# An Interpretable Recurrent Neural Network for Waterflooding Reservoir Flow Disequilibrium Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Material Balance Equation of the Waterflooding Reservoir

#### 2.2. Development of the Interpretable Recurrent Neural Network

Algorithm 1: The training pseudocode for IRNN. | |

Input:WIRs, water injection rates; BHP_{I} and BHP_{P}, BHP data of the injectors and producers, respectivelyOutput: Estimated liquid production rates, $\widehat{LPRs}$ | |

1 | / Start training / |

2 | While stop criteria is not met |

3 | / Drainage module computation / |

Calculate the fluid change rate among the drainage volume,${D}_{t},$ according to Equation (9); | |

4 |
/ Inflow module computation / |

Calculate the total inflow rates for Q producers,${I}_{t},$ according to Equation (12); | |

5 |
/ Modeled liquid production rates computation / |

Calculate the modeled liquid production rates, $\widehat{LPRs}$, according to Equation (13); | |

6 |
/ Loss evaluation / |

Compute the total loss of IRNN, $L\left(LPRs,\widehat{LPRs}\right)$, using Equation (14); | |

7 |
/ Model parameters optimization / |

Optimize weight matrices and the independent variables of Equation (10). | |

8 |
End while |

9 | / End training / |

## 3. Results and Discussion

#### 3.1. Three-Channel Model

^{−4}for IRNN and 0.0225 for MLP). It can be understood that the high-precision results of the proposed model are at the cost of a higher computing time cost. As listed in Table 3, both the training time costs of LPRs and the water cut of IRNN (208.9531 and 170.5313 s, respectively) are higher than those of MLP (24.1563 and 16.3875 s, respectively), but the prosed model still can finish the training process within several minutes. Thus, the accuracy of the proposed model is significantly improved at the expense of a part of the calculation time.

#### 3.2. Olympus Model

^{−4}, which are only 24.26% and 3.12% of those obtained via MLP. In addition, the history matching and prediction errors of the water cut of IRNN (0.0012 and 1.4271 × 10

