# Reservoir Characterization and Productivity Forecast Based on Knowledge Interaction Neural Network

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

**(1) Statistical and signal processing methods.**These methods are based on statistical analysis and signal processing techniques. Spearman rank correlations [9] were presented to measure the relationship between injectors and producers, while the authors also pointed out that this method was not completely robust and nor were the influence factors fully understood. Tian and Horne [10] proposed a modified Pearson’s correlation coefficient method to capture the influence from injectors to producers, showing a more precise inter-well characterization ability than the Spearman rank correlation method. Wavelet analysis was adopted to infer the connecting relationships [11], revealing new insights into the inter-well connectivity analysis. Some novel signal processing approaches, like cross-correlation, spectral analysis, magnitude-squared coherence, and periodogram were also applied to infer the inter-well communication [12]. Although these methods have high computational efficiency by analyzing the correlation and mapping relationships between injection and production signals, they are not established on the physical laws of waterflooding. Therefore, the robustness of these methods is hard to be guaranteed, and these models are often combined together or served as complemental methods to reduce the uncertainty [9,11,12].

**(2) Machine learning methods.**These methods usually quantify inter-well connectivity through model parameters. Panda and Chopra [13] proposed a related approach, using artificial neural networks (ANNs) to estimate the interactions between injectors and producers, while the geological and geostatistical data were required to determine the model parameters. Artun [14] evaluated the inter-well connectivity via the products of weight matrices in ANNs, providing a new perception of the inter-well connectivity analysis. While Jensen [15] commented that Artun’s ANNs model can’t reflect the physical mechanism of the waterflooding process, so it was unclear for ANN’s performance on the field disturbances (e.g., temporary shut-in or completion). Even though these machine learning (ML) methods are capable of inferring the inter-well connectivity via their strong nonlinear mapping abilities, they are considered as “black box” models, for their weakness in physical interpretation, limiting their practical applications.

**(3) Physical models.**These models are established on the physical process and derived from corresponding physical laws. Yousef et al. [16] used the capacitance resistance model (CRM or CRMIP) to reflect the connectivity and time lag between injector and producer pairs, which derived from the material balance equation and linear productivity model. Compared with the multiple linear regression (MLR) model [17], it considered the effects of compressibility and transmissibility. Based on the work of CRM, a series of models were introduced, such as the capacitance resistance model for the producer control volume (CRMP) [18] and the capacitance resistance model for a tank or field control volume (CRMT) [19] and. Different from CRMIP, CRMT and CRMP assign each drainage volume or each tank to a constant time delay, making the production signals react synchronously with the signals of all injectors. In addition, Sayarpour [20] proposed a CRM-blocks model, dividing the drainage volume into several blocks to calculate the flow rate. However, the complexity of the CRM-blocks model is inevitably increased, since it simulates the flowing process block by block, limiting its applications in complex cases. To infer the inter-well connectivity from multilayers, the multilayer CRM (ML-CRM) [21] was proposed, by modeling the injected fluids flowing across different layers with the help of production logging tools (PLT). Besides, Zhao et al. [22] presented an inter-well numerical simulation model (INSIM) to approximate the performance of waterflooding reservoirs. INSIM is derived from the mass material balance and front tracking equations, which consist of inter-well control units considering transmissibility and control pore volume. Moreover, considering more complex cases, such as the conversion from a producer to an injector, INSIM-FT [23] was designed; INSIM-FT-3D was proposed to simulate the flow in three dimensions with gravity [24]. Recently, INSIM-FPT [25] has been presented to reveal the inter-well connectivity via history matching data instead of the reservoir petrophysical properties. These physical models have clear physical assumptions and can be applied in other aspects, such as production optimization [26,27,28,29].

