The data set considered in this paper as secondary value of the modeling, is prepared based on post-stack inversion that provide acoustic Impedance AI property volume. AI data consist more information compared to the seismic data, since it is integrating several sources of data; typically seismic information, well logs and seismic velocity information. Seismic inversion technique increases the resolution of seismic data by reducing the effects of wavelet tuning and attenuating the random noise. Sonic and density well logs are essential to generate synthetic seismogram required for AI inversion.

There could be some problems in well logs data, such as wellbore washout, casing point, and different loggings run. To compensate these errors, all suspicious values of logs are corrected or omitted and then the intervals with no logs data, are estimated from other logs, based on artificial neural network approach. Investigation of the seismic data and interpreted horizons revealed the suitable quality of these data as input of AI inversion. After quality control of the input data, the well to seismic correlation and wavelet extraction from post-stack seismic data can be performed.

In this study, one wavelet extracted using five wells.

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Finally after generation of low-frequency model and determination of inversion parameters, constrained sparse spike inversion CSSI process was utilized on the seismic data. The study area is located at the junction of the Arabian Shield and Iranian continental block that belong to two different lithospheric plates Arabian and Eurasian.

The area is located in the northwest Persian Gulf Fig. Albian stage occupies one of the most important places in the Mesozoic stratigraphy of the Persian Gulf with high-frequency tectonic changes. A rapid sea level drop in early Albian initiated deposition of Kazhdumi and Burgan over the Dariyan and Shuaiba carbonates. In the deeper part of the basin where break in sedimentation is documented between Dariyan carbonates, and the Kazhdumi, deposition may have commenced in the Late Aptian Ghazban During this time, the Arabian Shield became the source for the clastic sediments in that area.

From the Arabian Shield to the Iranian coast, these clastics pass into the neritic argillaceous limestone and organic-rich shales Fig. Thickness of the formation increases toward northwest and east of Persian Gulf and decreases on the middle part. There is an interval of clastic sediments claystone, siltstone, and sandstone in the lower part of the Kazhdumi formation.

This unit was deposited during the transgression on post Aptian unconformity surface in a deltaic environment. The sandy part of this unit is hydrocarbon reservoir in some part of north of Persian Gulf. However, strong facies changes of the layer on one hand and poorly distributed data on other hand are the main challenge for modeling this reservoir. The Kazhdumi formation is time equivalent of three stratigraphic units in the western Persian Gulf countries. But In this work, to handle Kazhdumi Facies estimation, it was divided into five zones, Kz-1 to Kz-5, based on well data.

These five zones are equivalent to three previously known stratigraphic parts of the formation. Kz-1 and Kz-2 are equivalent to the upper and middle Kazhdumi, respectively, but Kz-3, Kz-4, and Kz-5 zones are all equivalent to the lower Kazhdumi Burgan formation. These three zones have different distributions in the study area. However, target of the study is stochastic lithofacies modeling for the Kz-4 in the area. The dataset is composed of 11 well data upscaled into two meters located in layer no. The location map can be seen in Fig.

For instance, some lithofacies are presented only one time in a well. Location map of the wells in Kazhdomi formation showing the distribution of lithofacies. Because of irregularity of the sampling mesh and preventing any bias in the calculation process, the proportion of each lithofacies is determined before and after declustering. As can be inspired form declustered proportion, the most covered lithofacies in the region is Sand, Sh 70, 30 and the least one is the sand located in western part of the region.

According to the geological interpretation, the lithofacies and should be at the left side of the region and the right side, the lithofacies 83 and are more dominated. This consideration is a key to validate the results. The contact relationship according to the geological consideration is defined as: alrgLst, Sh 70, 30 is in contact with Sh. Therefore, it is supposed that there should not be any geological relationship between the first lithofacies code: 83 and code: , , , and As mentioned earlier, in the case of complex contact relationship, the hierarchical plurigaussian simulation provides such a flexibility to handle the process of modeling.

