# Waterflooding Interwell Connectivity Characterization and Productivity Forecast with Physical Knowledge Fusion and Model Structure Transfer

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## Abstract

**:**

## 1. Introduction

- (1)
- A novel neural network named PKFNN is developed, cooperating with ODE (the material balance equation) to control the approximation of the waterflooding process, thereby revealing the physical principle of the data and guaranteeing the rationality of estimation.
- (2)
- A physical evaluation function is built to ensure the physical boundaries of inter-well connectivity and avoids the complex computation resulting from constraint optimization.
- (3)
- The physical knowledge transfer and model structure transfer are employed in PKFNN to cope with the continuity and homogeneity of geological properties, increasing the interactions between models.

## 2. Preliminary Knowledge

#### 2.1. Transfer Learning

_{1}, x

_{2}, …, x

_{k}, …,x

_{K}} belongs to x; $\mathcal{T}$ = {y, f(x)} is a task, where y is a label space, and 𝑓(x) is the predictive function. When the source domain $\mathcal{D}$

_{s}, source task $\mathcal{T}$

_{s}and target domain $\mathcal{D}$

_{t}, target task $\mathcal{T}$

_{t}are given, where $\mathcal{D}$

_{s}≠ $\mathcal{D}$

_{t}, and/or $\mathcal{T}$

_{s}≠ $\mathcal{T}$

_{t}, transfer learning would help the predictive function f

_{t}(.) to improve its performance by learning the knowledge from $\mathcal{D}$

_{s}and $\mathcal{T}$

_{s}. The pre-training strategy is widely used in the transfer learning of neural networks. When a neural network finishes its task, its learned structures can be transferred to another neural network as the initialization weights, which means the knowledge learned by the first network can be transferred to the second one. Thus, pre-training can improve the generalization ability and shorten the training time of neural networks.

#### 2.2. Inter-Well Connectivity Characterization Based on the Material Balance Equation

## 3. Physical Knowledge Fusion-Based Neural Network (PKFNN)

#### 3.1. Knowledge-Distillation Block

_{1}, i

_{2}, …, i

_{m}, …, i

_{M}]

^{T}. Followed with the input signals, the physical evaluation is constituted for the evaluation of the contributions between each injector to the considered producer. Summing the dot product of the given input vectors and their corresponding physical evaluation values, the output of the knowledge-distillation block yields:

_{1}, i

_{2}, …, i

_{m}, …, i

_{M}]

^{T}denotes the WIRs of M injectors, and Q = [q

_{1}, q

_{2}, …, q

_{o}, …, q

_{O}]

^{T}is the LPRs of O producers; $\rho \left(I,Q\right)$ and $cov(I,Q)$ are the Pearson correlation coefficients and the covariance matrix calculated from I and Q; ${\sigma}_{I}$, ${\sigma}_{Q}$ represent the standard deviations of I and Q, respectively.

#### 3.2. Mapping-Transfer Block

#### 3.3. Model Framework and Learning Strategy

**Algorithm**1. Based on WIR and LPR data and the BHP data (if available) of injectors and producers, PKFNN aims to approximate and forecast the productivity of producers and characterize the inter-well connectivity. First of all, for each producer from 1 to O, the physical evaluation function is initialized by Equations (6) and (8). In this way, the multiplicity of the inter-well connectivity characterization can be reduced by transferring the physical information in the knowledge-distillation block. Moreover, the model structure transfer learning is applied in the mapping transfer block to inherit the weights learned from the target domain. For the first model, the initialized connecting weights obey the normal distribution with 0 mean value and 0.25 variance, $\mathcal{N}$(0, 0.25). When the history matching task is finished, the optimized weights would be transferred to the second model, according to Equation (19), and so on until all models finish training. Thus, the topology structures learned from the source domain can be inherited from the current model in the target domain. In this process, the injection data are in both the source task and the target task, and the production data (BHP data and liquid production rate data) are selected according to the analyzed producer. Then, the outputs of the knowledge-distillation block and the mapping-transfer block can be calculated by Equations (5) and (13) (or Equation (14)), respectively. Next, the model output can be generated by the material balance equation, Equation (16) (or Equation (17)), and the model loss can be computed by Equation (18). Finally, the network weights, $\theta $, and the independent variables of the physical evaluation function, $\gamma $, can be optimized by their gradients with respect to the loss function.