## 2. Methods

## 3. Knowledge Interaction Neural Network (KINN)

#### 3.1. Injection Regulator Module (IRM)

**I**= [

**i**]

_{1,}i_{2, …,}i_{k, …,}i_{M}^{T}is the well water injection rate (WIR) data of M injectors, and

**Q**= [

**q**]

_{1,}q_{2, …,}q_{j, …,}q_{N}^{T}is the well liquid production rate (LPR) data of N producers, where

**i**and

_{k}**q**are vectors. As shown in Figure 1, the input data of IRM is

_{j}**I**, followed by a gate function layer defined as:

#### 3.2. Control Volume Module (CVM)

#### 3.3. Production Monitor Module (PMM)

#### 3.4. Reservoir Characterization and Productivity Prediction

**Q**, and the connecting weights in CVM also need an initialization. Afterward, guided by the material balance equation, IRM and CVM would cooperatively simulate the influence caused by water injection and compressibility in the waterflooding process. In the IRM part, the input comes from the water injection rate data of all injectors, and each vector would multiply a gate function, which is the inter-well connectivity indicator and has to be optimized during the training procedure. The output of IRM is the total inflow rate, the sum of all multiplied vectors. In the CVM part, a number of fully connected layers are utilized to realize the map from liquid production rate to the fluid change rate of the control volume, whose connecting weight matrices need optimization in the learning process. Afterward, both IRM and CVM would be combined by the material balance equation to calculate the model liquid production rate for producer j, and the loss between model output and target output can be measured by the loss function. It must be noted that the model loss comes from the outputs of both IRM and CVM, hence their physical knowledge would interact cooperatively with the model training process. During the optimization process, all the model parameters would be updated at the same time. When the stop criterion is satisfied, KINN would stop training, and the inter-well connectivity values can be inferred directly by gate functions.

Algorithm 1: Knowledge Interaction Neural Network (KINN) | |

Input: I, WWIR for M injectors, and Q, WLPR for N producers | |

Output:${\widehat{q}}_{j}$ | |

/ *** start KINN training *** / | |

1 | Initialization${\lambda}_{M\times N}$: Compute ${\gamma}_{M\times N}$ using database I and Q by Equations (6) and (7), and initialize the parameters of ANN in CVM |

2 | For $j$ = 1 to N do |

3 | While convergence tolerance is not met |

4 5 | / *** IRM calculation *** /Select the ${j}_{th}$ column, ${\gamma}_{M\times 1}$, in ${\gamma}_{M\times N}$ as the independent variable of gate function; Calculate the output of IRM, ${\Gamma}_{j}$, with ${\gamma}_{M\times 1}$ and I, using Equation (5) |

6 | / *** CVM calculation *** /Calculate the output of CVM, $Net\left({q}_{j}\right)$, with ${q}_{j}$, using Equation (13) |

7 | / *** PMM calculation *** /Calculate the output of PMM,${\widehat{q}}_{j}$, using Equation (14) |

8 9 10 11 | / *** parameters update *** /Evaluate the loss function using Equation (15) Update ${\gamma}_{M\times 1}$ and weight matrices of CVM via gradient descent algorithm End WhileEnd For/ *** end KINN training *** / |

## 4. Results

#### 4.1. The Streak Reservoir Case

#### 4.2. The Braided River Reservoir Case

#### 4.3. Egg Reservoir Case

^{−2}within 200 iterations. Meanwhile, the initial MSE for SLFNN is much bigger than that of KINN-tansig and KINN-Gaussian, and so is the converged error. As illustrated in Table 6, KINN-tansig, KINN-Gaussian, and SLFNN all demonstrate significant computation efficiency, taking 1.1282 s, 0.8539 s, and 0.3361 s, to finish training, respectively. As expected, both KINN-tansig and KINN-Gaussian are capable of producing more accurate results in history matching (0.0022 and 0.0035) and productivity prediction (0.0171 and 0.02263, respectively) than those obtained by SLFNN (0.0097 and 0.0426).

