# Iterative static modeling of channelized reservoirs using history-matched facies probability data and rejection of training image

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## Abstract

Most inverse reservoir modeling techniques require many forward simulations, and the posterior models cannot preserve geological features of prior models. This study proposes an iterative static modeling approach that utilizes dynamic data for rejecting an unsuitable training image (TI) among a set of TI candidates and for synthesizing history-matched pseudo-soft data. The proposed method is applied to two cases of channelized reservoirs, which have uncertainty in channel geometry such as direction, amplitude, and width. Distance-based clustering is applied to the initial models in total to select the qualified models efficiently. The mean of the qualified models is employed as a history-matched facies probability map in the next iteration of static models. Also, the most plausible TI is determined among TI candidates by rejecting other TIs during the iteration. The posterior models of the proposed method outperform updated models of ensemble Kalman filter (EnKF) and ensemble smoother (ES) because they describe the true facies connectivity with bimodal distribution and predict oil and water production with a reasonable range of uncertainty. In terms of simulation time, it requires 30 times of forward simulation in history matching, while the EnKF and ES need 9000 times and 200 times, respectively.

## Keywords

History-matched facies probability map Training image rejection Iterative static modeling Channelized reservoirs Multiple-point statistics History matching## 1 Introduction

For reliable decision-making in the petroleum industry, reservoir characterization is implemented to estimate the distribution of reservoir parameters of interest. Conventional reservoir characterization uses static and dynamic data in consecutive order. After initial reservoir models are generated from static data, they are simulated to predict reservoir performance, which should be compared with observed data. Dynamic data are used to modify initial models to minimize the difference in reservoir performance. Here, static data such as core data and well logs have a constant value with time, while dynamic data vary with time such as 4D seismic data and well oil production rate (WOPR).

In the case of static data, they represent reservoir parameters at certain location (hard data) or are closely related to them (soft data). These types of data are applied to build prior reservoir models via geostatistical methods. On the other hand, dynamic data are assimilated by inverse algorithms because dynamic data are indirectly related to reservoir parameters. In the case of channelized reservoirs, it is hard to generate reliable reservoir models by geostatistics and inverse modeling due to the following issues (Wang and Li 2011; Hou et al.2015; Kim et al. 2016a; Jo et al. 2017; Jung et al. 2017; Kang et al. 2017; Lee et al. 2017a): First, reservoir performances are complicated by a unique pattern of sand facies. Second, a histogram of reservoir properties has a bimodal distribution, not Gaussian distribution, because there is a stark contrast in the properties between sand and background facies.

To replicate the spatial connectivity of channelized sand facies, multiple-point statistics (MPS) are more appropriate than two-point statistics (TPS) (Strebelle 2002). The two statistics need spatial information, hard data, and soft data as input data. Here, hard data mean direct information about the reservoir parameters such as core and well log data, whereas soft data are indirect information such as facies probability and vertical proportion. The main difference between the two statistics is how spatial relation can be represented. Training image (TI) and variogram are used for MPS and TPS, respectively. TI contains a geological concept from the interpretation of the depositional environment, while variogram is based on mathematical calculations of spatial correlation (Journel 2002).

One of the advantages in MPS over TPS is that conditional probability from TI is easily coupled with soft data through the tau model (Kashib and Srinivasan 2006). TI gives an approximate pattern of facies distribution, while soft data provide constraint for each grid. However, if there are no available seismic data and sufficient geological interpretation, it is difficult to determine channel geometry for TI. This is why the previous studies used several TIs to consider uncertainty in a geological concept (Jafarpour and McLaughlin 2009; Scheidt and Caers 2009a, b; Lorentzen et al. 2012; Lee et al. 2013b, 2016). Therefore, characterization of TI and soft data is crucial, since the reliability of MPS highly depends on their quality.

Recently, a new paradigm has arisen where geostatistical input parameters are obtained by dynamic data (Agbalaka and Oliver 2011; Jafarpour and Khodabakhshi 2011; Astrakova and Oliver 2014; Tavakoli et al. 2014; Sebacher et al. 2015; Chang et al. 2016; Lee et al. 2017b; Kim et al. 2017). Conventional history matching is to characterize model parameters of interest, but updated models cannot preserve static information because inverse algorithms may ignore given hard data, soft data, and geological concept (Jafarpour and Khodabakhshi 2011; Hu et al. 2013; Hou et al. 2015; Satija et al. 2017). To solve this problem, updated models from inverse modeling are used to generate pseudo-static data which are coupled with static data given to rebuild reservoir models. This procedure has the same effect as history matching because regenerated models are based on both static data given and history-matched static data.

