## 1 Introduction

In the petroleum industry, the essence of exploration is to convert undiscovered resources to recoverable reserves. Petroleum potential modeling is an important preliminary step in petroleum exploration. There are several approaches for assessing petroleum potential of an area, which can be generally classified as data-driven, knowledge-driven, and hybrid approaches. Data-driven models involve quantitative analysis of spatial relationships between discovered petroleum pools in a region of interest and indirect evidence of petroleum potential. The resulting relationships are used to determine the parameters of the model by which evidence data sets are integrated into a single petroleum potential map. In contrast, knowledge-driven methods rely on the judgments of experts who evaluate the relative importance of input data sets and the model parameters [1].

Despite the fundamental differences, there are the methodological similarities between the mineral prospectivity mapping (MPM) and petroleum potential modeling. It is accordingly worthwhile to first provide a brief summary of the commonly used data- and knowledge-driven methods for predicting mineral prospectivity that can potentially be used to evaluate petroleum resource potential. In recent decades, several data-driven methods have been developed and successfully applied in MPM such as logistic regression [2], weights of evidence (WofE) [3], fuzzy WofE [4], boost WofE [5], support vector machine [6], artificial neural networks [6], Bayesian network classifiers [7], decision tree analysis [8], random forests [9], isolation forest [10], certainty factor [11], extreme learning machines [12], and maximum entropy [13]. Boolean logic, index overlay [14], wildcat mapping [15], fuzzy logic [16], data envelopment analysis [17], PROMETHEE [1], ELECTRE [18], AHP [19], and TOPSIS [20] are examples of knowledge-driven methods. It is noteworthy that some of the methods like evidential belief functions (EBFs) can be implemented in both frameworks in order to map mineral prospectivity (i.e., data-driven EBF [21] and knowledge-driven EBF [22]).

In the past few years, some researchers have attempted to evaluate petroleum potential using various algorithms. Zargani et al. [23] applied the WofE method successfully to delineate the areas with the highest potential in the Murzuq Basin of Libya for more investigation. Source rock, reservoir rock, migration, and trapping as the main geological factors were combined in a geographic information system (GIS) framework to produce a probability map of hydrocarbon occurrence. The resulting posterior probability map showed that 64% of petroleum accumulations were located in the favorable regions. An approximate fuzzy assessment (AFA) system on the strength of the possibility and fuzzy set theories for petroleum resource evaluation was developed by Tounsi [24]. Although the AFA system considered all essential petroleum system elements and could efficiently deal with imprecise information and incomplete data, it investigated petroleum favorability in a nonspatial framework. Bingham et al. [25] introduced a GIS-based fuzzy multicriteria evaluation model for producing the petroleum potential map. The proposed method was used for frontier exploration areas of northern South America because it works without relying on prior probabilities of the presence of petroleum accumulations. Arab Amiri et al. [26] utilized two data-driven methods, EBF and frequency ratio, to predict the potential distribution of petroleum resources in the Williston Basin. Seven subcriteria of the Red River petroleum system elements were selected as input data sets. The findings indicated that the frequency ratio achieved slightly better performance than the EBF. Ziyong et al. [27] employed a GIS-based fuzzy logic for integrating geological, geophysical, and simulation data in order to investigate the gas potential of the Chu-Sarysu Basin in Kazakhstan. The results of the study were validated by in situ gas testing. A geochemical study was conducted by Lei et al. [28] to delineate the anomaly areas of the Sangnan field in northwestern China for predicting the spatial distribution of petroleum resources. Fifteen petroleum indices from surface soil samples were classified into three classes according to principal component analysis (PCA) and multifractality. Petroleum prospecting areas were then identified by the concentration-area fractal model. The results revealed that the potential areas correspond well with the discovered petroleum sites.