#### 4.4. Sensitivity to Noise

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Nomenclature | Explanations |

C_{t} | total compressibility, bar^{−1} |

i_{k} | water injection rate, m^{3}/Day |

J | productivity index, m^{3}/Day/bar |

M | number of injectors |

N | number of producers |

n | time-like variable |

$\overline{p}$ | average reservoir pressure, bar |

p_{wf} | bottom hole pressure, bar |

$\widehat{q}$ | estimated production rate, m^{3}/Day |

q_{j} | liquid production rate, m^{3}/Day |

t | time step, Day |

${V}_{p}$ | drainage pore volume, m^{3}/Day |

λ_{kj} | inter-well connectivity value |

γ_{kj} | independent variable of inter-well connectivity of intelligent connectivity model |

$\rho $ | Pearson correlation coefficient |

$\tau $_{i} | time constant of capacitance resistance model, Day |

${\Gamma}_{j}$ | comprehensive injection rate, m^{3}/ day |

k | injector index |

j | producer index |

## References

- Hashan, M.; Jahan, L.N.; Zaman, T.U.; Imtiaz, S.; Hossain, M.E. Modelling of fluid flow through porous media using memory approach: A review. Math. Comput. Simul.
**2020**, 177, 643–673. [Google Scholar] [CrossRef] - Mozolevski, I.; Murad, M.A.; Schuh, L.A. High order discontinuous Galerkin method for reduced flow models in fractured porous media. Math. Comput. Simul.
**2021**, 190, 1317–1341. [Google Scholar] [CrossRef] - Ma, X.; Zhang, K.; Zhang, L.; Yao, C.; Yao, J.; Wang, H.; Jian, W.; Yan, Y. Data-Driven Niching Differential Evolution with Adaptive Parameters Control for History Matching and Uncertainty Quantification. SPE J.
**2021**, 26, 993–1010. [Google Scholar] [CrossRef] - Xu, X.; Wang, C.; Zhou, P. GVRP considered oil-gas recovery in refined oil distribution: From an environmental perspective. Int. J. Prod. Econ.
**2021**, 235, 108078. [Google Scholar] [CrossRef] - Xu, X.; Lin, Z.; Li, X.; Shang, C.; Shen, Q. Multi-objective robust optimisation model for MDVRPLS in refined oil distribution. Int. J. Prod. Res.
**2021**, 5, 1–21. [Google Scholar] [CrossRef] - Yin, F.; Xue, X.; Zhang, C.; Zhang, K.; Han, J.; Liu, B.; Wang, J.; Yao, J. Multifidelity Genetic Transfer: An Efficient Framework for Production Optimization. SPE J.
**2021**, 26, 1614–1635. [Google Scholar] [CrossRef] - Zhang, K.; Zhang, J.; Ma, X.; Yao, C.; Zhang, L.; Yang, Y.; Wang, J.; Yao, J.; Zhao, H. History Matching of Naturally Fractured Reservoirs Using a Deep Sparse Autoencoder. SPE J.
**2021**, 26, 1700–1721. [Google Scholar] [CrossRef] - Xu, X.; Lin, Z.; Zhu, J. DVRP with limited supply and variable neighborhood region in refined oil distribution. Ann. Oper. Res.
**2022**, 309, 663–687. [Google Scholar] [CrossRef] - Heffer, K.J.; Fox, R.J.; McGill, C.A.; Koutsabeloulis, N.C. Novel Techniques Show Links between Reservoir Flow Directionality, Earth Stress, Fault Structure and Geomechanical Changes in Mature Waterfloods. SPE J.
**1997**, 2, 91–98. [Google Scholar] [CrossRef] - Tian, C.; Horne, R.N. Inferring Interwell Connectivity Using Production Data. In Proceedings of the SPE Annual Technical Conference and Exhibition, Dubai, United Arab Emirates, 26–28 September 2016. [Google Scholar] [CrossRef]
- Unal, E.; Siddiqui, F.; Rezaei, A.; Eltaleb, I.; Kabir, S.; Soliman, M.Y.; Dindoruk, B. Use of Wavelet Transform and Signal Processing Techniques for Inferring Interwell Connectivity in Waterflooding Operations. In Proceedings of the SPE Annual Technical Conference and Exhibition, Calgary, AB, Canada, 30 September–2 October 2019. [Google Scholar] [CrossRef]