This novel approach still depends on the results of inverse algorithms to generate pseudo-static data. In other words, this iterative static modeling can give a meaningful result only if the inversion results are reliable. However, it is difficult with channelized reservoirs to ensure the reliability of the inversion results (Kim et al. 2016b; Lee et al. 2014). Furthermore, it causes a heavy burden on simulation time during inverse algorithms and extensive iterations (Queipo et al. 2002; Kang et al. 2016; Kim et al. 2016c).

In this research, we propose a novel iterative static modeling scheme for channelized reservoirs, which have uncertainty in channel geometry. For each iteration, global facies probability from TI is managed by TI rejection and local facies probability is updated by history-matched soft data. According to TI rejection and the soft data, three strategies are tested in two channelized reservoir cases to optimize the iterative static modeling.

## 2 Methodology

### 2.1 Procedure of the proposed method

*k*-means clustering are used and verified the suitability in Sect. 2.4. After the clustering, a representative model for each cluster is implemented for reservoir simulation instead of all the models. The best representative model, which has the lowest root mean square (RMS) error between simulated and observed dynamic data, is determined (Fig. 3d). Finally, the facies models surrounding the best model in the metric space are selected (Fig. 3e), and the mean of them becomes pseudo-soft data or final models (Fig. 2).

*F*represents the event of occurrence of a certain facies. TI stands for the probability from the TI for given well data, while SD means pseudo-soft data.

*τ*is a weighting on the information from the TI and facies probability map. If

*τ*

_{2}is greater than

*τ*

_{1}, the probability from TI has less influence than pseudo-soft data.

In Lee et al. (2017b), there is no uncertainty in TI (one TI) and iteration concept. In this research, various TIs are utilized to consider uncertainty in channel geometry. Whenever facies models are regenerated, TIs are rejected according to the proportion of TIs, which is used to build the selected facies models. Furthermore, a proper practice to generate pseudo-soft data is proposed in this research.

### 2.2 Three strategies for iterative static modeling

Three strategies for iterative static modeling

Strategy | Ensemble ratio for TIs | History-matched pseudo-soft data |
---|---|---|

1 | The same ratio for all TIs (no TI rejection) | Average of all qualified models for all TIs (unified soft data) |

2 | Proportion of TI used for generating qualified models (TI rejection) | Average of qualified models derived from the same TI (separated soft data) |

3 | Proportion of TI used for generating qualified models (TI rejection) | Average of all qualified models for all TIs (unified soft data) |

Strategies 2 and 3 differ in the way of generating pseudo-soft data. Strategy 2 distinguishes the qualified models based on TI and generates the separated soft data for each TI. For example, the mean of the 6 models from the TI 1 becomes pseudo-soft data for the TI 1 and the mean of the 4 models from the TI 2 is utilized as the soft data for the TI 2 (Fig. 4c). However, strategy 3 makes the unified soft data using all 10 qualified models regardless of the TIs (Fig. 4b). Strategy 1 also follows the same practice with strategy 3 for the soft data.

*α*. The RMS is calculated from the following equation [Eq. (2)]:

*i*th grid and \(N_{\text{grid}}\) is the number of grids. The superscripts c and p represent current and previous, respectively. In this research,

*α*of 3.5 is set empirically because RMS values in Tables 6 and 10 are about 3.5 when the strategies 2 and 3 are converged by the first convergence criterion. The convergence of the strategy 1, which does not use the TI rejection concept, can be judged based on the RMS value even though the convergence of the strategies 2 and 3 is determined by both criteria.

### 2.3 Synthetic channelized reservoir cases

Geological parameters for the static data integration by geostatistics

Parameters | Values | ||
---|---|---|---|

TI | Grid system | 250 × 250 × 1 | |

Grid size, ft |
| ||

Pattern | Orientation, ° | Uniform (−10, 10) | |

Amplitude, ft | Triangular (50, 80, 100) | ||

Width, ft | Constant (50) | ||

Wavelength, ft | Triangular (500, 800, 1000) | ||

SNESim | Grid system | 25 × 25 × 1 | |

Grid size, ft |
| ||

Hard data | Sand facies for all wells | ||

Search template | Nodes | 60 | |

Ranges | (min, max) = (300, 300) | ||

Facies ratio | (sand, shale) = (0.6, 0.4) |

Uncertainty for orientation of channel geometry in Example A

Parameters | TI 1 (default) | TI 2 | TI 3 | TI 4 |
---|---|---|---|---|

Orientation of pattern, ° | Triangular (−10, 10) | Triangular (35, 55) | Triangular (80, 100) | Triangular (125, 145) |