Seraj and Delavar [29] proposed a methodology based on the Dempster–Shafer theory in a GIS framework for assessing risk of play-based petroleum exploration under spatial uncertainty in the Fars domain of Zagros Basin of Iran. In the methodology, the play risk is subdivided into the regional risk elements. The maps of the regional risk elements, referred to as common risk segment (CRS) maps, are produced. The CRS maps of a specific play are combined into a composite common risk segment (CCRS) map, which provides a basis for an estimate of play risk. The results indicated that about 26% of the study area was delineated as the high petroleum potential class by the classified CCRS map and 73.4% of the testing data were correctly predicted by the proposed model. The Dempster–Shafer theory of evidence and fuzzy sets were integrated by Seraj et al. [30] for spatially evaluating geological risk of petroleum exploration. The authors found that 35% of the area of the Fars sedimentary region of Iran has a high risk for petroleum exploration and 75.9% of the testing data confirms this finding. Various spatial approaches such as the fuzzy-belief networks [31], object-based stochastic model [32], multivariate and Bayesian statistical methods [33], geostatistical techniques and Monte-Carlo simulation [34], hybrid fuzzy-probabilistic method [35], Bayesian networks [36], and machine learning [37] have also been used for risk assessment in petroleum exploration.

The optimized fuzzy ELECTRE (OFE) approach is presented in this article for petroleum potential assessment, in which FE provides a robust decision-making framework that has the ability to handle data with uncertainty and the artificial bee colony (ABC) algorithm is used to optimize the model parameters (i.e., weights of the criteria). In order to validate the proposed approach, a case study was conducted in the Red River petroleum system of the Canadian side of the Williston Basin.

No study, to the best of our knowledge, has yet applied outranking or fuzzy outranking methods to petroleum exploration. Experts’ knowledge is incorporated as crisp values into the ELECTRE methods, which is not suitable for the petroleum potential modeling, especially in frontier areas, where information about the importance of the criteria is uncertain, incomplete, and ambiguous. Hence, one of the novelties of the current study is the use of the triangular fuzzy (TF) numbers to define weights of the criteria for petroleum resource assessment. Another point to note is that bias is a major challenge in the exploration of oil and gas. Biased expert-elicited values for the model parameters are strongly associated with unsuccessful outcomes and can lead to a dry hole or an uneconomic reservoir. Accordingly, another important novelty of the presented research work includes the use of the ABC algorithm to optimize the experts-assigned weights of the criteria.

## 2 Materials and methods

### 2.1 Geological setting

The Williston Basin is the archetypal intracratonic basin that straddles the United States–Canada border and covers an area of about 3,00,000 square miles [38] across three states in the United States and two provinces in Canada (Figure 1). The concept of petroleum system was first applied in the basin by Williams [39] and Dow [40], who identified three petroleum systems. Further studies reported the presence of seven other petroleum systems, one of which is the Red River, a self-sourced and highly prolific petroleum system that has significant remaining exploration potential [41].

The carbonate succession of the Bighorn Group in Saskatchewan comprises three formations, which are, in ascending order, the Red River, Stony Mountain and Stonewall (Figure 2). The Red River strata is divisible into the Yeoman and Herald Formations on lithological grounds [42]. Three brining-upward cycles are distinguishable in the strata. The lowest cycle, which is the interval of interest here, consists of the Yeoman Formation and the overlying Lake Alma Member of the Herald Formation. The uppermost third of the Yeoman Formation or Upper Yeoman is strongly mottled, commonly porous, and permeable. These have resulted in relatively good reservoir properties [43,44]. The Upper Yeoman also contains widely distributed, commonly laminated, relatively thin organic-rich interbeds, which are known as kukersites. These interbeds are the major source of hydrocarbons in the Red River petroleum system [45].

Most of the petroleum pools discovered to date in the Red River petroleum system occur mainly in structural traps with stratigraphic enhancement capped by dense evaporitic deposits [46]. The upper evaporitic unit of Lake Alma Member, informally named the Lake Alma anhydrite, forms an extensive and impermeable cap rock for the lowest cycle of the Red River strata [47].

### 2.2 Criteria identification

According to Magoon and Sánchez [48], “the petroleum system is the naturally occurring hydrocarbon–fluid system in the geosphere.” The petroleum system concept provides a basis for evaluating exploration opportunities and is efficiently used to find undiscovered commercial oil and gas accumulations [49]. The source rock, reservoir rock, seal rock, and overburden rock are the essential elements of a petroleum system. The processes affecting a petroleum system include trap formation and the generation–migration–accumulation of petroleum. The elements and processes of a petroleum system must be correctly placed in time and space so that organic matter of a source rock can be converted to oil and gas accumulations [50]. Some characteristics of the essential elements of petroleum system are used here as the criteria to evaluate petroleum potential of the Red River petroleum system. These are explained below.