Algorithm 1. Pseudocode of PKFNN training process. |

/Start PKFNN training/ |

For o = 1 to O do |

While stop criteria is not met |

/Physical knowledge transfer/Initialize physical evaluation function using the ${o}_{th}$ column, ${\gamma}_{o}$, according to Equations (6) and (8); |

/Model structure transfer/If $o=1$Initialize ${\theta}_{o}^{i}$ obeying normal distribution $\mathcal{N}$(0, 0.25); Else doAssign ${\theta}_{o}^{f}$ to ${\theta}_{o-1}^{i}$, according to Equation (19); |

/Knowledge-distillation block calculation/Calculate the total injected water flowing rate, ${\Gamma}_{o}$, according to Equation (5); /Mapping-transfer block calculation/Constant producing BHP case: calculate the fluid change rate of ${V}_{p}$ byEquation (13); Variable producing BHP case: calculate the fluid change rate of ${V}_{p}$ byEquation (14); |

/Loss evaluation/Generate the model output ${\widehat{q}}_{o}$ byEquation (16) (or Equation (17)); Calculate the loss function using Equation (18); |

/Parameters update/Update $\gamma $ and $\theta $ via their gradients with respect to the loss function, Equation (18). End ifEnd whileEnd for |

/End PKFNN training/ |

#### 3.4. Productivity Forecast

## 4. Results

#### 4.1. 8-Injector–8-Producer Case

#### 4.2. Braided River Case

^{0}without transfer learning, while the initial MSE is at least lower over one order magnitude than Figure 15a by transfer learning, as shown in Figure 15b. In addition, PRO-02 needs less iteration times (about 50 runs) under this transfer learning frame, and the converged MSE of the other producers is also about one order magnitude lower than that without transfer learning.

#### 4.3. Egg Case

**,**all four of these well pairs are also assigned to relatively big values by the physical evaluation function.

#### 4.4. Brugge Field Case

^{3}/D until the end of production. The BHP data of the five injectors are shown in Figure 25a, similar to the trends of the WIR curves in Figure 24, staying at 0 bar for the same periods and growing slowly from 160 bar to 230 bar during the injection process. From PRO-01 to PRO-05, the five producers are shut in at the first 60, 180, 210, 270, 330 days, respectively, as illustrated in Figure 25b. Then, except for PRO-02, the BHP of the other four production wells start at around 150 bar, maintain a very slight downward trend until about Day 750, and slowly increase to about 200 bar at Day 3030. PRO-02 starts production with BHP of 131 bar at Day 181, and then the BHP decreases to 92 bar at Day 690, and after that, it increases to 188 bar at Day 3030. Additionally, compared with the BHP curves of the other four producers, there are more subtle fluctuations in the BHP curve of PRO-02. Same as the training mode employed in the Egg case, in the Brugge field case, the BHP data of five injectors and five producers are also fed to PKFNN for the approximation and prediction of productivity and water cut, via the mapping-transfer block.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Nomenclature | Explanations |
---|---|

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

I_{m} | water injection rate, m^{3}/D |

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

M | number of injectors |

O | number of producers |

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

p_{wf} | bottom hole pressure, bar |

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

q | liquid production rate, m^{3}/D |

t | time step, D |

$BH{P}_{m}$ | bottom-hole pressure of injector, bar |

$BH{P}_{o}$ | bottom-hole pressure of producer, bar |

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

λ_{mo} | inter-well connectivity value |

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

$\rho $ | Pearson correlation coefficient |

${\Gamma}_{o}$ | comprehensive injection rate, m^{3}/D |

m | injector index |

o | producer index |

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**Figure 3.**The transfer learning used in PKFNN. (

**a**) Physical knowledge transfer of the physical evaluation function in the knowledge-distillation block; (

**b**) Model structure transfer of the neural network in the mapping-transfer block.

**Figure 6.**The observed and modeled production rates of 8-injector–8-producer case. The black lines denote the observed data, and the red lines represent the model outputs; the gray vertical lines separate the results into history-matching stage and productivity-forecast stage: (