Uncertainty for width and amplitude of channel geometry in Example B

Parameters | TI 1 (default) | TI 2 | TI 3 | TI 4 |
---|---|---|---|---|

Width, ft | Constant (30) | Constant (30) | | |

Amplitude, ft | Triangular (50, 80, 100) | | Triangular (50, 80, 100) | |

*x*and

*y*directions. It is assumed that WOPR data are observed every 20 days up to the current time of 900 days (Fig. 7b). There are 360 observation data: multiplying the 8 production wells and the 45 observation time steps. The standard deviation for the observation data is set to 0.01 STB/day. The production wells P2, P6, and P7 show a sharp decrease in WOPR about 200 days because of fast water breakthrough due to the connectivity between the water injection well and the production wells. Channel characterization is very important for the prediction of reservoir performances because fluid movement is dominant in channelized sandstone rather than background shale.

Petrophysical parameters for the reservoir simulation

Parameters | Values | |
---|---|---|

Initial conditions | Pressure, psi | 2000 at 2700 ft |

Oil–water contact depth, ft | 3000 | |

Initial porosity, fraction | 0.3 | |

Water saturation, fraction | 0.1 | |

Permeability, mD | Sand facies | 2000 |

Background facies | 20 | |

Oil formation volume factor, rb/STB | 1.012 at 0 psig | |

1.011 at 1000 psig | ||

1.01 at 2000 psig | ||

Water formation volume factor, rb/STB | 1 at 2000 psig | |

Water compressibility, 1/psi | 5.00E−07 | |

Viscosity of fluid, cP | Oil | 3 |

Water | 1 | |

Density of fluid, lb/ft | Oil | 48.623 |

Water | 62.313 | |

Rock compressibility, 1/psi | 3.00E−05 at 2000 psig | |

Well constraint | Injection well: rate, STB/day | 400 |

Production well: BHP, psi | 500 |

For the proposed method, the three strategies in Sect. 2.2 are tested in Example A and the best strategy is examined in Example B. Also, the standard ensemble-based methods, ensemble Kalman filter (EnKF) and ensemble smoother (ES), are applied to both examples for comparison. We verify the result of the proposed method in view of facies distribution, permeability histogram, production predictions on the existing production wells and a newly drilled production well, and simulation time.

### 2.4 Verification of the distance-based method

*x-*and

*y*-axes are calculated as follows Eq. (3):

*N*

_{m}means the number of pairs of facies model, which satisfy given lag distance.

*H*

_{m}is the Hausdorff distance of the pairs.

*q*(

*M*

_{i}) stands for production data from the

*i*th facies model

*M*

_{i}, and

*N*

_{d}indicates the number of observed production data. For example, in Fig. 8b, the second point from the left has 68.08 ft and 399,820 STB

^{2}/day as the Hausdorff distance and experimental variogram of WOPR, respectively. It is calculated from 33 pairs of facies models, which have a Hausdorff distance from 65 to 70 ft.

## 3 Example A

### 3.1 Characterization of channel connectivity

^{11}mD, while the smallest value is less than 1.74 × 10

^{−7}mD: both values are physically unrealistic. Also, the histograms follow a Gaussian distribution, not the bimodal distribution in the reference field (Fig. 10a) due to the inherent assumption in the ensemble-based methods. These results clarify the problem of conventional inverse algorithms mentioned in Sect. 1 because they do not consider the geological meaning of reservoir parameters. It has been reported that this problem can be solved by techniques such as localization in many studies (Watanabe and Datta-Gupta 2012; Luo et al. 2018; Jung et al. 2018). Since this research is not a study to improve the ensemble-based method, only the standard ensemble-based methods are used as a comparison of the proposed method.