Effective source rocks satisfy three requirements including quantity, quality, and thermal maturity [51] (Table 1). The amount of petroleum generated and expelled from source rocks is controlled by the total organic carbon (TOC) content, thickness of source rock (*D*_{s}), and thermal maturity (*T*_{max}) [52]. Quality of organic matter in source rocks is measured by the hydrogen index (HI) parameter [53]. In the present study, TOC, *D*_{s}, *T*_{max}, and HI were selected as criteria in order to assess the source rock potential of the Upper Yeoman.

(a) Generative potential (quantity) of immature source rock, (b) kerogen type (quality), and (c) thermal maturity [51]

Potential (quantity) | TOC (wt%) | Kerogen (quality) | HI (mg HC/g TOC) | Maturity | T_{max} (°C) |
---|---|---|---|---|---|

Poor | <0.5 | I | >600 | Immature | <435 |

Fair | 0.5–1 | II | 300–600 | Early mature | 435–445 |

Good | 1–2 | II/III | 200–300 | Peak mature | 445–450 |

Very good | 2–4 | III | 50–200 | Late mature | 450–470 |

Excellent | >4 | IV | <50 | Postmature | >470 |

(a) | (b) | (c) |

There are two hydrocarbon-volume attributes that determine the volume of hydrocarbon in the prospect: depth to reservoir and the thickness of reservoir rock (*D*_{r}) [54]. Unlike *D*_{r}, there is no clear relationship between potential of reservoir rock and reservoir depth. In the absence of permeability and porosity data of the Upper Yeoman, *D*_{r} was considered as the criterion to evaluate the reservoir rock potential.

The probability of structures that may have trapping potential being present at a given point is indirectly related to curvature (*C*_{r}) and roughness (*R*_{r}) of reservoir top surface [26,55,56]. Traps in the Red River petroleum system are predominantly structural [46], so *C*_{r} and *R*_{r} were adopted as the criteria for assessing the potential of traps formed by anticlinal folds and faults, respectively.

A proper cap rock should not only be impervious, but also should form a barrier above and around the reservoir rock. Two other aspects that must be considered in seal evaluation are the extension and thickness of the cap rock (*D*_{c}) [57]. The Lake Alma anhydrite extends across almost the entire study area and the sealing ability of this cap rock mainly depends on the thickness parameter [47]. So, *D*_{c} was taken into account in analyzing the sealing potential of the Lake Alma anhydrite.

### 2.3 Data sets and data preparation

The OFE approach can be implemented either pixel or voxel based. This study was carried out in two-dimensional space; all data sets were, therefore, converted to raster grids with spatial resolution of 1,000 m per pixel.

Data sets required for petroleum potential modeling were selected on the basis of the criteria outlined in Section 2.2. Rock-Eval pyrolysis was undertaken on 183 core samples of the source rock to determine TOC, *T*_{max}, and HI values. These values were then interpolated by the tension spline method (Figure 3a–c). Isopach and structure data were collected from the exploratory and production wells across the Canadian Williston Basin and interpolated by the ordinary kriging method. The resulting isopach grids of the Upper Yeoman (*D*_{s}), the reservoir zone of the lowest cycle of the Red River strata (*D*_{r}) and the Lake Alma anhydrite (*D*_{c}) were used as input data sets for the model (Figure 3d–f).

Surface roughness can be estimated using techniques like roughness index (RI). RI quantifies surface heterogeneity by the following equation [58]:

*R*

_{r}data set that is the result of applying RI to structure grid of the reservoir rock.

Curvature is the second derivative of a surface fitted to elevation [59] and measures the convexity or concavity of a surface at a particular pixel. Thus, it can be used for evaluating the trapping potential. *C*_{r} data set (Figure 3h) is the second derivative of the structure grid of the reservoir rock.

It should be noted that the sources of all data used in this article are Geoscience Data Repository of Earth Sciences Sector of Natural Resources Canada and the Williston Basin Targeted Geoscience Initiative database [60].

