Journal of Petroleum Exploration and Production Technology
https://doi.org/10.1007/s13202-019-00788-1
ORIGINAL PAPER - EXPLORATION GEOPHYSICS
Application of geostatistics in facies modeling of Reservoir‑E, “Hatch
Field” offshore Niger Delta Basin, Nigeria
H. T. Jika1 · K. M. Onuoha1 · C. I. P. Dim1
Received: 26 March 2019 / Accepted: 30 September 2019
© The Author(s) 2019
Abstract
Lithofacies are very influential in the transmission of fluids within the reservoir. The objective of this study is to use geosta-
tistical techniques of sequential indicator simulation (SISIM) a variogram-based algorithm (VBA), single normal equation
simulation (SNESIM) and filter-based simulation (FILTERSIM) of multiple-point geostatistics (MPG) in developing realistic
facies model. A reservoir sand package “Reservoir-E” was correlated across five wells in the field. Synthetic seismogram of
well HT-1 was generated, and Horizon E picked on seismic section to produce time and depth surfaces of the reservoir. The
conditional if statement to generate lithofacies was applied on the extracted volume of shale data within “Reservoir-E,” and
the data were inputted in Stanford Geostatistics Modeling Software for facies modeling. The first realization from SISIM was
converted to a training image used for MPG. Visually, the MPG algorithm of SNESIM and FILTERSIM produced realization
that is substantially better and more realistic than the VBA of SISIM. The magnitude of correlation coefficients of algorithms
was carried out using the mean and variance of realizations, the results revealed mean and variance magnitude of correla-
tion coefficients between SISIM and SNESIM with 0.8933 and 0.9637, SISIM and FILTERSIM with 0.8639 and 0.5097
and SNESIM and FILTERSIM with 0.9717 and 0.8603. The results revealed a very good mean and variance magnitude
of correlation coefficients between SISIM and SNESIM; good between SISIM and FILTERSIM; and very good mean and
variance correlation coefficient between SNESIM and FILTERSIM. The qualitative interpretation of the model built with
SNESIM and FILTERSIM clearly detects lithofacies in the field which makes them a better algorithm in facies modeling.
Keywords Filter-based simulation · Sequential indicator simulation · Single normal equation simulation · Training image
Introduction became the foundation of geostatistical modeling, indicator
simulation (Journel and Alabert 1989), object-based mod-
Facies model is an important part in reservoir characteri- eling (Haldorsen and Damsleth 1990). Despite having some
zation (Aliakbar et al. 2016). The connectivity of facies is shortcomings, variogram-based and object-based algorithms
very influential in the flow of fluids. As the quest for pro- became popular in facies modeling over the years after their
ducing reliable facies models in reservoir characterization inventions (Hashemi et al. 2014). The popular variogram-
increases, several facies models have been developed by based methods are called two-point statistics (Liu et al.
different authors in reservoir characterization, yet only a 2004; Zhang 2008b) or traditional two-point geostatistics
few have explicitly characterized these reservoirs in terms (Deutsch and Journel 1998).
of their heterogeneity. Over the past years, different geo- Since reservoir properties are tied to facies and since
statistical approaches have been invented to achieve this facies are tied to fluid flow, it will be more appropriate
goal. Matheron (1973) introduced the basis on an algorithm to develop a facies model for the middle Miocene Reser-
that is known today as Gaussian simulation. It was after the voir-E in the Hatch Field before distributing properties;
mid-1980 that several efforts started on the algorithm that it is based on this note that the researcher is applying
sequential indicator simulation (SISIM), single normal
* H. T. Jika equation simulation (SNESIM) and filter-based simula-
jikahilary@yahoo.com tion (FILTERSIM) methods of two-point statistics and
multiple-point geostatistics (MPG) in the Hatch Field off-
1
Department of Geology, University of Nigeria, Nsukka, shore Niger Delta. The essence of the study is to produce
Nigeria
13
Vol.:(0123456789)
Journal of Petroleum Exploration and Production Technology
a realistic facies model of the field that will be condition Geology of the study area
to reservoir property distribution. Since variogram only
measures linear continuity, variogram algorithms such The Hatch Field is an oilfield in Nigeria. It was located in
as SISIM cannot capture curvilinear structures (Hashemi License block OPL XXX offshore Niger Delta (Fig. 1). The
et al. 2014). This is the most critical setback associated field covers approximately 154.24 km2 in an average water