- Wang, Y.; Kabir, C.S.; Reza, Z. Inferring Well Connectivity in Waterfloods Using Novel Signal Processing Techniques. In Proceedings of the SPE Annual Technical Conference and Exhibition, Dallas, TX, USA, 24–26 September 2018. [Google Scholar] [CrossRef]
- Panda, M.N.; Chopra, A.K. An Integrated Approach to Estimate Well Interactions. In Proceedings of the SPE India Oil and Gas Conference and Exhibition, Society of Petroleum Engineers, New Delhi, India, 17–19 February 1998; p. SPE-39563-MS. [Google Scholar]
- 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. Neural Comput. Appl.
**2016**, 28, 1745–1746. [Google Scholar] [CrossRef] [Green Version] - 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] - Albertoni, A.; Lake, L.W. Inferring interwell connectivity only from well-rate fluctuations in waterfloods. SPE Reserv. Eval. Eng.
**2003**, 6, 6–16. [Google Scholar] [CrossRef] - Lake, L.W.; Liang, X.; Edgar, T.F.; Al-Yousef, A.; Sayarpour, M.; Weber, D. Optimization Of Oil Production Based On A Capacitance Model Of Production And Injection Rates. In Proceedings of the Hydrocarbon Economics and Evaluation Symposium, Dallas, TX, USA, 1–3 April 2007. [Google Scholar]
- Sayarpour, M.; Zuluaga, E.; Kabir, C.S.; Lake, L.W. The use of capacitance–resistance models for rapid estimation of waterflood performance and optimization. J. Pet. Sci. Eng.
**2009**, 69, 227–238. [Google Scholar] [CrossRef] - Sayarpour, M. Development and Application of Capacitance-Resistive Models to Water/CO₂ Floods; University of Texas at Austin: Austin, TX, USA, 2008. [Google Scholar]
- Mamghaderi, A.; Bastami, A.; Pourafshary, P. Optimization of Waterflooding Performance in a Layered Reservoir Using a Combination of Capacitance-Resistive Model and Genetic Algorithm Method. J. Energy Resour. Technol.
**2012**, 135, 013102–013110. [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, 23–25 February 2015. [Google Scholar]
- Guo, Z.; Reynolds, A.C. INSIM-FT in three-dimensions with gravity. J. Comput. Phys.
**2019**, 380, 143–169. [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, 10–11 April 2019; p. SPE-193841-MS. [Google Scholar] [CrossRef]
- Zhao, H.; Xu, L.; Guo, Z.; Zhang, Q.; Liu, W.; Kang, X. Flow-Path Tracking Strategy in a Data-Driven Interwell Numerical Simulation Model for Waterflooding History Matching and Performance Prediction with Infill Wells. SPE J.
**2020**, 25, 1007–1025. [Google Scholar] [CrossRef] - Kansao, R.; Yrigoyen, A.; Haris, Z.; Saputelli, L. Waterflood Performance Diagnosis and Optimization Using Data-Driven Predictive Analytical Techniques from Capacitance Resistance Models CRM. In Proceedings of the SPE Europec Featured at 79th EAGE Conference and Exhibition, Paris, France, 12–15 June 2017. [Google Scholar] [CrossRef]
- Olenchikov, D.; Posvyanskii, D. Application of CRM-Like Models for Express Forecasting and Optimizing Field Development. In Proceedings of the SPE Russian Petroleum Technology Conference, Moscow, Russia, 22–24 October 2019. [Google Scholar]
- Nguyen, A.P.; Lasdon, L.S.; Lake, L.W.; Edgar, T.F. Capacitance Resistive Model Application to Optimize Waterflood in a West Texas Field. In Proceedings of the SPE Annual Technical Conference and Exhibition, Denver, CO, USA, 30 October–2 November 2011. [Google Scholar]
- Guo, Z.; Reynolds, A.; Zhao, H. Waterflooding optimization with the INSIM-FT data-driven model. Comput. Geosci.
**2018**, 22, 745–761. [Google Scholar] [CrossRef] - Chen, B.; Pawar, R.J. Characterization of CO2 storage and enhanced oil recovery in residual oil zones. Energy
**2019**, 183, 291–304. [Google Scholar] [CrossRef] - Alimohammadi, H.; Rahmanifard, H.; Chen, N. Multivariate Time Series Modelling Approach for Production Forecasting in Unconventional Resources. In Proceedings of the SPE Annual Technical Conference and Exhibition, Virtual. 26–29 October 2020. [Google Scholar] [CrossRef]
- Chen, B.; Harp, D.R.; Lin, Y.; Keating, E.H.; Pawar, R.J. Geologic CO