### 2.4 Methods

### 2.4.1 The FE approach

The multicriteria analysis (MCA) is useful for problems like petroleum potential analysis in which there are a finite number of spatial elements (pixels or voxels) as alternatives to be assessed on the basis of several, sometimes conflicting, criteria featuring various forms of data and information. Among different methods of MCA, the ELECTRE outranking approach [61] is suitable for finding outranking relations and ranking alternatives [62]. The various ranking formats of the ELECTRE offer several advantages over existing techniques for modeling real-world decision-making problems, the most important of which is the nontotal compensation of multiple criteria aggregation [63]. Another advantage of the ELECTRE methods is that preference and indifference thresholds can easily be considered when modeling partial and imprecise knowledge, which is impossible with other algorithms such as the MAUT, AHP, TOPSIS, MACBETH, SMART, and methods based on fuzzy integrals. Moreover, fuzzy logic [64] is an effective mathematical tool for dealing with uncertainty [35] and is adequate to handle imprecise information and incomplete data in petroleum geology [65]. Consequently, integrating fuzzy logic and FE provides a flexible and effective knowledge-driven framework to model petroleum potential of an area.

The methods of the ELECTRE family are based on

Step 1: To begin with, a panel of experts is assembled. Members

Step 2: All the values assigned to the alternatives with respect to each criterion are used to form the decision matrix

Step 3: The decision matrix is normalized by using the following equations:

Step 4: The normalized decision matrix *R* is then multiplied by TF weights of the evaluation criteria to produce the weighted normalized decision matrix

Step 5: The concordance and discordance indices are computed for different TF weights of each criterion (i.e.,

The discordance index

Step 6: The final concordance and discordance indices are calculated by the formula (12), which can be considered as the defuzzification process [66].

The dominance relationship of

Step 7: In order to obtain the ranking of all actions by use of the FE algorithm, some kind of overall outranking needs to be measured. So, following Aouam et al. [67], the overall outranking intensity

### 2.4.2 The ABC algorithm

Most knowledge-driven models for petroleum potential assessment such as the FE approach are characterized by complex functional relationships and a large number of parameters that must be determined by experts. In most cases, the model parameters cannot be specified with precision and the results may suffer from expert subjectivity and bias, hence the need for the parameter optimization. It is a process in which model parameters are tuned for a particular problem. During the last few decades, several tuning techniques have been proposed for optimal tuning of model parameters. Among them, optimization algorithms based on swarm intelligence, known as metaheuristic algorithms, have gained popularity in solving complex and high-dimensional real-world problems. Metaheuristic algorithms generally overcome the drawbacks of conventional or deterministic optimization methods, that is, divergence situations and getting trapped in local optima, because most of them are independent of the initial solutions and are derivative-free [68]. The findings of several studies [69,70,71,72] show that the ABC algorithm [73] can outperform other existing techniques such as the genetic algorithms, particle swarm optimization, and differential evolution and is very promising and efficient in solving optimization problems. This is mainly due to the fact that the local and global search of the algorithm are conducted simultaneously in each iteration, so the probability of finding the optimal solutions is considerably increased [74].

The ABC is a simple and robust stochastic algorithm, which simulates the intelligent foraging behavior of honeybee swarms and can be mathematically described as follows.

Suppose

Step 1: A set of feasible solutions

*j*th dimension of the solution vector. The fitness value of each solution vector is then calculated and the top

*N*

_{e}best solutions are selected to form the initial population of the employed bees

Step 2: For an employed bee in the *n*th iteration

*k*, and

*j*are randomly generated, and

Step 3: The greedy selection operator

*T*

_{s}is not related to the iteration number

*n*.

Step 4: Each unemployed bee picks an employed bee out of the colony according to its fitness value. The probability distribution of the selection operator

Step 5: The neighborhood of the selected employed bee is explored by the unemployed bee to find new solutions. The updated best fitness value is detected as

Step 6: If the search of the neighborhood of an employed bee Bas ends at a specified stopping time

Step 7: If the current iteration number is greater than the predefined threshold (i.e.,

Step 8: It is the most important aspect of the ABC algorithm that makes it distinctly different from other metaheuristic optimization methods because it diversifies the population to avoid getting trapped in local optima. Obviously, this step can significantly increase the probability of finding an optimal solution [74].

## 3 Application and results

### 3.1 Implementation of the FE approach

Cell values of the eight raster data sets described in Section 2.3 were extracted into a table with eight fields as criteria and records as alternatives. In order to apply the FE approach, a vector containing TF weights must first be defined. To this end, the exploration experts participating in the present study were asked to assign one of the seven linguistic variables shown in Figure 4 to each criterion. This linguistic variable represents the relative contribution of each criterion to the overall petroleum potential as a TF number. The variables assigned by different experts were combined by fuzzy simple additive weighting [76] to obtain a TF weight for each criterion. The aggregate weights of the criteria are illustrated as TF numbers in Figure 5.