RMS between the mean of the qualified models in Example A

Strategy | Initial | Iteration 1 | Iteration 2 | Iteration 3 |
---|---|---|---|---|

1 | – | 5.02 | 4.08 | 3.35 |

2 | – | 6.17 | 3.61 | – |

3 | – | 6.15 | 3.70 | 3.36 |

The number of facies models per TI for total 200 models in Example A

Strategy | TI | Initial | Iteration 1 | Iteration 2 | Iteration 3 |
---|---|---|---|---|---|

1 | Ratio | 1:1:1:1 | 1:1:1:1 | 1:1:1:1 | 1:1:1:1 |

Number of models | 50:50:50:50 | 50:50:50:50 | 50:50:50:50 | 50:50:50:50 | |

2 | Ratio | 1:1:1:1 | 6:4:0:0 | 10:0:0:0 | – |

Number of models | 50:50:50:50 | 120:80:0:0 | 200:0:0:0 | – | |

3 | Ratio | 1:1:1:1 | 6:4:0:0 | 9:1:0:0 | 10:0:0:0 |

Number of models | 50:50:50:50 | 120:80:0:0 | 180:20:0:0 | 200:0:0:0 |

Regardless of the strategies in the proposed method, initial models are built from all TIs equally. As mentioned above, the strategy 1 generates 50 new models from each TI during iteration, but the other two strategies construct new facies models according to a certain ratio. The number of models from each TI is proportional to the qualified models in the previous step as explained in Sect. 2.2.

These differences result in final models in Fig. 9. The result of the strategy 1 has three vertical connectivities and shows the connection between the production well P6 and the injection well (Fig. 9e). Its histogram has a bimodal-like distribution in Fig. 10e. However, there is a huge uncertainty in the left area in the mean field, which indicates the variety of facies distribution in the final models. It results from usage of all TIs during iterative static modeling. It demonstrates the importance of a proper TI for MPS algorithms even though soft data can guide a local facies probability.

The final models from the strategies 2 and 3 have similar facies distribution with the reference field (Fig. 9f, g), and their histograms overcome the Gaussian problem in the ensemble-based methods (Fig. 10f, g). Technically, the results from the strategy 3 show better performance because the connection between the production well 6 and the injection well is clear, and the facies ratio in the histogram is similar to the reference field.

Insufficient results in the strategy 2 are caused by biased soft data for iterative static modeling in the early stage. The first-qualified models in Fig. 4a, which are chosen from the initial models, have much uncertainty because the initial models are generated with the 4 TIs and without integration of dynamic data. The reference field in Fig. 9a has mainly vertical channel streams (TI 1 in Fig. 5a) and 45° connection between the production well P6 and the injection well (TI 2 in Fig. 5a). Therefore, the first-qualified models consist of six models from the TI 1 and four models from the TI 2 (Fig. 4a). If pseudo-soft data are generated for the TIs 1 and 2 separately, the vertical TI is coupled with the vertical trend soft data and the 45° TI is used with the 45° trend soft data (Fig. 4c). This strategy 2 intensifies biased trends during iterative static modeling, and regenerated models can have improper facies distribution if the geological information is incorrect or has high uncertainty. The strategy 3 uses unified soft data in Fig. 4b, and it mitigates the robust tendency of the separated soft data in Fig. 4c.

### 3.2 Uncertainty quantification in production forecasts

The predictions from the initial models have a wide uncertainty range (Fig. 12a) because they are not integrated with observed dynamic data. Also, the average (the blue line) deviates from the true production trend due to uncertainty in the channel direction of the TIs. Especially, there are wide uncertainty ranges for WWCTs. In the case of the ES and EnKF, although the uncertainty ranges are reduced by the integration of dynamic data, the predictions are quite unreliable, such as WCT for both P4 and P6 wells (Fig. 12b, c), even worse than the result of the initial models. It results from the wrong updated models in Figs. 9 and 10. Example A is a very challenging inverse problem for the ensemble-based method because of the bimodal distribution and uncertainty in the TIs, which cause initial ensemble design issues (Jafarpour and McLaughlin 2009; Lee et al. 2013b, 2016).

All strategies for the proposed method demonstrate better predictions than both the initial models and ensemble-based methods (Fig. 12d–f). In the case of WWCT, the strategies 1 and 2 still have wide uncertainty ranges, even though the average can predict water breakthrough time and overall tendency properly. However, the predictions of the final models from the strategy 3 converged to the true productions with narrow bands. Also, they form a band of predictions without ensemble collapse problem, which all models become almost same after history matching. It is a natural result because the final models of the strategy 3 in Fig. 11b look similar to the reference field.