**a**) PRO-01; (

**b**) PRO-02; (

**c**) PRO-03; (

**d**) PRO-04; (

**e**) PRO-05; (

**f**) PRO-06; (

**g**) PRO-07; (

**h**) PRO-08.

**Figure 7.**The observed and modeled water cut of 8-injector–8-producer case. The black lines denote the observed data, and the red lines represent the model outputs; the gray vertical lines separate the results into history-matching stage and productivity-forecast stage: (

**a**) PRO-01; (

**b**) PRO-02; (

**c**) PRO-03; (

**d**) PRO-04; (

**e**) PRO-05; (

**f**) PRO-06; (

**g**) PRO-07; (

**h**) PRO-08.

**Figure 9.**The mean square error curves of 8-injector–8-producer case: (

**a**) PKFNN without model structure transfer; (

**b**) PKFNN with model structure transfer.

**Figure 12.**The observed and modeled production rates of braided river case. The black lines denote the observed data, and the red lines represent the model outputs; the gray vertical lines separate the results into history-matching stage and productivity-forecast stage: (

**a**) PRO-01; (

**b**) PRO-02; (

**c**) PRO-03; (

**d**) PRO-04; (

**e**) INJ-03.

**Figure 13.**The observed and modeled water cut of braided river case. The black lines denote the observed data, and the red lines represent the model outputs; the gray vertical lines separate the results into history-matching stage and productivity-forecast stage: (

**a**) PRO-01; (

**b**) PRO-02; (

**c**) PRO-03; (

**d**) PRO-04; (

**e**) INJ-03.

**Figure 14.**The inter-well connectivity characterization results of braided river case: (

**a**) inter-well connectivity heatmap before the well conversion operation; (

**b**) inter-well connectivity heatmap after the well conversion operation.

**Figure 15.**The mean square error curves of braided river case: (

**a**) PKFNN without model structure transfer; (

**b**) PKFNN with model structure transfer.

**Figure 19.**Injection rates of Egg case. The observed and modeled production rates of Egg case. The black lines denote the observed data, and the red lines represent the model outputs. The gray vertical lines separate the results into history-matching stage and productivity-forecast stage: (

**a**) PRO-01; (

**b**) PRO-02; (

**c**) PRO-03; (

**d**) PRO-04.

**Figure 20.**The observed and modeled water cut of Egg case. The black lines denote the observed data, and the red lines represent the model outputs; the gray vertical lines separate the results into history-matching stage and productivity-forecast stage: (

**a**) PRO-01; (

**b**) PRO-02; (

**c**) PRO-03; (

**d**) PRO-04.

**Figure 22.**The mean square error curves of Egg case. (

**a**) PKFNN without model structure transfer; (

**b**) PKFNN with model structure transfer.

**Figure 25.**BHP curves of Brugge field case: (

**a**) BHP curves of 5 injectors; (

**b**) BHP curves of 5 producers.

**Figure 26.**The observed and modeled production rates of Brugge field case. The black lines denote the observed data, and the red lines represent the model outputs; the gray vertical lines separate the results into history-matching stage and productivity-forecast stage: (

**a**) PRO-01; (

**b**) PRO-02; (

**c**) PRO-03; (

**d**) PRO-04; (

**e**) PRO-05.

**Figure 27.**The observed and modeled water cut of Brugge field case. The black lines denote the observed data, and the red lines represent the model outputs; the grey vertical lines separate the results into history-matching stage and productivity-forecast stage: (

**a**) PRO-01; (

**b**) PRO-02; (

**c**) PRO-03; (

**d**) PRO-04; (

**e**) PRO-05.

**Figure 29.**The mean square error curves of Brugge field case: (

**a**) PKFNN without model structure transfer; (

**b**) PKFNN with model structure transfer.

Hyperparameter | PKFNN |
---|---|

Learning rate | 0.02 |

Number of neurons in mapping-transfer block | 10 |

Number of layers in mapping-transfer block | 3 |

Activation function of the hidden layer in mapping-transfer block | $\frac{{e}^{x}-{e}^{-x}}{{e}^{x}+{e}^{-x}}$ |