The values of the overall outranking intensity index (*I*) were calculated by means of equations (2)–(12) and methodology proposed by Aouam et al. [67]. Finally,

### 3.2 Optimization of TF weights of the criteria by the ABC algorithm

The accuracy in determining the parameters of knowledge-driven models such as FE by the experts is dependent on many factors including complexity of the research object, expert qualification, the scale of evaluation, the number of criteria to be evaluated, etc. For example, complicated models with a large number of criteria may suffer from expert-elicited values of parameters. For experts who contributed to this study, there was noticeably higher ambiguity in determining weights of the criteria than others. Therefore, the experts-assigned weights were optimized using the ABC algorithm to obtain more accurate results.

Prior to optimization process, the vector layer of all thirteen discovered oil pools of the Red River petroleum system was rasterized to a 1,000 m cell size grid. Pixels of the grid were randomly subdivided into training and testing subsets with a 70%–30% ratio. The training subset was used in the optimization process to estimate the model parameters, and the testing subset was used to evaluate the predictive power of the model.

In order to determine the optimal values of the weights, the optimization objective function needs to be first defined. The objective function, in fact, is an indicator of agreement between actual and calculated values of the variable of interest. In this article, root mean square error (RMSE) was selected as the objective function to be minimized in the optimization process. The formula to calculate RMSE is:

*i*th alternative (pixel) of training subset,

A TF number can be represented by

The control parameters of the ABC algorithm including the colony size and maximum evaluation number were set to 50 and 2,000, respectively. The objective function values obtained by the ABC algorithm versus the number of iterations are depicted in Figure 6. Figure 7 also shows the optimized TF weights of the criteria.

### 3.3 Implementation of the OFE approach

The optimized TF numbers were used as weights of the criteria in FE to construct the OFE approach. The

### 3.4 Validation of the proposed approach

In order to validate the approach, *x*-axis and the cumulative percentage of the discovered pools area in the training subset on the *y*-axis. PRC was calculated in a similar way as SRC but using the testing subset. The area under the curve (AUC) was then summed to get an efficiency score. SRC and PRC are plotted in Figure 9. As shown in the figure, AUC for SRC (upper dashed line) is 0.851, which corresponds to 85.1% of success accuracy. The area under PRC (lower dashed line) is 0.814 implying a prediction accuracy of 81.4% for the petroleum potential model.

Attributes of petroleum potential classes

Class | Petroleum potential | Area (%) | Area of discovered pools located in class (%) | |
---|---|---|---|---|

N | No potential | 0.0–0.2 | 76.01 | 8.03 |

VL | Very low | 0.2–0.4 | 4.08 | 0 |

L | Low | 0.4–0.6 | 13.69 | 8.26 |

M | Moderate | 0.6–0.8 | 5.15 | 4.96 |

H | High | 0.8–1.0 | 1.07 | 78.75 |

## 4 Discussion

Exploration in the petroleum industry is an expensive, risky, and increasingly difficult, but necessary, operation. Therefore, to reduce both the costs and risks associated with exploration, a robust and efficient petroleum potential model is expected to lead to high-accuracy results. Both success and prediction accuracies of OFE lie in the range of 80–90%, indicating an excellent model performance [78,79]. In addition, overlaying the vector layer of the discovered oil pools on the reclassified map (Figure 10) showed 78.75% of the area of the pools has been located in the class “H.” Comparing the results of the FE and OFE approaches indicated that the optimization of the weights by the ABC algorithm has improved accuracy by approximately 15%, namely, a relatively higher success rate and lower risk in petroleum exploration.

It is difficult to compare the results of the current study to those carried out previously since geological conditions, type and number of criteria, number of potential classes, the area ratio of the potential classes, and techniques of validation vary greatly in the literature. Nevertheless, considering the five-class petroleum potential, the OFE approach with the success and prediction rates of 85.1% and 81.4%, respectively, outperforms the spatial models used in previous similar studies that have been cited here.