In the case of the proposed method and initial models, WWCTs increase sharply in the early stages because the injected water prefers to flow through high-permeability channel facies. However, water breakthrough time from both the ES and EnKF is delayed due to the Gaussian distribution (Fig. 12c). They fail to describe the fluid behavior of channel reservoirs with a bimodal distribution. This is why many previous studies have used transform techniques for the ensemble-based methods for channelized reservoir characterization.

Figure 13a shows the distribution of water saturation of the reference field at 900 days. The distribution is similar to the connectivity of sand facies because the injected water mainly moves through high-permeability sand facies. Therefore, understanding the channel distribution is critical to predict the performance of the new well P9. In the case of the initial models without any calibration, they have very diverse predictions and most of them start to produce water immediately at 900 days (Fig. 13b). It results from an overestimation of the channel connection between the injection well and the lower right of the field. The EnKF and ES cannot give a reliable prediction (Fig. 13c, d), which may lead to wrong decisions, because of the over- and undershooting problems and Gaussian distribution of the updated models.

The proposed method represents the movement of the injection water of the new well properly because it maintains the bimodal distribution with reasonable facies distribution. The strategy 1 makes meaningful WWCT predictions with significantly reduced uncertainty range compared to the initial models and the ensemble-based techniques (Fig. 13e). In the case of the strategies 2 and 3, water breakthrough time is predicted with a very high reliability (Fig. 13f, g). Also, its uncertainty is appropriate to provide a rational basis for decision-makings. For example, we can decide the location of an infill well based on these credible results. The outstanding success of the proposed method is proven through the facies distribution, its histogram, the preexisting production wells, and the infill well.

The number of forward simulations for the ensemble-based methods and the proposed method in Example A

Procedure | ES | EnKF | Strategies 1 and 3 | Strategy 2 |
---|---|---|---|---|

History matching | 200 | 9000 | 30 | 20 |

Prediction | 200 | 200 | 20 | 20 |

Total | 400 | 9200 | 50 | 40 |

However, the proposed method needs less than 30 forward simulations for history matching (Table 8). Only the center models are required to predict WOPRs for each iteration due to the distance-based clustering (see Fig. 2). The simulation number for history matching in the strategy 2 is less than the number in the strategies 1 and 3 because it stops iteration at the early stage due to the biased soft data (Table 7). The proposed method requires only 20 simulations in the prediction step (Table 8) as the below reasons. Instead of simulating all 200 regenerated models at the last iteration, 10 simulations for the center models are needed to find the final models. Then, future reservoir performances are predicted by only the final 10 models. Therefore, the proposed method is a very efficient methodology to integrate static and dynamic data and to estimate uncertainty ranges over the conventional methods.

## 4 Example B

### 4.1 Characterization of channel connectivity

^{21}mD, min: 5.381 × 10

^{−18}mD) in Fig. 15c. The mean of the updated models from the EnKF (Fig. 14d) has distinct vertical connections. It is a much better result than the result in Fig. 9d because both the uncertainty in TIs and initial ensemble design issue are relieved. Even though the ES is much faster than the EnKF (Table 8), EnKF shows much better performances than ES from the standpoint of an inverse stability. However, the result of the EnKF still cannot detect a detailed facies distribution such as the connection on the lower left area (Fig. 14d). Also, its histogram fails to preserve a bimodal distribution even if the over- and undershooting problems in the ES is settled down (Fig. 15d).

The number of facies models per TI for total 200 models in Example B

TI | Initial | Iteration 1 | Iteration 2 | Iteration 3 |
---|---|---|---|---|

Ratio | 1:1:1:1 | 5:2:3:0 | 8:0:2:0 | 10:0:0:0 |

Number of models | 50:50:50:50 | 100:40:60:0 | 160:0:40:0 | 200:0:0:0 |

Finally, the TI 1 (the true TI) generates 200 new models with the soft data from the second iteration (Fig. 16c) for the third iteration. The qualified models at the third iteration in Fig. 16f originated from the TI 1, of course. As TI converges, the iteration can be stopped and the qualified models become the final models. When the final models in Fig. 16f are compared with the qualified models from the initial stage in Fig. 16e, facies distribution has converged to have a specific tendency. It is a natural result because the soft data are assimilated by dynamic data and are integrated with static data given. Therefore, the mean of the final log-permeability models is quite similar to the true field (Fig. 14e), and its histogram can preserve a bimodal distribution of the channelized reservoirs (Fig. 15e).