So, one can see the conceptual hierarchical relationship of the underlying lithofacies in Fig. So, four sublayers can, therefore, be concluded as follow:. Hierarchical relationship between the lithofacies in Kazhdomi formation, layer no. First layer: lithofacies 83 is in direct relationship with lithofacies and the former one is in contact with its complement.

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## Open Geosciences

Second layer: lithofacies is in direct relationship with its lithofacies complements , , , and except lithofacies Third layer: lithofacies is in direct relationship with its lithofacies complements , , , and except lithofacies Fourth layer: lithofacies is in direct relationship with its complement , , , and except lithofacies Such a hierarchical contact based on the explained truncation rule involves five truncation thresholds, which can be defined corresponding to the declustered proportion Armstrong et al. Hence, Proportion of the first lithofacies threshold according to the Eq.

Indeed, the approximate anisotropy directions are inferred from two source of information: the seismic map of geophysical attribute and the nearest neighborhood interpolation map of the well data. Although the deterministic techniques such as nearest neighborhood cannot quantify the uncertainty at unsampled locations and provides just one exact boundary among two adjacent lithofacies, but they produce some intuitively useful consideration about the approximate layout of the lithofacies.

The next step is to fit theoretical variograms to the experimental variograms.

However, the experimental indicator variograms calculated from well data resulted in a very patchy sort of experimental points in the variogram graph and one cannot fit a proper model. This problem emerged due to the lack of adequate lithofacies data at well location. An idea is to use the seismic information instead, owing to the fact that it conveys abundant knowledge through the entire region Pyrcz and Deutsch As mentioned earlier, calibrated proportions by seismic information could propose the maps showing the probability for each lithofacies.

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Based on the most probable plot obtained from these probability maps, one can generate a categorical map Fig. This map bears a resemblance to the nearest neighborhood map and could also provide visual favorable information. The experimental indicator variograms are derived again, but this time from the most probable map so obtained.

The fitting process is a type of semi-automated algorithm presented by Emery It is worth mentioning that the fitted model of variogram, which is cubic, getting an impact from the smoothness of the contact lines between the lithofacies. This type of variogram leads to regular boundaries Lantuejoul Provided with the inferencing the model parameters truncation rule, truncation thresholds, and variogram analysis , to consideration whether the geophysical information has an impact on the probabilistic modeling of lithofacies domains, the geostatistical simulation of lithofacies has been implemented over three following cases:.

In this case, realizations have been generated, using a procedure applied by Maleki et al. In truncation rule, the declustered global proportions have been inserted to infer the truncation thresholds.

The truncation rule and variogram analysis remain also unchanged. In this case, realizations have been generated conditionally to the well data, but for truncation thresholds, the declustered global proportions have been used for lithofacies 83, , , , and for lithofacies and , the calibrated proportions have been used which acquired from seismic information. It should be noted that the proportions are then rescaled to sum to 1. In case I, the truncation rule is favorably recognized hierarchical relationship between lithofacies Figs. For instance, realization no. This phenomenon according to the geological consideration cannot be interpreted.

In case II, both realizations could not meet the truncation rule Figs. This problem arises in the interest of equal weighting thresholds from calibrated proportions in truncation rule. The heterogeneity for lithofacies somewhat is honored, but the realizations lack realism from the contact relationship point of view. But in case III, the results are more interesting, one can see from Figs. Furthermore, the extension of lithofacies 83 and as were expected agreed the geological understanding. Therefore, as mentioned earlier, the anticipation from geological consideration which met in this case is that, firstly, the lithofacies 83 should not have any relationship with others and, secondly, should demonstrate an extended region from north-east to south-east, sheltered by lithofacies Attention must be taken in this place that all the realization are in complete compatibility with well data locations Fig.