^{−4}, respectively) are more than one order of magnitude lower than those of MLP (0.0137 and 0.0054, respectively). As listed in Table 4, although MLP costs less computation time (3.9843 s for LPRs and 3.0625 s for water cut) than IRNN, the proposed model can finish the calculation within tens of seconds. Therefore, at the cost of an acceptable increase in the calculation time, both the fitting and the prediction abilities of IRNN are significantly enhanced.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Xu, J.; Zhan, S.; Wang, W.; Su, Y.; Wang, H. Molecular dynamics simulations of two-phase flow of n-alkanes with water in quartz nanopores. Chem. Eng. J.
**2022**, 430, 132800. [Google Scholar] [CrossRef] - Li, W.; Nan, Y.; Wen, X.; Wang, W.; Jin, Z. Effects of Salinity and N-, S-, and O-Bearing Polar Components on Light Oil–Brine Interfacial Properties from Molecular Perspectives. J. Phys. Chem. C
**2019**, 123, 23520–23528. [Google Scholar] [CrossRef] - Sheng, G.; Su, Y.; Wang, W. A new fractal approach for describing induced-fracture porosity/permeability/ compressibility in stimulated unconventional reservoirs. J. Pet. Sci. Eng.
**2019**, 179, 855–866. [Google Scholar] [CrossRef] - Wang, W.; Fan, D.; Sheng, G.; Chen, Z.; Su, Y. A review of analytical and semi-analytical fluid flow models for ultra-tight hydrocarbon reservoirs. Fuel
**2019**, 256, 115737. [Google Scholar] [CrossRef] - Cai, J.; Jin, T.; Kou, J.; Zou, S.; Xiao, J.; Meng, Q. Lucas–Washburn Equation-Based Modeling of Capillary-Driven Flow in Porous Systems. Langmuir
**2021**, 37, 1623–1636. [Google Scholar] [CrossRef] - Chen, G.; Li, Y.; Zhang, K.; Xue, X.; Wang, J.; Luo, Q.; Yao, C.; Yao, J. Efficient hierarchical surrogate-assisted differential evolution for high-dimensional expensive optimization. Inf. Sci.
**2021**, 542, 228–246. [Google Scholar] [CrossRef] - Yousef, A.A.; Gentil, P.H.; Jensen, J.L.; Lake, L.W. A Capacitance Model To Infer Interwell Connectivity From Production and Injection Rate Fluctuations. SPE Reserv. Eval. Eng.
**2006**, 9, 630–646. [Google Scholar] [CrossRef] - Naudomsup, N.; Lake, L.W. Extension of Capacitance/Resistance Model to Tracer Flow for Determining Reservoir Properties. SPE Reserv. Eval. Eng.
**2018**, 22, 266–281. [Google Scholar] [CrossRef] - Temizel, C.; Artun, E.; Yang, Z. Improving Oil-Rate Estimate in Capacitance/Resistance Modeling Using the Y-Function Method for Reservoirs Under Waterflood. SPE Reserv. Eval. Eng.
**2019**, 22, 1161–1171. [Google Scholar] [CrossRef] - Gubanova, A.; Orlov, D.; Koroteev, D.; Shmidt, S. Proxy Capacitance-Resistance Modeling for Well Production Forecasts in Case of Well Treatments. SPE J.
**2022**, 27, 3474–3488. [Google Scholar] [CrossRef] - Arouri, Y.; Lake, L.W.; Sayyafzadeh, M. Bilevel Optimization of Well Placement and Control Settings Assisted by Capacitance-Resistance Models. SPE J.
**2022**, 27, 3829–3848. [Google Scholar] [CrossRef] - Zhao, H.; Kang, Z.; Zhang, X.; Sun, H.; Cao, L.; Reynolds, A.C. INSIM: A Data-Driven Model for History Matching and Prediction for Waterflooding Monitoring and Management with a Field Application. In Proceedings of the SPE Reservoir Simulation Symposium, Houston, TX, USA, 23 February 2015. [Google Scholar]
- Guo, Z.; Chen, C.; Gao, G.; Vink, J. Enhancing the Performance of the Distributed Gauss-Newton Optimization Method by Reducing the Effect of Numerical Noise and Truncation Error With Support-Vector Regression. SPE J.
**2018**, 23, 2428–2443. [Google Scholar] [CrossRef] - Zhang, Y.; He, J.; Yang, C.; Xie, J.; Fitzmorris, R.; Wen, X.-H. A Physics-Based Data-Driven Model for History Matching, Prediction, and Characterization of Unconventional Reservoirs. SPE J.
**2018**, 23, 1105–1125. [Google Scholar] [CrossRef] - Guo, Z.; Reynolds, A.C. INSIM-FT-3D: A Three-Dimensional Data-Driven Model for History Matching and Waterflooding Optimization. In Proceedings of the SPE Reservoir Simulation Conference, Society of Petroleum Engineers, Galveston, TX, USA, 20–22 February 2019; p. SPE–193841-MS. [Google Scholar]
- Li, Y.; Onur, M. INSIM-BHP: A physics-based data-driven reservoir model for history matching and forecasting with bottomhole pressure and production rate data under waterflooding. J. Comput. Phys.
**2023**, 473, 111714. [Google Scholar] [CrossRef] - Artun, E. Characterizing interwell connectivity in waterflooded reservoirs using data-driven and reduced-physics models: A comparative study. Neural Comput. Appl.
**2017**, 28, 1729–1743. [Google Scholar] [CrossRef] - Jensen, J. Comment on “Characterizing interwell connectivity in waterflooded reservoirs using data-driven and reduced-physics models: A comparative study” by E. Artun DOI 10.1007/s00521-015-2152-0. Neural Comput. Appl.
**2016**, 28, 1–2. [Google Scholar] [CrossRef] - Mo, S.; Zhu, Y.; Zabaras, N.; Shi, X.; Wu, J. Deep Convolutional Encoder-Decoder Networks for Uncertainty Quantification of Dynamic Multiphase Flow in Heterogeneous Media. Water Resour. Res.
**2019**, 55, 703–728. [Google Scholar] [CrossRef] - Carpenter, C. Machine Learning Improves Accuracy of Virtual Flowmetering and Back-Allocation. J. Pet. Technol.
**2019**, 71, 77–78. [Google Scholar] [CrossRef] - Zhang, Z.; He, X.; AlSinan, M.; Kwak, H.; Hoteit, H. Robust Method for Reservoir Simulation History Matching Using Bayesian Inversion and Long-Short-Term Memory Network-Based Proxy. SPE J.
**2022**, 1–25. [Google Scholar] [CrossRef] - Chakraborty, R.; Pal, N.R. Feature Selection Using a Neural Framework With Controlled Redundancy. IEEE Trans. Neural Netw. Learn. Syst.
**2015**, 26, 35–50. [Google Scholar] [CrossRef] - Chen, G.; Zhang, K.; Xue, X.; Zhang, L.; Yao, J.; Sun, H.; Fan, L.; Yang, Y. Surrogate-assisted evolutionary algorithm with dimensionality reduction method for water flooding production optimization. J. Pet. Sci. Eng.
**2020**, 185, 106633. [Google Scholar] [CrossRef] - Chen, G.; Zhang, K.; Zhang, L.; Xue, X.; Ji, D.; Yao, C.; Yao, J.; Yang, Y. Global and Local Surrogate-Model-Assisted Differential Evolution for Waterflooding Production Optimization. SPE J.
**2020**, 25, 105–118. [Google Scholar] [CrossRef] - Fonseca, R.-M.; Rossa, E.; Emerick, A.; Hanea, R.G.; Jansen, J.-D. Overview Of The Olympus Field Development Optimization Challenge. In Proceedings of the 16th European Conference on the Mathematics of Oil Recovery, ECMOR 2018, Barcelona, Spain, 3–6 September 2018. [Google Scholar]