RMS between the mean of the qualified models in Example B

Parameters | Initial | Iteration 1 | Iteration 2 | Iteration 3 |
---|---|---|---|---|

RMS | – | 6.17 | 3.62 | 3.40 |

### 4.2 Uncertainty quantification in production forecasts

*y*-axis in Fig. 17 is the ratio of simulated WWCT to the true WWCT at the end of the prediction time (1800 days). The black horizontal dotted line indicates the true WWCT. The

*x*-axis consists of the initial models, the ensemble-based methods, and the three strategies in both examples. For example, the box plots of the Example A’s initial models (Ex.A_Initial) are further from the black line than the ones of the Example B’s initial models (Ex.B_Initial) due to the higher uncertainty of channel geometry (Fig. 17).

In terms of the existing production wells, WWCTs of the production well P6 have larger uncertainties than those of the production well P4 (Fig. 17a, b) because the connectivity between P6 and the injection well is a critical feature in the reference field. If the connection is not considered, the WWCT of P6 must be underestimated compared to the true WWCT at 1800 days. Therefore, most cases in Fig. 17b predict lower than 1, which means smaller simulated WWCTs than the true values. In the case of the ES and EnKF, the results give completely wrong WWCTs for both existing wells even though it takes much more time to integrate dynamic data than the proposed method.

For the proposed method, the importance of the TI rejection scheme can be seen from the box plots of the strategy 1. They show better performance than the ensemble-based methods but still have too wide an uncertainty because the strategy 1 uses all TIs during the iteration. However, the strategies 2 and 3 in Example A and the strategy 3 in Example B reduce the uncertainty range significantly, and they can reflect the true WWCT. Also, uncertainty ranges for the new well are wider than ones of the existing wells because the new well is not included for determining soft data. Nevertheless, the proposed method can provide a trustworthy WWCT for decision-making.

## 5 Conclusions

In this research, a novel idea, iterative static modeling using history-matched soft data, is proposed and successfully applied to synthetic channelized reservoirs. The three strategies are tested to optimize the iteration procedure for the following two issues: usage of a TI rejection scheme and the unified or separated soft data. The iteration can be terminated according to the convergence of the TI or soft data. The distance-based clustering, which consists of the Hausdorff distance, MDS, and *k*-means clustering, is utilized to reduce the number of forward simulations.

We use several TIs to reflect the uncertainty in channel geometry. Example A deals with the effect of channel direction, and Example B considers the effect of channel amplitude and width. Among the parameters for channel geometry, the facies distribution and reservoir performances are influenced by the order of direction, amplitude, and width. In Example A, the unified soft data with TI rejection, the strategy 3, shows the best performance compared to the strategies 1 and 2. The concept of TI rejection can manage global trend of channel streams such as main channel direction. The unified soft data can mitigate the effect of biased information in the separated soft data at the early iteration.

The proposed method solves the problems in the conventional inverse algorithms. The strategy 3 for both examples can make a reliable facies distribution and conserve static data given such as bimodal distribution, facies ratio, hard data, and geological concepts in TI. The standard ensemble-based methods, ES and EnKF, fail to characterize channel fields and show over- and undershooting problems. From the standpoint of simulation time, the proposed method has an advantage over the ES and EnKF. In the case of the strategy 3, it requires only 30 times of forward simulation for history matching, while the EnKF and ES need 9000 times and 200 times, respectively.

The performances of the production wells are predicted from the updated models using the ensemble-based methods and the final models using the proposed method. WWCTs of the updated models from the ES and EnKF cannot mimic a sharp increase after water breakthrough due to a Gaussian distribution. Even if the ensemble-based methods take a long time for history matching, the predictions may lead to erroneous decisions because they give even worse predictions than the ones from initial models, which use static data only. However, the final models from the proposed method can provide reliable predictions with reduced uncertainty for both the existing production wells and the newly drilled production well.

From the result of the strategy 1, the significance of TI in MPS is confirmed once again because soft data such as a facies probability map can provide information about the guideline level only. If the quality of TI is very low, setting tau1 to be less than tau2 may increase the effect of pseudo-soft data over TI on facies modeling.

## Notes

### Acknowledgements

This work was supported by Korea Institute of Geoscience and Mineral Resources (Project No. GP2017-024) and Ministry of Trade and Industry [Project No. NP2017-021 (20172510102090)]. B. Min was funded by National Research Foundation of Korea (NRF) Grants (Nos. NRF-2017R1C1B5017767, NRF-2017K2A9A1A01092734). The authors also thank the Institute of Engineering Research at Seoul National University, Korea.

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