**4.2. Results**

The first sets of 32 images of acoustic impedances are generated with the direct sequential simulation conditioned to well data (AI) and the chosen variogram model. In the first iteration, the generated acoustic models (figure 5) are not constrained to any soft data, so

**Figure 5.** Acoustic Impedance model of simulation #1 (left) and #15 (right).

**Figure 6.** Correspondent Synthetic Seismic model of simulation #1 (left) and #15 (right).

only the wells are reproduced. This causes a high variability between the synthetic models (figure 6) and wide range standard deviation when calculated using all 32 simulations.

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**4.2. Results** 

As one can notice, there is a very good match with the synthetic amplitude (red line on right side) calculated with the acoustic impedance (white line in the middle) from the wells and the real seismic amplitude extracted from the seismic cube (cyan line in the right side).

The first sets of 32 images of acoustic impedances are generated with the direct sequential simulation conditioned to well data (AI) and the chosen variogram model. In the first iteration, the generated acoustic models (figure 5) are not constrained to any soft data, so

**Figure 5.** Acoustic Impedance model of simulation #1 (left) and #15 (right).

**Figure 6.** Correspondent Synthetic Seismic model of simulation #1 (left) and #15 (right).

As it can be noticed, the two models have a totally different spatial distribution, although the histograms and variograms are the same.

That different spatial distribution is visible in the correlation cubes between the synthetic models and the real seismic (figure 7).

**Figure 7.** Correlation cubes between the synthetic seismic model and the real seismic, of simulation #1 (left) and #15 (right).

**Figure 8.** Average (left) and Standard Deviation (right) of acoustic impedance of all 32 simulations of this iteration.

To confirm this variability, the average cube and standard deviation (figure 8) of all 32 simulations were determined and the influence of the wells in the average of the unconditioned simulations is highly visible principally around well 1, as the variogram is reproducing the well log spatial variability.

As perceived in the average model, the small scale variability has disappeared, however this cube can be considered the Low Frequency Model, because it reproduces the main trends that are represented in the wells logs. Also noticeable the practically constant value of standard deviation, representing the variability of unconditional stochastic simulations, except the small decline of values in the middle of the figure, which is the influence of the well 1.

Still, these first 32 simulations were used to build the "Best Correlation Cube" and "Best Acoustic Impedance cube" that will be used as soft data for the next iteration (figure 9).

**Figure 9.** "Best Acoustic Impedance cube" (left) and "Best Correlation Cube" (right) derived from first iteration.

It is visible the delineation of the structural layers in the Best Acoustic Impedance model (BAI) and in the Best Correlation Cube (BCC) a selection of correlation coefficients high values.

This assembled acoustic impedance model, has lost all the spatial distribution that the original acoustic impedance models had (figure 5), but this is a simple intermediate result and not the final one since these will be used as secondary or soft data for the next iteration that will impose the variogram spatial distribution and wells global histogram.

The algorithm has made six iterations, one of them, the first, was unconditional to any secondary or soft data, only the last five had the Best Correlation Cube and Best Acoustic Impedance cube as secondary or soft data imposition.

Since the algorithm will always choose the best genes from each iteration, the patterns that are in the real seismic will start to become more visible in each iteration and the correlation will became higher and more continuous in all cube positions.

The convergence of the process is inevitable until a local maximum correlation is attained (figure 10). This maximum does not represent the best that the original seismic can produce, but rather a simulation that the entire process has created. If one has run the same process with the same data but with a different seed for the generation of the acoustic impedance model the result could be a different one, but not so totally different.

**Figure 10.** Convergence progress of the algorithm.

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well 1.

iteration.

values.

reproducing the well log spatial variability.

To confirm this variability, the average cube and standard deviation (figure 8) of all 32 simulations were determined and the influence of the wells in the average of the unconditioned simulations is highly visible principally around well 1, as the variogram is

As perceived in the average model, the small scale variability has disappeared, however this cube can be considered the Low Frequency Model, because it reproduces the main trends that are represented in the wells logs. Also noticeable the practically constant value of standard deviation, representing the variability of unconditional stochastic simulations, except the small decline of values in the middle of the figure, which is the influence of the

Still, these first 32 simulations were used to build the "Best Correlation Cube" and "Best Acoustic Impedance cube" that will be used as soft data for the next iteration (figure 9).

**Figure 9.** "Best Acoustic Impedance cube" (left) and "Best Correlation Cube" (right) derived from first

It is visible the delineation of the structural layers in the Best Acoustic Impedance model (BAI) and in the Best Correlation Cube (BCC) a selection of correlation coefficients high

This assembled acoustic impedance model, has lost all the spatial distribution that the original acoustic impedance models had (figure 5), but this is a simple intermediate result and not the final one since these will be used as secondary or soft data for the next iteration

The algorithm has made six iterations, one of them, the first, was unconditional to any secondary or soft data, only the last five had the Best Correlation Cube and Best Acoustic

that will impose the variogram spatial distribution and wells global histogram.

Impedance cube as secondary or soft data imposition.

The 32 images of acoustic impedances of iteration 5 (considering that the first iteration is called 0, because it is an unconditional to soft data) are generated with the direct sequential co-simulation conditioned to well data (acoustic impedances), the chosen variogram model and the soft or secondary data (BAI and BCC) of the forth iteration.

One can clearly see the fast convergence from iteration zero to iteration 1 and afterwards the process starts to stabilize. The algorithm chooses the parts that have higher correlation values in the end of iteration 0 and after iteration 1, almost every simulation has its correlation values around 1, making the selection of each part of simulation, a very detailed event (sometimes in the third of forth decimal of the correlation value).

The obtained final results demonstrate that the conditional data is imposing a very strong effect in all 32 simulations. The variability of different models generated with different seeds is now almost none existing (figure 11 and 12) and they practically look the same.

As it can be noticed the two models have an almost equally spatial distribution, but some differences can be found, since they are two independent realizations.

**Figure 11.** Acoustic Impedance model of simulation #3 (left) and #28 (right).

**Figure 12.** Correspondent Synthetic Seismic model of simulation #3 (left) and #28 (right).

Those small differences are easier to distinguish in the correlation cubes between the synthetic models and the real seismic (figure 13), since the correlation coefficient is very sensible to little variations in patterns.

In these examples the layer set sizes were big enough to show the differences.

Integration of Seismic Information in Reservoir Models: Global Stochastic Inversion 135

**Figure 11.** Acoustic Impedance model of simulation #3 (left) and #28 (right).

**Figure 12.** Correspondent Synthetic Seismic model of simulation #3 (left) and #28 (right).

In these examples the layer set sizes were big enough to show the differences.

sensible to little variations in patterns.

Those small differences are easier to distinguish in the correlation cubes between the synthetic models and the real seismic (figure 13), since the correlation coefficient is very

**Figure 13.** Correlation cubes between the synthetic seismic model and the real seismic, of simulation 3 (left) and 28 (right).

To confirm this lack of variability (figure 14), the average cube is almost equal to a simulation, and validated by the lower values of standard deviation of all 32 simulations.

**Figure 14.** Average and standard deviation of acoustic impedance of all 32 simulations of this iteration.

In figure 14 the same color scale for standard deviation, was used, for comparison with the standard deviation of the first iteration model (figure 8).

The values of standard deviation for the final iteration only vary from 0 to 6900, with an average of 1600 which is a reduction of almost 80% in variability, comparing with the variation between 0 and 11500 and average of 8000 of the first iteration.

The influence of the wells is no longer visible because all data of wells are integrated in the full model.