_{2}sequestration monitoring design: A machine learning and uncertainty quantification based approach. Appl. Energy**2018**, 225, 332–345. [Google Scholar] [CrossRef] - Li, Y.; Wang, G.; McLellan, B.; Chen, S.-Y.; Zhang, Q. Study of the impacts of upstream natural gas market reform in China on infrastructure deployment and social welfare using an SVM-based rolling horizon stochastic game analysis. Pet. Sci.
**2018**, 15, 898–911. [Google Scholar] [CrossRef] [Green Version] - Boret, S.E.B.; Marin, O.R. Development of Surrogate models for CSI probabilistic production forecast of a heavy oil field. Math. Comput. Simul.
**2019**, 164, 63–77. [Google Scholar] [CrossRef] - Fumagalli, A.; Zonca, S.; Formaggia, L. Advances in computation of local problems for a flow-based upscaling in fractured reservoirs. Math. Comput. Simul.
**2017**, 137, 299–324. [Google Scholar] [CrossRef] - Xu, X.; Hao, J.; Zheng, Y. Multi-objective Artificial Bee Colony Algorithm for Multi-stage Resource Leveling Problem in Sharing Logistics Network. Comput. Ind. Eng.
**2020**, 142, 106338. [Google Scholar] [CrossRef] - Huang, D.S. Radial Basis Probabilistic Neural Networks: Model and Application. Int. J. Pattern Recognit. Artif. Intell.
**1999**, 13, 1083–1101. [Google Scholar] [CrossRef] - Huang, D.; Du, J. A Constructive Hybrid Structure Optimization Methodology for Radial Basis Probabilistic Neural Networks. IEEE Trans. Neural Netw.
**2008**, 19, 2099–2115. [Google Scholar] [CrossRef] - Karpatne, A.; Watkins, W.; Read, J.; Kumar, V. Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling. arXiv
**2017**, arXiv:1710.11431. [Google Scholar] - Raissi, M.; Perdikaris, P.; Karniadakis, G.E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys.
**2019**, 378, 686–707. [Google Scholar] [CrossRef] - Tartakovsky, A.M.; Marrero, C.O.; Perdikaris, P.; Tartakovsky, G.D.; Barajas-Solano, D. Physics-Informed Deep Neural Networks for Learning Parameters and Constitutive Relationships in Subsurface Flow Problems. Water Resour. Res.
**2020**, 56, e2019WR026731. [Google Scholar] [CrossRef] - Dehghani, H.; Zilian, A. A hybrid MGA-MSGD ANN training approach for approximate solution of linear elliptic PDEs. Math. Comput. Simul.
**2021**, 190, 398–417. [Google Scholar] [CrossRef] - Wang, N.; Zhang, D.; Chang, H.; Li, H. Deep learning of subsurface flow via theory-guided neural network. J. Hydrol.
**2020**, 584, 124700. [Google Scholar] [CrossRef] [Green Version] - Csiszár, O.; Csiszár, G.; Dombi, J. Interpretable neural networks based on continuous-valued logic and multicriteria decision operators. Knowl.-Based Syst.
**2020**, 199, 105972. [Google Scholar] [CrossRef] - Huang, C.; Xu, H.; Xu, Y.; Dai, P.; Xia, L.; Lu, M.; Bo, L.; Xing, H.; Lai, X.; Ye, Y. Knowledge-aware Coupled Graph Neural Network for Social Recommendation. In Proceedings of the 35th AAAI Conference on Artificial Intelligence (AAAI), Virtual. 2–9 February 2021. [Google Scholar] [CrossRef]
- Zandvliet, M.J.; Bosgra, O.H.; Jansen, J.-D.; Van den Hof, P.; Kraaijevanger, J.B.F.M. Bang-bang control and singular arcs in reservoir flooding. J. Pet. Sci. Eng.
**2007**, 58, 186–200. [Google Scholar] [CrossRef] - Khan, Z.; Chaudhary, N.I.; Zubair, S. Fractional stochastic gradient descent for recommender systems. Electron. Mark.
**2019**, 29, 275–285. [Google Scholar] [CrossRef]

