Journal of Petroleum Science and Engineering 29 , , Journal of Electronic Materials 37 6 , , Journal of Petroleum Science and Engineering 46 , , Stochastic simulation of patterns using distance-based pattern modeling M Honarkhah, J Caers Mathematical Geosciences 42 5 , , Representing spatial uncertainty using distances and kernels C Scheidt, J Caers Mathematical Geosciences 41 4 , , The probability perturbation method: a new look at Bayesian inverse modeling J Caers, T Hoffman Mathematical geology 38 1 , , Some channels are not able to be distinguished from each other.

Furthermore, the simulated facies do not agree with the real situation compared with figure 3 a. Usually, changing the random seed can acquire various facies models. Generally, selecting a satisfactory result needs hundreds of simulations when implementing inversion, according to the usual practice.

## Training images library - Multiple-point Geostatistics stochastic modeling with training images

The amount of calculations is large, and the simulation process is slow. Two realizations based on conditional simulation with two-point geostatistics. Two realizations based on conditional simulation with multi-point geostatistics. Different products are simulated by changing random seeds. Therefore, based on conditional simulation with multi-point geostatistics, we perturb the probability distribution to produce a new facies model. Then, we transform it into porosity volumes by random sampling, compute the impedance volumes by rock physics, synthesize seismic records by forward modeling, and compare these records with actual seismic data.

The rock physics relationship of the Stanford reservoir model can be analyzed easily by statistical means according to the training data A and B. However, the filed data is much more complicated. Utilizing a proper rock physics model is an alternative approach. The facies inverted figure 7 by the proposed method are output with 36 iterations.

Compared with simulated results figure 6 , the products after utilizing the probability perturbation method are more accurate because of the constraint by seismic data. Channel distribution has an excellent consistency with the model, and channel shapes are rational and easily identifiable. Furthermore, it agrees with the well data. At the same time, a pseudo porosity volume figure 8 a corresponding to the inverted facies figure 7 is also emerged. Figure 8 b is the error section between this porosity volume and the porosity of the model figure 3 b. It can be noted that this porosity volume is similar to the model map, but there are still some errors especially in the boundaries, where the errors are large.

Furthermore, the continuity of this volume is poor, and the characteristics of porosity in channels are not reflected well because we use random sampling without any modification and optimization when converting the facies into porosity.

If this volume is directly used as the inverted porosity, it will be detrimental to further modeling, accurate reservoir prediction, and fine reservoir description. Therefore, the porosity volume is not an excellent inverted result. We use it as an initial guess toward intelligent optimization. Although intelligent optimization does not rely on the initial model, a preferable initial model would contribute to narrowing the search range and reducing the amount of computation.

Considering its computational efficiency and accuracy, the quantum annealing algorithm is recommended. Inverted facies using the presented method.

### Description

The pseudo porosity volume generated in the probability perturbation stage a , and b its error relative to the real porosity. The difference between this porosity volume and the real map is apparent.

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The continuity is very poor and the error is great. Figure 9 displays the optimized porosity using the quantum annealing algorithm, and its error. We notice that the optimized porosity has a preferable consistency with the real situation, and the channels can be clearly distinguished. Compared with figure 8 , the boundary of the channel can be effectively recognized from figure 9 a , and the errors are decreased figure 9 b.

The continuity of this optimized porosity map is improved, and is more similar to reality. The optimized porosity presents various details, and is more accurate than the pseudo porosity volume generated in the probability perturbation stage, although not as perfect as the porosity map of the model figure 3 b around the boundaries. The channels can be well distinguished, and the continuity is improved. The inversion results are similar to the actual map.

We also test the effect of the simulated annealing algorithm, and analyze the convergence of the simulated and quantum annealing algorithms. Figure 10 shows the optimized porosity by the simulated annealing algorithm, and its error.

## Multiple-point Geostatistics: Stochastic Modeling with Training Images

These two optimized algorithms are performed based on the same initial model generated by the probability perturbation method figure 8 a , and with identical search ranges. Although the continuity is improved in figure 10 compared with figure 8 , the optimized porosity map is not as good as the optimized result by quantum annealing algorithm figure 9. The continuity of the porosity map optimized by the quantum annealing algorithm is better and more similar to the real porosity.

Figure 11 shows the convergent differences between these two algorithms. It can be noted that the objective function of the quantum annealing algorithm can converge to a smaller value, and is still decreasing as the number of iterations increases. However, the objective function of the simulated annealing algorithm only converges to a relatively large value, and then no longer changes. The histograms of errors between the results and real porosity with porosity produced after probability perturbation, quantum annealing, and simulated annealing are shown in figure The error in figure 12 b is clearly the most aggregated.

Although the channels can be identified, it is obvious that the continuity is worse than that in figure 9. The optimization is inferior to that obtained by using the quantum annealing algorithm. Value of objective function versus the number of iterations for quantum red and simulated blue annealing algorithms. It is obvious that the convergence of the quantum annealing algorithm is faster than that of the simulated annealing algorithm.

The simulated annealing algorithm gets into a local minimum, while the objective function value of the quantum annealing algorithm is still decreasing as the number of iterations increases. Histograms of error between real porosity and the porosity volume generated after a using the probability perturbation method, b optimized porosity by the quantum annealing algorithm, and c optimized porosity by the simulated annealing algorithm.

Finally, we apply the improved method to a two-dimensional seismic section from a Chinese onshore exploration area. The target area is dominated by sandstone, siltstone, and mudstone. Wells A and B are available, and they are employed as training data for conditional simulation with multi-point geostatistics. The training image is obtained by facies discriminant analysis according to the previous research about this work area. The seismogram is shown in figure 13 , and the black circles indicate locations of target reservoirs. Figures 14 and 15 show the actual logging curves and lithofacies interpretation information of two wells.

Three main facies can be identified: shale, brine sandstone, and oil sandstone. Due to the stationary hypothesis, we also utilize seismic probability information as secondary information when executing the SNESIM algorithm. Seismic constraint information can be obtained through many approaches such as Bayesian discriminant analysis e.

## Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics

Liu et al and so on. The relationship between reservoir properties and elastic attributes is established by the soft sandstone model Mavko et al Post stack profile of the field data from a Chinese work area. The black lines represent the well locations, and black circles denote the locations of the target reservoirs. Logging curves and interpreted facies of well A.

Logging curves and interpreted facies of well B. The final inverted results of facies and porosity figure 16 are produced with 52 iterations due to the complexity of the actual situation. It can be seen that the inverted result of lithofacies reflects the geological features of the area, and the target reservoirs can be predicted black circles. In addition, the target reservoir can also be identified from the inverted porosity map. The porosity value of oil sandstone is the biggest, and that of shale is the smallest. The inverted porosity can well distinguish the differences of porosity among different lithofacies.

### Shop with confidence

Moreover, the continuity has been improved compared with the pseudo porosity section generated by the probability perturbation method figure Figure 16 c is the synthetic seismogram; it is similar to the actual seismogram, which demonstrates the effectiveness of this improved method. The application to the field data fully demonstrates the feasibility and effectiveness of the proposed method.

Inverted facies and porosity: a facies; b porosity; and c synthetic seismogram.