitted well by a Gaussian
model (blue dashed lines). Therefore, calculating the experimental
semivariogram based on slice-by-slice porosity is an appropriate
tool for variogram interpretation, a tool that for the best of our
knowledge was not used in previous studies using CT data.

I believe the authors fail to recognize that they were trying to
establish a geostatistical model for the pore space variability which
ends up violating fundamental assumptions. In the case of the new
12_new pictures, it should be evident that a gaussian variogram with
such a small effective correlation length (0.1 mm) is not adequate to
describe - in geostatistical sense - the sample at the scale they
would like to. Basically the authors say that trespassing a distance
h=0.1 mm, knowing the slice porosity at location x does not correlate
with the slice porosity located at x+h. How can such a model be
meaningful for a sample which is 3x3x4 mm large? Zooming in the
experimental variogram at such small scale to highlight the actual
"s-shaped" form of the experimental variogram near the origin actually
reinforces my argument. With the available sample discretization and
the slice-by-slice evaluation of variogram, the only actually sensible
conclusion is that we are in presence of pure nugget effect with some
nested periodic fluctuations in directions x- and y-, and that there
is a macroscopic stratification in z- that makes impossible to assume
stationarity of order two of the underlying random variable (slice
porosity).

2) We think, therefore, that regularizing is not needed for
semivariogram calculation based on slice-by-slice porosity of CT
data.

Again I strongly disagree. The regularization *by slice* is not suited
to describe variability in this sample. Possibly, defining a
3-dimensional support (a cube!) for the evaluation of porosity,
support which must be large enough to let some continuity in the
computed porosity when ranging across the whole sample, would give
meaningful results and be suitable for variography at the scale of the
sample, provided that it captures some spatial variability which can
be successfully described by an order two stationary RV.

3) The average distance from peak-to-trough in the semivariogram is
about 0.1 mm, with a similar value to the range of correlation that
was calibrated with the Gaussian model for all samples and in all
directions.

This statement again reinforces my opinion: the chosen support for the
description of local porosity is not adequate to represent spatial
variability at the sample scale. This means that the resulting model
is at best useless in geostatistical sense (since it is basically a
pure nugget with some periodic stratifications). Try and apply, e.g.,
kriging with this model: if you are at a distance of around h < 0.2 mm
you can estimate the next slice as being roughly equal to the known
one, and afterwards there is no usable spatial correlation (since the
variogram goes abruptly to the sill, which is equal to the population
variance). Basically, with this model, kriging would give the same
results as a IDW estimation.

Furthermore, the theory of variogram maintains that it is a monotonic
function of distance. If this is not the case, as with the large hole
effects the authors present in their response, it means that something
fundamental such as, i.e., stationarity of order 2 is violated. A hole
effect is a typical example of that. That can be addressed in many
ways, none of which the authors adequately pursued in the paper
revisions which I received.

4) Author response 5

I meant 3D support, not voxel-by-voxel, which is however already a
better approach. The vx-by-vx approach suffers from some of the same
flaws as the "slice-by-slice" approach, since a voxel is clearly way
too small to be locally meaningful. A 3D support is a cube of voxels,
whose sides are to be adjusted in order to make a "continuous"
porosity emerge when the cube moves within the whole sample. The
resulting "cube porosity" could be possibly used as basis for
variography and inform a spatial model at the sample scale.

Nevertheless, with respect to the new figure 21 in the response,
suddenly there is no gaussian / s-shaped behavior of the experimental
variograms near the origin, contradicting their own previous results,
and actually confirming once more my reserves. Did the authors try and
understand why is that? Maybe the apparent experimental gaussian slice
variogram results from an unsuitable choice of support for the
porosity...?

5) Nevertheless, the semivariogram is a useful tool to estimate
distances of correlation and lack of correlation between
measurements. Determining this range of correlation is useful to
estimate the scale of heterogeneity, and is not directly related to
the directional REV. It is more an apparent range, influenced by
cementation, clay minerals and grains. For S3, this apparent range
is shorter compared to the REV determined by the classical
approach. The shorter apparent range from the semivariogram is in
our opinion clearly related to the grain packing, the only
structural phenomenon at the size of the sample domain under
investigation. As a rule of thumb, one would use at least four (4)
grain diameters as REV edge size.

I clearly agree that variography IS indeed a useful tool, BUT it must
be done properly in order to lead to solid conclusions. I disagree
however with the "physical" explanation of why the author's results
came out like they did, in S3 and in the other stone samples. There is
no doubt that the variograms obtained in this way are not usable,
because they are not representative of a spatial variability at the
desired scale, hence not meaningful, irrespective of the degree of
cementation and clay fractions in their samples. Instead of
"discussing away" my technical critiques to an unusable model with a
rather philosophical physical interpretation, the authors should
improve the fundaments of their variographic model itself. As we all
know, "all models are wrong, some of them are useful".

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ED: Publish as is (05 Feb 2021) by Joachim Gottsmann

ED: Publish as is (05 Feb 2021) by Joachim Gottsmann (Executive editor)

AR by Regina Katsman on behalf of the Authors (09 Feb 2021)

Short summary

In this paper, we evaluate a multi-methodological approach for the comprehensive characterization of reservoir sandstones. The approach enables identification of links between rock permeability and textural and topological rock descriptors quantified at microscale. It is applied to study samples from three sandstone layers of Lower Cretaceous age in northern Israel, which differ in features observed at the outcrop, hand specimen and micro-CT scales, and leads to their accurate characterization.