**Figure 2.**The curve of the gate function and its independent variables: (

**a**) independent variables of the gate function; (

**b**) curve of the gate function.

**Figure 3.**The inflow performance relationship (IPR) curves. The black line is the linear IPR prediction curve, and the red dashed line is the IPR curve of unstable flow.

**Figure 6.**History matching results of the streak reservoir case by KINN-tansig, KINN-Gaussian and SLFNN. The black line with squares is the result obtained by KINN-tansig method; the blue line with triangles represents the result gotten by KINN-Gaussian method; the green line with stars is the results obtained by SLFNN; the red line with circles is the liquid production rate gotten by ECLIPSE; the grey vertical dashed line makes a separation of history matching period and the productivity forecast period: (

**a**) P1; (

**b**) P4.

**Figure 7.**The training error (MSE) curves of KINN-tansig, KINN-Gaussian and SLFNN for the streak reservoir case. The black line with squares is the error curve of KINN-tansig; the blue line with triangles represents the error curve of KINN-Gaussian; and the green line with stars represent the error curve of SLFNN.

**Figure 8.**The heatmaps of the inter-well connectivity analysis by three models for the streak reservoir case: (

**a**) KINN-tansig; (

**b**) KINN-Gaussian; (

**c**) SLFNN.

**Figure 10.**History matching results of the braided river reservoir case by KINN-tansig, KINN-Gaussian and SLFNN. The black line with squares is the result obtained by KINN-tansig method; the blue line with triangles represents the result gotten by the KINN-Gaussian method; the green line with stars is the results obtained by SLFNN; the red line with circles is the liquid production rate gotten by ECLIPSE; the grey vertical dashed line makes a separation of history matching period and the productivity forecast period: (

**a**) P1; (

**b**) P2; (

**c**) P3; (

**d**) P4.

**Figure 11.**The training error (MSE) curves of KINN-tansig, KINN-Gaussian and SLFNN for the braided river reservoir case. The black line with squares is the error curve of KINN-tansig; the blue line with triangles represents the error curve of KINN-Gaussian; and the green line with stars represent the error curve of SLFNN.

**Figure 12.**The heatmaps of the inter-well connectivity analysis by KINN-tansig and KINN-Gaussian for the streak reservoir case: (

**a**) KINN-tansig; (

**b**) KINN-Gaussian; (

**c**) SLFNN.

**Figure 13.**The permeability and oil saturation distribution of Egg reservoir case: (

**a**) Permeability distribution; (

**b**) oil saturation distribution.

**Figure 14.**History matching results of the Egg reservoir case by KINN-tansig, KINN-Gaussian and SLFNN. The black line with squares is the result obtained by KINN-tansig method; the blue line with triangles represents the result gotten by KINN-Gaussian method; the green line with stars is the results obtained by SLFNN; the red line with circles is the liquid production rate gotten by ECLIPSE; the grey vertical dashed line makes a separation of history matching period and the productivity forecast period: (

**a**) P1; (

**b**) P2; (

**c**) P3; (

**d**) P4.

**Figure 15.**The training error (MSE) curves of KINN-tansig, KINN-Gaussian, and SLFNN for the Egg reservoir case. The black line with squares is the error curve of KINN-tansig; the blue line with triangles represents the error curve of KINN-Gaussian, and the green line with stars represents the error curve of SLFNN.

**Figure 16.**The heatmaps of the inter-well connectivity analysis by KINN-tansig and KINN-Gaussian for the Egg reservoir case: (

**a**) KINN-tansig; (

**b**) KINN-Gaussian; (

**c**) SLFNN.

**Figure 17.**The average absolute error of connectivity values by KINN-tansig and KINN-Gaussian in the noise measurement case. The black line is the error curve of KINN-tansig and the blue line is the error curve of KINN-Gaussian.

**Figure 18.**A cross plot of the connectivity values using KINN-tansig and KINN-Gaussian in the wells shut-in case against the basic case results. The black squares and blue circles represent the connectivity values of KINN-tansig and KINN-Gaussian, respectively, and the dashed is $45\xb0$ line.