with variogram-based algorithm. It is based on this limi- depth of 1000 m. The field was discovered in 1996, with
tation of variogram-based algorithms that the SNESIM government approval for its development given in 2002. The
and FILTERSIM algorithm methods of MPG will be Niger Delta clastic wedge is believed to be formed along a
incorporated in the work to actually capture these cur- failed arm of the triple junction that originally evolved dur-
vilinear structures. The MPG algorithms determine geo- ing the breakup of South American and African plates in the
logical uncertainty, template scanning and nonstationar- late Jurassic ((Burke et al. 1972) and (Whiteman 1982)). The
ity (Eskandaridalvand and Srinivasan 2010) in geological Niger Delta is a wave and tidal dominated delta (Weber and
scenarios. Daukoru 1975; Doust and Omatsola 1990). The Niger Delta
is divided into three formations, namely Benin, Agbada
and Akata Formations (Fig. 2), representing prograding
Fig. 1 Map showing location of Hatch Field offshore Niger Delta
13
Journal of Petroleum Exploration and Production Technology
Fig. 2 Stratigraphic column
showing formations in the Niger SOUTHWEST NORTHEAST
Delta (Modified from Owoyemi Quaternary
2004) Continental Alluvia Sand
Pliocene Deltaic Facies
(Benin FM)
(Agbada FM)
Late
Afam Clay
Soku Oprema
Middle Clay Channel
Complex
Buguma
Clay
Agbada
Clay
Early
(Agbada FM)
Marine Shales
(Akata FM)
Oligocene
Late Deltaic Facies
Middle
Early
Paleocene
Late
Cret.
AB-TU
Extent of erosional Truncation
depositional facies that are distinguished by most authors the environment of deposition into sand and shale the lithofa-
on the basis of sand/shale ratios. The type sections of these cies were also classified into sand and shale. The conditional
formations are described in (Short and Stauble 1967), and if statement used is Facie is equal to if volume of shale is less
the summary is given in numerous papers (Avbovbo 1978; than 0.05, 0.3, 0.1, 0.4 and 0.2 assign sand else shale for HT-1,
Doust and Omatsola 1990; Kulke 1995). HT-2, HT-4ST1, HT-3ST1 and HT-5. A simulation grid of
dimension 100*100*10 was designed to meet the required reso-
lution need of simulation. The first realization of SISIM was
Method of study converted to a training image that was used for SNESIM and
FILTERSIM using Stanford Geostatistical Modeling Software.
The data used for this study were constrained to well log data Facies models were developed from the computed volume of
for five wells (5) with their log suits, deviation data, check shot, shale using Eqs. 1 and 2 (Asquith and Krygowski 2004).
core data and seismic data. The seismic cube is rectangular with
GRLOG − GRMin
inline range of 2650–3809 and crossline range of 2520–3370. IGR = (1)
GRMax − GRMin
A reservoir package named Reservoir-E was delineated and
correlated across all the wells in the field. A seismic-to-well
IGR
tie was performed on the seismic section, and the Reservoir-E VSh = (2)
top Horizon picked to show the reservoir package on a seismic 3 − 2 ∗ IGR
section. The volume of shale was calculated across all the wells, where IGR = gamma ray index; GRlog = gamma ray reading
and conditional if statement was used to calculate the facies from log; GRmin = minimum gamma ray; GRmax = maximum
in all the wells. In the formulation of the facies model since gamma ray; and Vsh = volume of shale.
the plot of brine permeability versus core porosity classified
13
Journal of Petroleum Exploration and Production Technology
lithology type to occupy the empty pixel point, proceeding
to step one and repeating the process until the entire simula-
tion grid is simulated (Zhang 2008a; Yu and Li 2012). The
SISIM is not memory demanding like the multiple-point
geostatistics methods and takes less time for simulation as
compared to multiple-point geostatistics methods (Manchuk
et al. 2011). The SISIM generates models that the highs
are maximally disconnected from the lows, thus produc-
ing maximum entropy in their generated model (Caers and
Fig. 3 Training image from first realization of SISIM for Reservoir-E Zhang 2004).
in Hatch Field
Single normal equation simulation (SNESIM)
The magnitude of correlation coefficient (Zayed 2017) The single normal equation simulation algorithm (SNESIM)
between realization of two different algorithms A and E is developed by Strebelle (2000, 2002) is a multiple-point geo-
given by statistics method, from the use of training images. It is an
� � efficient pixel-based sequential simulation algorithm. The
� ∑ ∑ �
� m n (Amn − ̄ mn − E)
A)(E ̄ � training image exported can be conditioned to hard or soft
�
�𝛾� = � � � (3)
∑ ∑ ∑ ∑ � data or both.