Initialization for connecting weights, $\theta $ | $\mathcal{N}$(0, 0.25) |

Initialization for independent variables of the physical evaluation function, $\gamma $ | Reciprocal of Pearson correlation |

Optimization method | Backpropagation algorithm and transfer learning |

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

Model scale | 100 × 100 × 1 grid |

Grid size | 75 × 75 × 10 ft |

Depth of reservoir top | 1800 m |

Initial reservoir pressure | 145 bar |

Initial reservoir temperature | 100 °C |

Pore compressibility | 1.45 × 10^{−5} bar^{−1} |

Porosity | 0.16 |

Initial oil saturation | 0.70 |

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

Viscosity of oil | 2.2 cP |

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

Density of water | 1000 kg/m^{3} |

Viscosity of water | 0.5 cP |

Water compressibility | 1.0 × 10^{−6} bar^{−1} |

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

Model scale | 100 × 100 × 1 grid |

Grid size | 80 × 80 × 10 ft |

Depth of reservoir top | 2200 m |

Initial reservoir pressure | 140 bar |

Initial reservoir temperature | 100 °C |

Pore compressibility | 1.45 × 10^{−5} bar^{−1} |

Porosity | 0.15 |

Initial oil saturation | 0.72 |

Density of oil | 850 kg/m^{3} |

Viscosity of oil | 2.3 cP |

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

Density of water | 1000 kg/m^{3} |

Viscosity of water | 0.5 cP |

Water compressibility | 1.0 × 10^{−6} bar^{−1} |

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

Model scale | 100 × 99 × 1 (6910 active) grid |

Grid size | 8 × 8 × 4 m |

Depth of reservoir top | 4000 m |

Initial reservoir pressure | 400 bar |

Initial reservoir temperature | 100 °C |

Pore compressibility | 1.45 × 10^{−5} bar^{−1} |

Porosity | 0.2 |

Initial oil saturation | 0.90 |

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

Viscosity of oil | 10.0 cP |

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

Density of water | 1000 kg/m^{3} |

Viscosity of water | 0.5 cP |

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

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

Model scale | 139 × 48 × 9 grid |

Grid size | 75 × 75 × 2.5 m |

Depth of reservoir top | 1700 m |

Initial reservoir temperature | 100 °C |

Pore compressibility | 5.08 × 10^{−5} bar^{−1} |

Porosity | |

Initial oil saturation | |

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

Viscosity of oil | 1.294 cP |

Oil compressibility | 1.34 × 10^{−4} bar^{−1} |

Density of water | 1000 kg/m^{3} |

Viscosity of water | 0.32 cP |

Water compressibility | 4.35 × 10^{−5} bar^{−1} |

Model scale | 139 × 48 × 9 grid |

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## Share and Cite

**MDPI and ACS Style**

Jiang, Y.; Zhang, H.; Zhang, K.; Wang, J.; Han, J.; Cui, S.; Zhang, L.; Zhao, H.; Liu, P.; Song, H.
Waterflooding Interwell Connectivity Characterization and Productivity Forecast with Physical Knowledge Fusion and Model Structure Transfer. *Water* **2023**, *15*, 218.
https://doi.org/10.3390/w15020218

**AMA Style**

Jiang Y, Zhang H, Zhang K, Wang J, Han J, Cui S, Zhang L, Zhao H, Liu P, Song H.
Waterflooding Interwell Connectivity Characterization and Productivity Forecast with Physical Knowledge Fusion and Model Structure Transfer. *Water*. 2023; 15(2):218.
https://doi.org/10.3390/w15020218

**Chicago/Turabian Style**

Jiang, Yunqi, Huaqing Zhang, Kai Zhang, Jian Wang, Jianfa Han, Shiti Cui, Liming Zhang, Hanjun Zhao, Piyang Liu, and Honglin Song.
2023. "Waterflooding Interwell Connectivity Characterization and Productivity Forecast with Physical Knowledge Fusion and Model Structure Transfer" *Water* 15, no. 2: 218.
https://doi.org/10.3390/w15020218