**Figure 3.**The liquid production rate curves of the three-channel model. The red lines with triangles represent the actual results obtained by the simulator, the black lines with circles represent the results obtained via IRNN, and the blue lines with dots are the results obtained via MLP. The history matching and predicting periods are divided by the grey vertical dashed line. From PRO-01 to PRO-09, the liquid production rates of 9 producers are demonstrated in (

**a**–

**i**) successively.

**Figure 4.**The water cut curves of the three-channel model. The red lines with triangles represent the actual results obtained by the simulator, the black lines with circles represent the results obtained via IRNN, and the blue lines with dots are the results obtained via MLP. The history matching and predicting periods are divided by the grey vertical dashed line. From PRO-01 to PRO-09, the water cut curves of 9 producers are demonstrated in (

**a**–

**i**) successively.

**Figure 6.**The disequilibrium factors of the three-channel model. (

**a**): The results obtained via IRNN; (

**b**) The results obtained via MLP.

**Figure 8.**The liquid production rate curves of Olympus model. The red lines with triangles represent the actual results obtained by the simulator, the black lines with circles represent the results obtained via IRNN, and the blue lines with dots are the results obtained via MLP. The history matching and predicting periods are divided by the grey vertical dashed line. From PRO-1 to PRO-11, the liquid production rates of 11 producers are demonstrated in (

**a**–

**k**) successively.

**Figure 9.**The water cut curves of the Olympus model. The red lines with triangles represent the actual results obtained by the simulator, the black lines with circles represent the results obtained via IRNN, and the blue lines with dots are the results obtained via MLP. The history matching and predicting periods are divided by the grey vertical dashed line. From PRO-1 to PRO-11, the water cut curves of 11 producers are demonstrated in (

**a**–

**k**) successively.

**Figure 11.**The disequilibrium factors of Olympus model. (

**a**): The results obtained via IRNN; (

**b**) The results obtained via MLP.

Hyperparameter | IRNN |
---|---|

Hidden nodes of GRU block | 15 |

Hidden nodes of queries, keys, and values of self-attention block | 10 |

Hidden nodes of fully connected layer | 15 |

Batch size | 10 |

Initial learning rate | 0.03 |

Coefficient of regularization item | 0.02 |

Reservoir Properties | Values |
---|---|

Model scale | 25 × 25 × 1 |

Grid size | 100 × 100 × 10 ft |

Depth | 4800 ft |

Initial pressure | 4000 psi |

Initial temperature | 100 °C |

Pore compressibility | 6.9 × 10^{−5} psi^{−1} |

Porosity | 0.20 |

Initial oil saturation | 0.80 |

Oil density | 900 kg/m^{3} |

Oil viscosity | 2.2 cP |

Water density | 1000 kg/m^{3} |

Water viscosity | 0.5 cP |

**Table 3.**The computation error (MSE) and time (seconds) of both MLP and IRNN of the three-channel model.

MLP | IRNN | |
---|---|---|

History matching error of LPRs | 0.0933 | 0.0124 |

Prediction error of LPRs | 0.2465 | 0.0203 |

History matching error of water cut | 0.0242 | 0.0017 |

Prediction error of water cut | 0.0225 | 4.9051 × 10^{−4} |

Training time of LPRs | 24.1563 s | 208.9531 s |

Training time of water cut | 16.3875 s | 170.5313 s |

MLP | IRNN | |
---|---|---|

History matching error of LPRs | 0.0272 | 0.0066 |

Prediction error of LPRs | 0.0320 | 1.4986 × 10^{−4} |

History matching error of water cut | 0.0137 | 0.0012 |

Prediction error of water cut | 0.0054 | 1.4271 × 10^{−4} |

Training time of LPRs | 3.9843 s | 71.5156 s |

Training time of water cut | 3.0625 s | 64.9062 s |

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## Share and Cite

**MDPI and ACS Style**

Jiang, Y.; Shen, W.; Zhang, H.; Zhang, K.; Wang, J.; Zhang, L.
An Interpretable Recurrent Neural Network for Waterflooding Reservoir Flow Disequilibrium Analysis. *Water* **2023**, *15*, 623.
https://doi.org/10.3390/w15040623

**AMA Style**

Jiang Y, Shen W, Zhang H, Zhang K, Wang J, Zhang L.
An Interpretable Recurrent Neural Network for Waterflooding Reservoir Flow Disequilibrium Analysis. *Water*. 2023; 15(4):623.
https://doi.org/10.3390/w15040623

**Chicago/Turabian Style**

Jiang, Yunqi, Wenjuan Shen, Huaqing Zhang, Kai Zhang, Jian Wang, and Liming Zhang.
2023. "An Interpretable Recurrent Neural Network for Waterflooding Reservoir Flow Disequilibrium Analysis" *Water* 15, no. 4: 623.
https://doi.org/10.3390/w15040623