� ( ̄ )(
2 ̄ ��
�
� m n (Amn − A) m n (Emn − E)
� For every location K = (x, y) along a random path, spa-
tial configuration of the local data values termed data
where Ā = mean 2(A), and Ē = mean 2(E). event is recorded. Replicate that matches this event is done
by scanning the training image. The replicates correspond-
ing to the central node values are used to calculate the
Sequential indicator simulation (SISIM)
The work flow for SISIM involves picking a pixel where a Table 1 Top and base of Reservoir-E in all the wells in Hatch Field
lithology type is unknown, identifies a neighboring pixel
with known lithology type, assigns weights to the neigh- Well name HT-1 HT-2 HT-3ST1 HT-4ST1 HT-5
boring points, constructs a local cumulative distribution Top (ft) 8076.39 9112.08 9541.75 8692.93 10,226.50
function (CDF) for the lithology type probability from the Base (ft) 8170.55 9150.50 9593.18 8754.08 10,329.50
neighboring lithology type, extracts from the CDF of single
Fig. 4 Reservoir-E correlation across wells in Hatch Field
13
Journal of Petroleum Exploration and Production Technology
Fig. 5 Seismic-to-well tie using checkshot of HT-1 Hatch Field
Fig. 6 Reservoir-E top picked on a 3D seismic volume of Hatch Field
13
Journal of Petroleum Exploration and Production Technology
Fig. 7 Horizon E time surface map for Hatch Field
conditional probability of the central value, given the data is far much less memory demanding, and it can handle
event. The implementation of SNESIM requires significant both categorical and continuous variables during simula-
CPU efficiency by performing this scanning before simula- tion. FILTERSIM uses linear filters to classify training
tion and saving the conditional probabilities in a dynamic patterns in a filter score space of reduced dimensions. The
data structure that is referred to as search tree. The method algorithm reduces pattern dimensions by applying spe-
is used successfully in facies model (Park et al. 2013). The cial designed filters and lowering the dimensional space
issue of memory is one among the limitations of SNESIM of the patterns. Coded patterns are then clustered, and a
algorithm. In complex 3D, multi-facies cases, where large prototype is chosen for each grid node. This speeds up the
training images with strong data connectivity, the RAM search process and reduces the run time of the algorithm
demand may exceed the current hardware system, thus pre- (Wu et al. 2008).
venting the algorithm from running. This limitation can The proposed FILTERSIM algorithm is actualized in
be overcome eventually by the continuously increasing three major steps (Wu et al. 2008): filter score calcula-
computer memory (Strebelle and Cavelius 2014). tion, pattern classification and pattern simulation. The pro-
cess works by applying a set of filters to the template data
obtained from scanning the training image. This produces
Filter‑based simulation (FILTERSIM) a set of filter score maps, with each training pattern repre-
sented by a vector of score values. This is done to actually
FILTERSIM is an MPS algorithm called filter-based simu- reduce the pattern data dimension from the template size to a
lation (Zhang et al. 2006), the algorithm was proposed smaller number of filter scores. The similar training patterns
to solve the challenges posed by SNESIM which handles are clustered into a so-called prototype class, each of this
only categorical variable in simulation, and it is memory class is being identified by a point-wise average pattern. In
demanding when the training image is large with a large the course of sequential simulation process, the conditioning
variety of different pattern. The FILTERSIM algorithm data event is retrieved with a search template of same size
13
Journal of Petroleum Exploration and Production Technology
Fig. 8 Horizon E depth surface map for Hatch Field
like the one used for scanning the training image. The pro- Training image
totype class similar to the conditioning data event is selected
using some distance function. A training pattern sampled Construction of 3D training image is challenging since
from that pattern class and pasted on the simulation grid. most geological pattern is either in 1D or 2D; nevertheless,
This simulation is actually based on pattern similarity. a code in MATLAB was used in the conversion of our train-
ing image from jpeg format to SGeMS format that will be
Fig. 9 Cross-plot of deposi-
tional facies in Hatch Field
13
Journal of Petroleum Exploration and Production Technology
Fig. 10 Upscale facies for
Reservoir-E for the five wells in
Hatch field
Fig. 11 SISIM realization for Reservoir-E Hatch Field Niger Delta
13
Journal of Petroleum Exploration and Production Technology
Fig. 12 SNESIM realization for Reservoir-E Hatch Field Niger Delta
acceptable by the Stanford Geostatistical Modeling software Results and discussion
in our modeling. In this study, the first realization of SISIM
was converted from jpg format to SGeMS format and was Well correlation
used as a training image (Fig. 3).
