# Progress and Challenges of Integrated Machine Learning and Traditional Numerical Algorithms: Taking Reservoir Numerical Simulation as an Example

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## Abstract

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## 1. Introduction

## 2. Numerical Methods in the Reservoir Numerical Simulation

#### 2.1. Finite Difference Method

#### 2.2. Finite Element Method

#### 2.3. Finite Volume Method

#### 2.4. Meshless Method

#### 2.5. Boundary Element Method

## 3. Application of Machine Learning in Numerical Methods

#### 3.1. Machine Learning to Update the Basic Functions of Multi-Scale Methods and Perform Coarse Grid Calculations

#### 3.2. Machine Learning Replaces Phase Equilibrium Calculations during Numerical Reservoir Simulation

_{2}repulsion, the phase equilibrium calculation process is carefully divided into three parts: supercritical phase determination, subcritical phase stability analysis, and phase separation problems. Relevance vector machines (RVMs) were chosen to train two classifiers for critical state determination using component model pressure and composition data. The first classifier is used to identify whether the input conditions correspond to a supercritical region. By quickly identifying the supercritical phases, the properties of the supercritical fluids, such as density and viscosity in simulations involving CO

_{2}oil drive, can be accurately estimated. In addition to the supercritical classifier, a second classifier is used to identify the number of stable phases in the subcritical region. RVMs are used to determine the posterior probabilities for each class, which are used to construct criteria for predicting phase states. An ANN is trained to anticipate the equilibrium K value for a given pressure and combination, which is used to compute the phase splitting problem to eliminate expensive iterations in the flash evaporation problem.

## 4. Deep Learning Methods for Solving Partial Differential Equations

#### 4.1. Data-Driven Deep Learning Approach for Solving Reservoirs Partial Differential Equations

_{2}saturation and pressure field prediction to the carbon storage field.

_{2}saturation field distributions in carbon storage. The mapping relationship between reservoir permeability field and water saturation is established using the cDC-GAN network structure model to realize the regression from image (permeability field distribution) to image (water content saturation distribution). The cDC-GAN network contains a pair of generative discriminative models. The generative model learns the relationship between input and output so that the generated output is as close to the training data as possible. The discriminative model distinguishes the trained output from the real data, enabling the cDC-GAN to learn the real data distribution features. This network structure model used multiple output channels to achieve water content saturation output for multiple time steps. Based on the above work, Zhong et al. [90] modified the cDC-GAN model to build a Co-GAN model with multiple outputs, as shown in Figure 7. The model consists of two parts: the generative model and the discriminative model. The same spatial feature extraction model is used to predict saturation and pressure fields in the generative model part. Separate spatial feature learning models are used to achieve simultaneous prediction of the pressure and saturation fields. The model also uses multiple heads and output channels to output the prediction results at different time steps.

_{2}replacement development for shale oil and were able to predict production curves quickly and improve the model computation speed. Sagheer et al. [93] used a genetic algorithm to optimize the LSTM network architecture for the problem that the LSTM network structure model needs to be set manually. The best network structure model was selected to predict the reservoir well production, and the model was applied to the realized reservoir. Song et al. [94] used the particle swarm optimization (PSO) algorithm to optimize the LSTM network model. Fan et al. [95] proposed a hybrid ARIMA-LSTM model by combining the advantages of the LSTM for high accuracy in forecasting production curves with nonlinear variations and the autoregressive integrated moving average model (ARIMA) for forecasting linear trends. The model can filter the nonlinear production curve trends and transfer them to the LSTM model for prediction. The results show that the hybrid model performs better than the separate models, and the combined algorithm of different models will significantly improve the accuracy and computational efficiency of the model.

#### 4.2. Physics-Driven Deep Learning Approach for Solving Oil Reservoir Partial Differential Equations

#### 4.3. Physical-Constraints Deep Learning Approach for Solving Reservoir Simulation Problem

## 5. Conclusions

## 6. Future Development and Prospects

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\mathcal{B}\left(\cdot \right)\left(BC\right)$ | the boundary conditions of the reservoir |

$B(\widehat{y})$ | the boundary loss of the predicted value of the deep learning model |

${b}_{1}$ | the bias vectors of $1\times q$ |

${b}_{2}$ | the bias vectors of $1\times k$ |

$D(x,y,t)$ | smooth function used to encode the output of the neural network |

${b}^{*}$ | optimization objectives of the neural network |

$f$ | the source fields |

${f}_{\theta 2}(x,y,t)$ | the neural network prediction |

$\mathcal{I}\left(\cdot \right)\left(IC\right)$ | the initial conditions of the reservoir |

$k$ | dimensional output of a single hidden layer neural network |

$m$ | static parameters |

$n$ | the number of uniform grid points of the Sobel multiplier |

$\mathcal{N}\left(\cdot \right)$ | differential operator |

$\tau $ | $\left[{\tau}_{1},{\tau}_{2}\right]$ |

$s$ | production regime |

$\widehat{u}$ | the model prediction value |

${u}_{0}$ | the initial value |

$u(t,x;\theta )$ | the approximate solution of the equation |

$u(t,x)$ | the labeled data required for the data-driven approach |

${\widehat{u}}^{*}(t,x;W,b)$ | the neural network predictions |

$V(\widehat{y};x)$ | the error of the equation in the form of the residual parametrization of the partial differential equations or the generalized variational function |

