# Data-Driven Proxy Models for Improving Advanced Well Completion Design under Uncertainty

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## Abstract

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^{®}reservoir simulator. The simulation speed and accuracy of the data-driven proxy models compared to physic-driven models were then evaluated. The evaluation showed that while the developed proxy models are 350 times faster, they can predict the production of oil and unwanted fluids through advanced wells with a mean error of less than 1% and 4%, respectively. As a result, the data-driven proxy models can be considered an efficient tool for uncertainty analysis where several simulations need to be performed to cover all possible scenarios. In this study, the developed proxy models were applied for uncertainty quantification of oil recovery from advanced wells completed with different types of downhole flow control devices (FCDs). According to the obtained results, compared to other types of well completion design, advanced wells completed with autonomous inflow control valve (AICV) technology have the best performance in limiting the production of unwanted fluids and are able to reduce the associated risk by 91%.

## 1. Introduction

^{TM}. The Latin hypercube sampling approach (LHS), which is a pseudo-random sampling method, was applied for the design of experiment (DOE). Moreover, by enlisting a new algorithm, the developed proxy models were able to predict cumulative oil and water production through advanced wells by passing the time. The developed data-driven proxy models were utilized to efficiently and accurately investigate the performance of advanced wells with different completion designs. Therefore, the developed methodology enables petroleum engineers to improve the design of advanced wells under uncertain geological parameters.

## 2. Data and Methods

#### 2.1. Development of the Physics-Based Integrated Well–Reservoir Model

^{3}/day. To maintain the liquid production rate below its threshold, the pressure drawdown is lowered when the rate of liquid production rises over 1000 m

^{3}/day. A schematic of the horizontal well and reservoir is shown in Figure 2.

^{3}. Table 1 provides information on the reservoir’s characteristics and their range of uncertainty. The ratio of vertical permeability to horizontal permeability is defined as permeability anisotropy.

^{3}/h. ${\rho}_{mix}^{}$ and ${\mu}_{mix}$ are the fluid mixture density and viscosity, respectively, and they were calculated using Equation (3) based on the water cut $\alpha $. The derived mathematical models for AICDs and AICVs vs. the experimental data are shown in Figure 7:

#### 2.2. Development of the Data-Driven Proxy Model

#### 2.2.1. Data Set Generation

_{i}, the sensitivity coefficient, Φ

_{i}, with respect to the desired output Y is calculated by Equation (4) as [26]:

_{i}/Y is added to normalize the coefficient by taking the effect of units out of the equation. For small changes in the input parameter, by neglecting the nonlinearities, the partial derivative in Equation (4) can be approximated as a finite difference and Equation (4) can be simplified as:

_{i}= ±10% and measuring the percentage change in the outputs (here cumulative oil and water production), the sensitivity coefficient of each reservoir parameter for the cumulative oil and water production was calculated. The obtained results are depicted as a tornado diagram in Figure 10. According to the presented results, among all the reservoir parameters, for both oil and water production, the sensitivity coefficient for the top six parameters is considerable. The sensitivity coefficient of the seventh parameter (maximum relative permeability of oil) compared to the top six parameters is considerably lower, and the last three parameters have a very small sensitivity coefficient as well. As a result, the top six parameters, were determined as the most impactful input variables for predicting both oil and water production. These input variables are given in Table 2 and are considered the model inputs for the proxy model development.

^{6}) data sets were generated for the training and improvement of the proxy models. In the same way, for each proxy model, 3 different samples for each input variable were used to generate 729 (3

^{6}) data sets for validation and testing of the developed proxy models. Two-thirds of these 729 data sets were used for validation and one-third of them (240 data sets) were considered for the test.

#### 2.2.2. Architecture of the Proxy

#### 2.2.3. Training and Testing the Proxy

## 3. Results and Discussions

#### 3.1. Performance of the Developed Proxy Models

#### 3.2. Evaluating the Performance of Advanced Wells under Uncertainty

^{6}= 117,649 possible scenarios were specified using the LHS approach. Using the simulation results for all these scenarios, the probability distribution function (PDF) and cumulative distribution function (CDF) of the total oil and water production from advanced wells with different FCD completions after 10 years were determined. Figure 14 and Figure 15 show PDF and CDF for oil and water production for different completions after 10 years, respectively. Moreover, the mean (average), P10 (low estimation), P50 (best estimation), and P90 (high estimation) and P10/P90 range (risk) were calculated and are given in Table 4.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic of advanced well completion with FCD and SCS [2].