The five wells HT-1, HT-2, HT-3ST1, HT-4ST1 and HT-5
Hard data conditioning were loaded in Petrel environment displaying measured
depth, gamma ray (black) and resistivity (red) logs, respec-
Primary data are direct measurement of targeted reservoir tively. The gamma ray log was used for our lithofacies, and
properties; for example, well log data are typical example the resistivity log was used to identify the presence of hydro-
of a primary data sets. The training image used in this study carbons to confirm it as a reservoir. Reservoir-E was deline-
was conditioned to these hard data using Stanford Geosta- ated and correlated across all the wells (Fig. 4) to enable
tistical Modeling Software. us produce a realistic facies model of the Hatch Field. The
Reservoir-E tops and base in all the wells are presented in
Table 1.
13
Journal of Petroleum Exploration and Production Technology
Fig. 13 FILTERSIM realization for Reservoir-E Hatch Field Niger Delta
Seismic horizon interpretation while those with high permeability are classified as sand.
The plot reveals that the reservoir is dominated by sand
A seismic-to-well tie was generated (Fig. 5) with a good tie facies with smaller fractions of shale. This was the basis
and Horizon E picked as Reservoir-E top (Fig. 6). The hori- for our interpretation of classifying the lithofacies into sand
zons picked were used to create time and depth surfaces for and shale.
Reservoir-E (Figs. 7, 8). The surface map revealed a ridge like The volume of shale calculated gave a proportion of sand
structure (anticline) in the North-east and South-west that is to shale as 0.7232:0.2768. The result clearly shows that
separated by a syncline that trends North-west to South-east Reservoir-E is composed of 72.32% of sand and 27.68% of
that divides the anticlinal structures (Fig. 8). The anticlinal shale. These percentages reaffirm Reservoir-E as a potential
structures are areas of interest for hydrocarbon exploitation. reservoir in Hatch Field Niger Delta.
Volume of shale analysis SISIM, SNESIM and FILTERSIM algorithm
interpretations
The plot of brine permeability versus core porosity reveals
two facies in the depositional environment (Fig. 9). The The data loaded in Stanford Geostatistics Modeling Soft-
areas with lower permeability values are classified as shale, ware were up-scaled (Fig. 10), and the facies properties
13
Journal of Petroleum Exploration and Production Technology
Table 2 Mean and variance SISIM Realization 000 Realization 001 Realization 002 Realization 003 Realization 004
realization statistics for SISIM algorithm
algorithm statistics
Mean 0.28999 0.56274 0.56501 0.23354 0.36771
Variance 0.205898 0.246066 0.245776 0.179001 0.232502
Table 3 Mean and variance SNESIM algo- Realization 000 Realization 001 Realization 002 Realization 003 Realization 004
realization statistics for rithm statistics
SNESIM algorithm
Mean 0.55366 0.52457 0.54341 0.49417 0.53387
Variance 0.247123 0.249399 0.248118 0.249969 0.248855
Table 4 Mean and variance FILTERSIM Realization 000 Realization 001 Realization 002 Realization 003 Realization 004
realization statistics for algorithm sta-
FILTERSIM algorithm tistics
Mean 0.49685 0.5334 0.48363 0.49358 0.47783
Variance 0.249993 0.248887 0.249735 0.249961 0.249511
Table 5 Variance magnitude of correlation coefficient between SISIM and SNESIM, SISIM and FILTERSIM, and SNESIM and FILTERSIM
Realization SISIM SNESIM SISIM FILTERSIM SNESIM FILTERSIM
0 0.205898 0.247123 0.205898 0.249993 0.247123 0.249993
1 0.246066 0.249399 0.246066 0.248887 0.249399 0.248887
2 0.245776 0.248118 0.245776 0.249735 0.248118 0.249735
3 0.179001 0.249969 0.179001 0.249961 0.249969 0.249961
4 0.232502 0.248855 0.232502 0.249511 0.248855 0.249511
|Corrcoef(SISIM, SNESIM)| = 0.9637 |Corrcoef(SISIM, FILTERSIM)| = 0.5097 |Corrcoef(SNESIM, FILTER-
SIM)| = 0.8603
distributed using SISIM, SNESIM and FILTERSIM. The yielded a value of 0.9637 for mean and 0.8933 for variance,
SISIM realization revealed poor connectivity in the litho- 0.5097 and 0.8639 for SISIM and FILTERSIM, and 0.8603
facies distribution as compared to SNESIM and FILTER- and 0.9717 for SNESIM and FILTERSIM.