$(W,b)$ | the optimal set of network parameters |

${W}_{1}$ | weight matrices of $d\times q$ |

${W}_{2}$ | weight matrices of $q\times k$ |

${W}^{*}$ | optimization objectives of the neural network |

$x$ | the input data of the neural network |

$x$ | the parameter of the deep learning model input |

$x{r}_{k}$ | relative permeability parameters |

$y$ | output data of the neural network |

$\widehat{y}$ | the predicted value of the deep learning model |

$\sigma \left(\xb7\right)$ | nonlinear model |

$\odot $ | the element-by-element product |

$\theta $ | the parameter corresponding to the approximate solution in the equation |

$\theta $ | the set of network parameters $\left\{W,b\right\}$ |

$\lambda $ | the weight of soft forced boundary condition (Lagrange multiplier) |

$\eta $ | the step size of the $i$th selected generation |

${\nabla}_{\theta}J$ | gradient of the loss function concerning the model parameter |

$\nabla u$ | the pressure field |

$\left[{u}_{h},{u}_{v}\right]$ | the two gradient images along the horizontal and vertical directions estimated by the Sobel filter |

$\tau $ | the flow field |

${\tau}_{1}$ | the horizontal components of the flow gradient field |

${\tau}_{2}$ | the vertical components of the flow gradient field |

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**Figure 5.**The network architecture of multiple input feature DenseED [85].

**Figure 6.**Network architecture of multiple input–output DenseED [87].

**Figure 7.**Network architecture of Co-GAN [90].

**Figure 8.**The network structure of LSTM [91].

**Figure 9.**Network architecture of R-U-Net [96].

**Figure 10.**Network architecture of recurrent R-U-Net [96].

**Figure 11.**Network architecture of DCML-NN [100].

**Figure 12.**MLGRU-based history matching method [102].

**Figure 14.**The neural network structure of physics-driven DenseED [103].

**Figure 15.**The workflow of AS-net [106].

**Figure 16.**The network structure of AS-net [106].

**Figure 17.**CNN structure-based single-step transient Darcy oil and gas flow simulation in porous media for two-dimensional reservoirs [108].

**Figure 18.**DCNN structure-based multi-step transient Darcy oil and gas flow simulation in porous media for two-dimensional reservoirs [108].

**Figure 19.**DCNN structure-based multi-step transient Darcy oil and gas flow simulation in porous media for two-dimensional reservoirs with spatial distribution K(x, y).

**Figure 20.**Single-step forecasting process [108].

**Figure 22.**A physics-guided autoregressive model [118].

**Figure 23.**The network structure of TGNN [123].

**Figure 24.**The network structure of TgCNN [131].

**Figure 25.**The network structure of coupling TGNN [133].

The Numerical Methods | Merits | Limitations |
---|---|---|

Finite Difference Method | The method exhibits strong intuitiveness, facilitates straightforward operation, ensures rapid computation, is easily implemented, and is adaptable to various physical problems and boundary conditions. | The method may encounter discretization errors, issues related to numerical stability, and challenges in addressing intricate geological structures, irregular meshes, and pronounced nonlinearities. |

Finite Element Method | The approach demonstrates significant flexibility when addressing intricate geometric configurations and diverse boundary conditions. | Specialized pre-processing and post-processing techniques are requisite. The method has computational complexity. |

Finite Volume Method | The approach ensures the conservation of physical quantities and accommodates irregular meshes. | Particular discretization strategies may be necessary, and discretization errors can arise. Accuracy depends on appropriate mesh density, and accurately describing heterogeneous and nonlinear reservoir characteristics can be challenging. |

Meshless Method | They are effectively handling highly dynamic problems. | Requires highly complex search algorithms and interpolation techniques. |

Boundary Element Method | They are effectively addressing infinite or semi-infinite problems. | They are challenging to apply for non-steady-state or nonlinear problems. |

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## Share and Cite

**MDPI and ACS Style**

Chen, X.; Zhang, K.; Ji, Z.; Shen, X.; Liu, P.; Zhang, L.; Wang, J.; Yao, J.
Progress and Challenges of Integrated Machine Learning and Traditional Numerical Algorithms: Taking Reservoir Numerical Simulation as an Example. *Mathematics* **2023**, *11*, 4418.
https://doi.org/10.3390/math11214418

**AMA Style**

Chen X, Zhang K, Ji Z, Shen X, Liu P, Zhang L, Wang J, Yao J.
Progress and Challenges of Integrated Machine Learning and Traditional Numerical Algorithms: Taking Reservoir Numerical Simulation as an Example. *Mathematics*. 2023; 11(21):4418.
https://doi.org/10.3390/math11214418

**Chicago/Turabian Style**

Chen, Xu, Kai Zhang, Zhenning Ji, Xiaoli Shen, Piyang Liu, Liming Zhang, Jian Wang, and Jun Yao.
2023. "Progress and Challenges of Integrated Machine Learning and Traditional Numerical Algorithms: Taking Reservoir Numerical Simulation as an Example" *Mathematics* 11, no. 21: 4418.
https://doi.org/10.3390/math11214418