1. Maria Volkova, Mikhail Perepechkin, Evgeniy Kovalevskiy*
A Comparative Analysis of Geostatistical Methods for a Field with a Large Number of Wells
Maria Volkova, Mikhail Perepechkin, Evgeniy Kovalevskiy*
Gridpoint Dynamics JSC, Moscow, Russia

2. Trial Field Trial field, Geometrical about 200 wells Framework
As a trial field was used a part of an existing large field in Western Siberia. The picture shows a trial field with about 200 wells. Next, the picture shows a geometrical framework of a geological 3D model (as a pile of stratigraphical surfaces), resulting from the correlation of the well log data and the subsequent interpolation of the stratigraphic marks. The frame is shown in the paleoreconstruction where the surfaces are transformed to the horizontal planes. The frame is used as a base for constructing a stratigraphic grid of 100x100x170 (170 cells on Z).
Trial field,
about 200 wells
Geometrical
Framework
(paleoreconstructed
view)

3. Initial Data ASP (normalized SP) well-log values
Used as initial data for interpolation are ASP (normalized SP) well-log values that are closely linked with the porosity parameter and are presented along the entire section of each well. Hereinafter, the stratigraphic grid of the model and initial data will remain unchanged. What will change are the ASP cube interpolation methods.
ASP (normalized SP)
well-log values

4. Starting point: deterministic interpolation
The result of a deterministic quasi 3D (layer-by-layer) interpolation of the ASP well-logging curves calculated with a 1/R2 weighting. Red colour approximately correspond to sands and green colour – to clays

5. Deterministic interpolation: the quality is very low
First - this interpolation does not take into account the categorical nature of the environment, being therefore incapable of reproducing a well data histogram. The biggest error is seen for the AV4 horizon.
Second, this interpolation incorrectly displays the variability of the environment. If the range of the horizontal variogram for borehole data is about 1000 m, its range for the interpolated values is about 2000 m.

6. Stochastic Interpolation by Sequential Gaussian Simulation
with Normal Score Transformation
SGS with NS enables the exact reproduction in space of histogram and variogram of initial data

7. Stochastic Interpolation by Sequential Gaussian Simulation
(with Normal Score Transformation) : the quality is high, but…
This method allows fitting of the histograms only for the entire cube. To prove that this method can lead to errors, it is quite enough to look at the local results it can deliver

8. The Essence of the Normal Score Transformation

9. Stochastic Interpolation by Multiple Point Statistics
The calculation was carried out using the Direct Sampling algorithm (Mariethoz G., Renard Ph., & Straubhaar J., 2010). As a training image was used a fragment of the deterministic model with the highest density of wells

10. Stochastic Interpolation by Multiple Point Statistics: the quality is high, but…
The interpolation by MPS excellently reproduces variogram and gives satisfactory reproduction of histogram (even locally). The quality of interpolation depends on the quality of Training Image. The quality of our TI can be argued, but it is the best TI we can obtain

11. Stochastic Interpolation by Multiple Point Statistics: the quality is high, but…
Error
Error
The AV1 horizon is modeled almost correctly. The method allows dealing with nonstationary data and a nonstationary training image. At the same time, the method leads to some errors. The well data does not indicate that the AV5 horizon reservoir has a gap

12. Stochastic Interpolation Based on the Fuzzy Model
The uncertainty arises locally from the contradiction in the surrounding data. In other details the algorithm is very similar to a standard sequential simulation

13. Stochastic Interpolation Based on the Fuzzy Model
The fuzzy model realization looks like “categorical” interpolation of quantitative data. Note, values in space are contrast

14. Stochastic Interpolation Based on the Fuzzy Model: the quality is high, but…
The well data histograms and variograms are reproduced well, even locally. The problem arises when the data are spaced very irregularly. At this circumstances we have to restrict evidence interpolation radius. After that the model begins to reproduce the local statistics

15. Deterministic interpolation. The meandering facies are present
Deterministic interpolation. The meandering facies are present. Realization of SGS. The meandering facies are absent. SGS erases deterministic features, presented in data! SGS realizations are “sterile”. That is why geologists do not like them. Realization of MPS. The meandering facies are present Realization of the fuzzy model. The meandering facies are present. Facies are not statistically uniform.

16. Two realizations (SGS, MPS, Fuzzy Model)

17. Sequential Gaussian Simulation vs Fuzzy Model

18. Multiple Point Statistics vs Fuzzy Model

19. Sequential Gaussian Simulation vs Multiple Point Statistics

20. A Comparative Analysis of Interpolation Methods for a Field with a Large Number of Wells
Conclusions
1. The deterministic interpolation does not reproduce histograms, nor variograms of well data The interpolation by SGS with NS excellently reproduces variogram and total histogram. At the same time, if local histograms have distinctions, the method gives errors. 3. The interpolation by MPS excellently reproduces variogram and gives satisfactory reproduction of histogram (even locally). The quality of interpolation depends on the quality of Training Image. The method can produce artifacts The interpolation as realizations of the fuzzy model excellently reproduces variogram and histogram (even locally). The method seems to be very effective for a field with a large number of wells.

21. Maria Volkova, Mikhail Perepechkin, Evgeniy Kovalevskiy*
A Comparative Analysis of Geostatistical Methods for a Field with a Large Number of Wells
Maria Volkova, Mikhail Perepechkin, Evgeniy Kovalevskiy*
Thank you for attention
Gridpoint Dynamics JSC, Moscow, Russia