SIM (Fig. 11a–e). Qualitatively, the visual interpretation
of the MPG algorithm of SNESIM and FILTERSIM pro-
duces realizations that are distinctly better than the popular Conclusion
variogram-based model (two-point statistics) ((Fig. 12a–e)
and (Fig. 13a–e)). The five realizations for SNESIM and The study shows that Reservoir-E is characterized by chan-
FILTERSIM clearly show good connectivity in lithofacies nel sand with the presence of shale as seen from the realiza-
distribution within Reservoir-E of the Hatch Field. The mean tion of SNESIM and FILTERSIM. These sands will allow
and variance realization of the algorithms are presented in transmission of fluid from one point to another. The SNE-
Tables 2, 3 and 4. The magnitude of correlation coefficient SIM and FILTERSIM algorithm show good continuity in
of algorithms was calculated using variance and mean of lithofacies distribution when compared to the SISIM algo-
their realization as shown in Tables 5 and 6. The magni- rithm which considers X and Y direction neglecting the Z
tude of correlation coefficient between SISIM and SNESIM direction. MPG simulations produce explicit facies models
13
Journal of Petroleum Exploration and Production Technology
Table 6 Mean magnitude of correlation coefficient between SISIM and SNESIM, SISIM and FILTERSIM, and SNESIM and FILTERSIM
Realization SISIM SNESIM SISIM FILTERSIM SNESIM FILTERSIM
0 0.28999 0.55366 0.28999 0.49685 0.55366 0.49685
1 0.56274 0.52457 0.56274 0.5334 0.52457 0.5334
2 0.56501 0.54341 0.56501 0.48363 0.54341 0.48363
3 0.23354 0.49417 0.23354 0.49358 0.49417 0.49358
4 0.36771 0.53387 0.36771 0.47783 0.53387 0.47783
|Corrcoef(SISIM, SNESIM)| = 0.8933 |Corrcoef(SISIM, FILTERSIM)| = 0.8639 |Corrcoef(SNESIM, FILTER-
SIM)| = 0.9717
with sharp display of realistic geological constraints that Eskandaridalvand K, Srinivasan S (2010) Reservoir modelling of com-
are honored by sampling training image TI that was gener- plex geological systems—a multiple-point perspective. J Can Pet
Technol 49(8):59–68
ated using SISIM. The multiple-point geostatistics is able to Haldorsen H, Damsleth E (1990) Stochastic modeling. J Pet Technol
capture the X, Y and Z direction of Reservoir-E, thus giving 42:404–412
more realistic geologic images than the mostly used SISIM Hashemi S, Javaherian A, Ataee-pour M, Khoshde H (2014) Two-
algorithm. point versus multiple point geostatistics: the ability of geosta-
tistical methods to capture complex geobodies and their facies
associations—an application to a channelized carbonate reservoir,
Acknowledgements We wish to acknowledge the Almighty God for southwest Iran. J Geophys Eng. https://doi.org/10.1088/1742-
the strength and grace for carrying out this research. We are equally 2132/11/6/065002
grateful to Petroleum Technology Development Fund (PTDF) for their Journel AG, Alabert F (1989) Non-Gaussian data expansion in the
sponsorship not leaving out Department of Petroleum Resources (DPR) earth Sciences. Terra Nova 1:123–134
for their assistance in data gathering from international oil and gas Kulke H (1995) Regional petroleum geology of the world, part II:
companies in Nigeria. Africa, America, Australia and Antarctica. Gebrüder Borntrae-
ger, Berlin, pp 143–172
Open Access This article is distributed under the terms of the Crea- Liu XM, Xu JM, Zhang MK, Huang JH, Shi JC, Yu XF (2004) Appli-
tive Commons Attribution 4.0 International License (http://creativeco cation of geostatistics and GIS technique to characterize spatial
mmons.org/licenses/by/4.0/), which permits unrestricted use, distribu- variabilities of bioavailable micronutrients in paddy soils. Environ
tion, and reproduction in any medium, provided you give appropriate Geol 46:189–194
credit to the original author(s) and the source, provide a link to the Manchuk JG, Lyster SJ, Deutsch CV (2011) A comparative study of
Creative Commons license, and indicate if changes were made. simulation techniques with multiple point statistics: the MPS
beauty contest. Centre for Computational Geostatistics Report
13, 107. University of Alberta, Canada
Matheron G (1973) Intrinsic random functions and their applications.