These realizations obtained from each case can then be post processed to calculate the uncertainty in the lithofacies at a local scale within the blocks , by means of probability maps. These maps present a hint for the risk of finding a lithofacies at unsampled and sampled locations Fig. The frequency of occurrence of each lithofacies can be calculated for each block in complement of geological interpretation.

The blue areas are those with high uncertainty and a low probability for a given lithofacies. In contrast, the red areas showing low uncertainty and high probability for finding that type of lithofacies. If you provide a validation field, things get a little complicated. In geostatistics, there is a distinction between the true process and the measured process. We assume that each location does have a true value, but when you attempt to measure it, you introduce measurement error. We say that the measured value is the sum of the true value plus random noise.

When you provide a validation field, that field is assumed to contain measured values, not true values. To validate against them correctly, you have to use the standard error of the measured process, which is always larger than the standard error of the true process. Twenty-two such data sets of size are simulated. If the user changes any of the default parameters of Then, the grid method is used to compute the em- the semivariogram and covariance models lag size, pirical semivariogram for each data set, and a spheri- range, nugget, and partial sill , then estimates are cal semivariogram model is subsequently fit to the recalculated beginning at stage 2.

## Peer reviewed publications by Michael Pyrcz

Additionally, REML is used to fit ods of fitting spatial semivariogram models exist, a spherical model to the empirical covariance, and the including maximum likelihood, restricted maximum result is subsequently converted to a semivariogram. Table 1.

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The average fitted semi- the spherical semivariogram model on each of the variogram is shown as the red curve in Figure 9, 22 simulated data sets using modified WLS. The pa- with lower and upper bounds given by the blue rameters are scattered around the true values given curves. These curves are calculated using the fol- in magenta.

The figure also suggests that the parame- lowing formulas: ters are correlated e. Estimates of the correlation between semivario- Figure 9 shows that, on average, both algorithms for gram parameters obtained using the modified WLS fitting the semivariogram worked quite well. How- dicated in the Table 1. An algorithm for obtaining a smooth semivariogram 1 X 22 X surface is presented below. For example, MISE 0,0. Three-dimensional scatter diagram of 22 sets of parameter estimates of the spherical semivariogram model.

Each estimate of the nugget, range, and partial sill is given in green, and the true value is given in magenta. Pa- implemented in commercial software packages e. Figure 11 presents examples dence, the algorithm gives very good defaults, which of empirical and smooth semivariogram surfaces.

This is particularly true for surface calculation. However, the user can control this situation by selecting different lag sizes with graphical tools. This chapter describes grid-based empirical semi- Cross-validation is also an important diagnostic in variograms and covariances, along with model fitting these instances. If the lag size is selected properly using modified WLS.

These procedures have been commonly, it is obvious how to manually improve Table 2. Estimates of correlation between the estimated parameters of the spherical semivariogram model for 22 simulated data sets. Figure Computer screen shot of an empirical semivariogram cloud and surface using a the grid method and b all semivariogram pairs and a smooth semivariogram surface using the same data set. The only serious drawback to this tech- of spatial data, in M. Moore, ed. Johnston, K. Ver Hoef, K. Krivoruchko, and N. Huijbregts, , Mining geo- statistics: London, Academic Press, p. Cressie, N. Vega, T.

Cardoso, and A. Shelly, least squares: Mathematical Geology, v. Unix: New York, Springer-Verlag, p. University Press, p. Kolmogorov, A. Botha, , Variogram fitting with sponding ordinary kriging predictors: Technometrics, a general class of conditionally nonnegative definite v. Related Papers. Space-time dependency modeling using general classes of flexible stationary variogram models. Modified weighted least squares semivariogram and covariance model fitting algorithm. By Konstantin Krivoruchko. By Guillaume Larocque. A comparison of approaches for valid variogram achievement.

By Manuel Febrero Bande. Spatiotemporal modeling and prediction of solar radiation. By Ernesto Lopez-Baeza. Download pdf. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link.