Hyperparameter | KINN-Tansig | KINN-Gaussian |
---|---|---|

Learning rate | 0.05 | 0.05 |

Number of hidden layers in CVM | 3 | 3 |

Number of neurons of each layer in CVM | [1, 10, 1] | [1, 10, 1] |

Activation function in CVM | tansig function | Gaussian kernel function |

Initialization range of weights in CVM | [0, 0.25] | [0, 0.25] |

Initialization method of γ in IRM | Pearson Correlation | Pearson Correlation |

Optimization algorithm | Gradient descent method | Gradient descent method |

Convergence error (MSE) | ${10}^{-6}$ | ${10}^{-6}$ |

Properties | Value |
---|---|

Model Size | 31 × 31 × 1 |

Depth | 2000 m |

Initial pressure | 2000 psi |

Porosity | 0.18 |

Initial water saturation | 0.3 |

Density of oil | 900 kg/m^{3} |

Viscosity of oil | 2.0 cp |

Oil compressibility | 5.0 × 10^{−6} bar^{−1} |

**Table 3.**The time consumption, training error (MSE) and testing error (MSE) of KINN-tansig, KINN-Gaussian and SLFNN in the streak reservoir case.

Methods | KINN-Tansig | KINN-Gaussian | SLFNN |
---|---|---|---|

Computation time (training and testing) | 0.3702 s | 2.3393 s | 1.2737 s |

Error of history matching (training error) | 0.0046 | 0.0047 | 0.0976 |

Error of prediction (testing error) | 0.0223 | 0.0256 | 0.1832 |

**Table 4.**The time consumption, training error (MSE) and testing error (MSE) of KINN-tansig, KINN-Gaussian and SLFNN in the braided river reservoir case.

Methods | KINN-Tansig | KINN-Gaussian | SLFNN |
---|---|---|---|

Computation time (training and testing) | 0.7417 s | 3.4679 s | 2.4602 s |

Error of history matching (training error) | 0.0052 | 0.0058 | 0.0104 |

Error of prediction (testing error) | 0.0071 | 0.0065 | 0.0142 |

Properties | Value |
---|---|

Model Size | 100 × 99 × 1 |

Depth | 4000 m |

Initial pressure | 5765 psi |

Porosity | 0.2 |

Initial water saturation | 0.1 |

Density of oil | 900 kg/m^{3} |

Viscosity of oil | 2.0 cp |

Oil compressibility | 1.0 × 10^{−5} bar ^{−1} |

**Table 6.**The time consumption, training error (MSE), and testing error (MSE) of KINN-tansig, KINN-Gaussian, and SLFNN in the Egg reservoir case.

KINN-Tansig | KINN-Gaussian | SLFNN | |
---|---|---|---|

Computation time (training and testing) | 0.1282 s | 0.8539 s | 0.3361 s |

Error of history matching (training error) | 0.0022 | 0.0035 | 0.0097 |

Error of prediction (testing error) | 0.0171 | 0.0263 | 0.0426 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jiang, Y.; Zhang, H.; Zhang, K.; Wang, J.; Cui, S.; Han, J.; Zhang, L.; Yao, J.
Reservoir Characterization and Productivity Forecast Based on Knowledge Interaction Neural Network. *Mathematics* **2022**, *10*, 1614.
https://doi.org/10.3390/math10091614

**AMA Style**

Jiang Y, Zhang H, Zhang K, Wang J, Cui S, Han J, Zhang L, Yao J.
Reservoir Characterization and Productivity Forecast Based on Knowledge Interaction Neural Network. *Mathematics*. 2022; 10(9):1614.
https://doi.org/10.3390/math10091614

**Chicago/Turabian Style**

Jiang, Yunqi, Huaqing Zhang, Kai Zhang, Jian Wang, Shiti Cui, Jianfa Han, Liming Zhang, and Jun Yao.
2022. "Reservoir Characterization and Productivity Forecast Based on Knowledge Interaction Neural Network" *Mathematics* 10, no. 9: 1614.
https://doi.org/10.3390/math10091614