References Adv Appl Probab 5(3):439–468
Owoyemi AOD (2004) The sequence stratigraphy of Niger Delta field,
Aliakbar B, Omid A, Abbas B, Meysam T (2016) Facies modeling of offshore Nigeria. MSc thesis, Texas A&M University
heterogeneous carbonates reservoirs by multiple point geostatis- Park H, Scheidt C, Fenwick D, Boucher A, Caers J (2013) History
tics. J Pet Sci Technol 6(2):56–65 matching and uncertainty quantification of facies models with
Asquith G, Krygowski D (2004) Basic well log analysis: methods in multiple geological interpretations. Comput Geosci 17:609–621
exploration series. AAPG 16:31–35 Short KC, Stauble AJ (1967) Outline of geology of Niger Delta. AAPG
Avbovbo AA (1978) Tertiary lithostratigraphy of Niger Delta. Geo- Bull 51(5):761–779
logic notes. AAPG Bull 62(2):297–306 Strebelle S (2000) Sequential simulation drawing structures from train-
Burke KC, Dessauvagie TFJ, Whiteman AJ (1972) Geological his- ing images. PhD thesis, Department of Geological and Environ-
tory of the Benue valley and adjacent areas. In: Dessauvagie TFJ, mental Sciences, Stanford University
Whiteman AJ (eds) African geology. Ibadan University Press, Strebelle S (2002) Conditional simulation of complex geological struc-
Ibadan, pp 187–205 tures using multiple-point statistics. Math Geol 34(1):1–21
Caers J, Zhang T (2004) Multiple-point geostatistics: a quantitative Strebelle S, Cavelius C (2014) Solving speed and memory issues in
vehicle for integrating geologic analogs into multiple reservoir multiple-point statistics simulation program SNESIM. Math Geo-
models. In: Grammer GM, Harris PM, Eberli GP (eds) Integra- sci 46:171–186
tion of outcrop and modern analogs in reservoir modeling. AAPG Weber KJ, Daukoru EM (1975) Petroleum geology of the Niger Delta.
Memoir. American Association of Petroleum Geologists, Tulsa, In: 9th World Petroleum Congress proceedings, Tokyo, vol 2, pp
pp 384–394 209–221
Deutsch CV, Journel AG (1998) Geostatistical software library Whiteman A (1982) Nigeria: its petroleum geology, resources, and
(GSLIB), 2nd edn. Oxford University Press, Oxford potential, vol 1–2. Graham and Totter, London
Doust DM, Omatsola E (1990) Divergent/passive margin basin. In: Wu J, Boucher A, Zhang T (2008) A SGeMS code for pattern simula-
Edwards JD, Santogrossi PA (eds) Niger Delta. AAPG Memoir, tion of continuous and categorical variables: FILTERSIM. Com-
vol 48. American Association of Petroleum Geologists, Tulsa, put Geosci 34(12):1863–1876
pp 239–248 Yu X, Li X (2012) The application of sequential indicator simula-
tion and sequential Gaussian simulation in modeling a case in
13
Journal of Petroleum Exploration and Production Technology
Jilin Oilfield. In: Deng W (ed) Future control and automation. Zhang T (2008b) Incorporating geological conceptual models and
Lecture notes in electrical engineering, vol 173. Springer, Berlin, interpretations into reservoir modeling using multiple-point geo-
pp 111–118 statistics. Earth Sci Front 15:26–35
Zayed MR (2017) Using entropy and 2-D correlation coefficient as Zhang T, Switzer P, Journel AG (2006) Filter-based classification of train-
measuring indices for impulsive noise reduction techniques. Int J ing image patterns for spatial simulation. Math Geosci 38(1):63–80
Appl Eng Res 12(21):11101–11106
Zhang J (2008a) Indicator and multivariate geostatistics for spatial Publisher’s Note Springer Nature remains neutral with regard to
prediction. Geospat Inf Sci 11(4):243–246 jurisdictional claims in published maps and institutional affiliations.
13