Another point to be taken into account is the ratio of the class “H” area to the total area under study, which has an inverse relationship with the screening ability of the petroleum potential model. A smaller ratio is indicative of a smaller area, which needs to be investigated in more detail to find the undiscovered petroleum accumulations. Generally, the lower the area, the lower the cost of exploration. As shown in Table 2, the ratio value for the class “H” of the OFE approach is about 0.01, which is significantly lower compared to those obtained in earlier studies.

The resultant map in Figure 10 clearly shows that the central part of the Canadian side of the Williston Basin has the highest petroleum potential. As summarized in Table 3, the entire area of nine discovered pools including the Bromhead, South Midale, Midale, West Midale, Froude, Mansur, Hartaven, Tyvan, and Minton are located in the class “H.” The Montmartre has the moderate petroleum potential. The South Hardy and Bemersyde lie in the class “L,” and the Chapleau Lake is mostly located in no-potential class. The box plots of the key criteria for the discovered pools (Figure 11) clearly show that the Montmartre, Bemersyde, and Chapleau Lake are likely to have different trapping mechanisms and the South Hardy probably has a source rock other than the kukersites of the Upper Yeoman.

Attributes of petroleum pools used in the verification process

No. (circled numbers in Figure 10) | Pool name | Area (km^{2}) | Class | ||
---|---|---|---|---|---|

1 | Minton | 16.254 | 0.833–0.961 | 0.878 | H (100%) |

2 | South Hardy | 4.547 | 0.502–0.588 | 0.569 | L (100%) |

3 | Bromhead | 2.601 | 0.898–0.951 | 0.927 | H (100%) |

4 | South Midale | 6.476 | 0.891–0.979 | 0.956 | H (100%) |

5 | Midale | 10.286 | 0.937–0.981 | 0.961 | H (100%) |

6 | West Midale | 3.243 | 0.885–0.908 | 0.900 | H (100%) |

7 | Froude | 3.244 | 0.873–0.911 | 0.899 | H (100%) |

8 | Mansur | 7.136 | 0.890–0.968 | 0.922 | H (100%) |

9 | Hartaven | 4.513 | 0.907–0.921 | 0.913 | H (100%) |

10 | Bemersyde | 3.879 | 0.518–0.559 | 0.544 | L (100%) |

11 | Montmartre | 5.190 | 0.753–0.779 | 0.766 | M (100%) |

12 | Chapleau Lake | 8.383 | 0.027–0.756 | 0.051 | N (97.5%) and M (2.5%) |

13 | Tyvan | 19.456 | 0.871–0.965 | 0.916 | H (100%) |

## 5 Conclusions

The present study proposes a new approach based on combining an outranking method, fuzzy logic, and the ABC optimization algorithm for assessing petroleum potential using the characteristics of petroleum system elements in a spatial framework. The proposed OFE approach brings together flexibility and simplicity for DMs to solve petroleum exploration problem using experts’ knowledge and the information associated with the discovered petroleum pools simultaneously. Having completed the necessary preprocessing steps, eight data sets related to the selected key criteria were integrated by the OFE approach to create a map that makes it possible to identify the areas with the highest petroleum resource potential and the lowest exploration risk.

All discovered oil accumulations in the Canadian part of the Red River petroleum system were used in the verification process. The verification results showed that the proposed method can deal effectively with incomplete data and imprecise information and can be efficiently applied in petroleum exploration. Furthermore, a comparison between the results of using the experts-assigned weights and optimized weights indicated that the ABC algorithm has the ability to successfully estimate the model parameters and lead to a considerable gain in the accuracy of the model.

In future studies, the model can be improved using other types of preference functions and by optimizing all the model parameters. Moreover, utilizing additional data sets such as permeability and porosity of reservoir rock, which was inaccessible in the present study, is expected to increase the accuracy of the model. The optimal placement of exploratory and production wells requires a robust ranking algorithm with fine-tuned parameters that has the capability to handle data with high uncertainty. Therefore, it is recommended that further studies be undertaken to determine optimal well locations using the proposed approach.

The authors would like to thank Mr. Kirk Osadetz from CMC Research Institutes, Inc., and Ms. Jessica Flynn from Saskatchewan Geological Survey for much helpful advice on the geological setting of the Red River petroleum system. Thanks also to the anonymous reviewers and the editor for the helpful comments on an earlier version of this manuscript.

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