**Figure 5.**Illustration of the MSW model [20].

**Figure 6.**Performance curves of ICDs, AICDs, and AICVs for oil, water, and gas [21].

**Figure 8.**Cumulative oil and water production from open-hole and advanced wells with different FCD completions for the base case.

**Figure 11.**Schematic diagram of a neural network with two hidden layers [29].

**Figure 13.**Prediction mean and relative error for the developed proxy models for the ICD, AICD, and AICV cases.

Parameter | Min | Mean | Max | Unit |
---|---|---|---|---|

Porosity | 0.15 | 0.21 | 0.27 | - |

Absolute permeability | 100 | 350 | 800 | mD |

Irreducible water saturation | 0.1 | 0.15 | 0.2 | - |

Residual oil saturation | 0.05 | 0.1 | 0.15 | - |

Maximum relative permeability of water | 0.2 | 0.4 | 0.5 | - |

Maximum relative permeability of oil | 0.85 | 0.95 | 1 | - |

Permeability anisotropy | 0.7 | 0.3 | 0.1 | - |

Initial water saturation | 0.12 | 0.2 | 0.25 | - |

Capillary pressure | 2 | 2.7 | 4 | Bar |

Aquifer productivity index | 2000 | 10,000 | 15,000 | m^{3}/d/bar |

Parameter | Min. | Mean | Max. |
---|---|---|---|

Porosity | 0.15 | 0.21 | 0.27 |

Initial water saturation | 0.12 | 0.2 | 0.25 |

Irreducible water saturation | 0.1 | 0.15 | 0.2 |

Residual oil saturation | 0.05 | 0.1 | 0.15 |

Maximum relative permeability of water | 0.2 | 0.4 | 0.5 |

Absolute permeability | 100 | 350 | 800 |

Case | Mean Error | |
---|---|---|

Oil | Water | |

ICD | 0.96% | 3.79% |

AICD | 0.48% | 1.98% |

AICV | 0.46% | 1.98% |

Case | Production [m^{3}] | Mean | P10 | P50 | P90 | P10/P90 Range |
---|---|---|---|---|---|---|

OPENHOLE | Oil | 2.04 × 10^{5} | 1.56 × 10^{5} | 2.01 × 10^{5} | 2.47 × 10^{5} | 9.10 × 10^{4} |

Water | 1.72 × 10^{6} | 1.24 × 10^{6} | 1.74 × 10^{6} | 2.01 × 10^{6} | 7.70 × 10^{5} | |

ICD | Oil | 1.97 × 10^{5} | 1.55 × 10^{5} | 1.94 × 10^{5} | 2.38 × 10^{5} | 8.30 × 10^{4} |

Water | 1.43 × 10^{6} | 1.14 × 10^{6} | 1.42 × 10^{6} | 1.58 × 10^{6} | 4.40 × 10^{5} | |

AICD | Oil | 1.92 × 10^{5} | 1.54 × 10^{5} | 1.89 × 10^{5} | 2.29 × 10^{5} | 7.50 × 10^{4} |

Water | 1.06 × 10^{6} | 9.36 × 10^{5} | 1.04 × 10^{6} | 1.10 × 10^{6} | 8.26 × 10^{4} | |

AICV | Oil | 1.88 × 10^{5} | 1.53 × 10^{5} | 1.83 × 10^{5} | 2.22 × 10^{5} | 6.90 × 10^{4} |

Water | 5.70 × 10^{5} | 4.98 × 10^{5} | 5.30 × 10^{5} | 5.66 × 10^{5} | 6.80 × 10^{4} |

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**MDPI and ACS Style**

Moradi, A.; Tavakolifaradonbe, J.; Moldestad, B.M.E.
Data-Driven Proxy Models for Improving Advanced Well Completion Design under Uncertainty. *Energies* **2022**, *15*, 7484.
https://doi.org/10.3390/en15207484

**AMA Style**

Moradi A, Tavakolifaradonbe J, Moldestad BME.
Data-Driven Proxy Models for Improving Advanced Well Completion Design under Uncertainty. *Energies*. 2022; 15(20):7484.
https://doi.org/10.3390/en15207484

**Chicago/Turabian Style**

Moradi, Ali, Javad Tavakolifaradonbe, and Britt M. E. Moldestad.
2022. "Data-Driven Proxy Models for Improving Advanced Well Completion Design under Uncertainty" *Energies* 15, no. 20: 7484.
https://doi.org/10.3390/en15207484