Recent advances in deep neural networks have shed new light on physics, engineering, and scientific computing. Reconciling the data-centered viewpoint with physical simulation is one of the research hotspots. The physics-informed neural network (PINN) is currently the most general framework, which is more popular due to the convenience of constructing NNs and excellent generalization ability. The automatic differentiation (AD)-based PINN model is suitable for the homogeneous scientific problem; however, it is unclear how AD can enforce flux continuity across boundaries between cells of different properties where spatial heterogeneity is represented by grid cells with different physical properties. In this work, we propose a criss-cross physics-informed convolutional neural network (CC-PINN) learning architecture, aiming to learn the solution of parametric PDEs with spatial heterogeneity of physical properties. To achieve the seamless enforcement of flux continuity and integration of physical meaning into CNN, a predefined 2D convolutional layer is proposed to accurately express transmissibility between adjacent cells. The efficacy of the proposed method was evaluated through predictions of several petroleum reservoir problems with spatial heterogeneity and compared against state-of-the-art (PINN) through numerical analysis as a benchmark, which demonstrated the superiority of the proposed method over the PINN.

Flow and transport in porous media play a crucial role in a variety of subsurface energy and environmental applications, such as reservoir recovery, carbon sequestration, etc. Many of these problems can be characterized as partial differential equations derived from the principles of conservation. However, due to the intricate nature of PDEs and the absence of rigorous theoretical analysis techniques, the approximation of most PDEs is only resorted to based on discretized methods, such as finite difference, finite volume, and finite element. After decades of development, these methods have become robust and flexible. However, for considerably large problems, numerical simulations can become increasingly slow and sometimes prohibitively so [

In recent years, with the increase in GPU computing power and the generalization of deep learning frameworks [_{2} storage sites, which eliminates nonlinearity by decomposing nonlinear PDEs into elementary differential operators. Li et al. [

The auto-differentiation-based PINN model is suitable for a homogeneous scientific problem, however, it is unclear how AD can enforce flux continuity across boundaries between cells of different properties where spatial heterogeneity is represented by grid cells with different physical properties [

The remainder of this paper is structured as follows.

In this study, we introduce the general oil-water two-phase Darcy flow model (Single phase model can be obtained by simplifying the two-phase model) in the porous media. We consider the fluids are slightly compressible and immiscible, and there is no mass transfer between the phases. The governing equation can be written as follows:

where the

where the

where the

The convolutional neural network (CNN or ConvNet) is a deep learning structure that can learn directly from image data without manually extracting features [

where

where

where

Despite the tremendous advancements the traditional data-driven method has made, it is inevitably considered to be a ‘black box’, which means that the system does not embody any physical meaning of the dataset and the predictions may be physically inconsistent or implausible [

where α is o for oil and w for water;

where

To achieve the seamless enforcement of flux continuity and integration of physical meaning into CNN, a predefined 2D convolutional layer with a criss-cross kernel layer is proposed, specifically as shown in

Therefore the discretized residual of the control equation (

where

For boundary condition constraints, the Cov2D operation of CNN is used to implement the constant pressure and closed boundary conditions, as shown in

The initial conditions are added to the loss function as a penalty item (

where

Similar to conventional neural networks, the MSE between prediction and truth data can also be added to the loss function of neural networks.

where

As a result, the total loss function can be as follows:

Typically, weights are assigned to each subterm of the loss function. However, there is no systematic analysis of weight determination or optimization in the literature, and these weights are commonly tuned by hand based on experience or trial and error. They remain constant during the training process [

Dimension | Convolutional kernel | |
---|---|---|

Input | (1, 1, 51, 51) | / |

Conv 1 | (1, 16, 49, 49) | k3s2p0 |

Conv 2 | (1, 32, 24, 24) | k5s2p1 |

Conv 3 | (1, 64, 11, 11) | k5s2p1 |

Conv 4 | (1, 128, 9, 9) | k3s1p0 |

Linear 1 | (100) | / |

Linear 2 | (100) | / |

Linear 3 | (10368) | / |

Dconv1 | (1, 64, 11, 11) | k3s2p0 |

Dconv2 | (1, 32, 23, 23) | k5s2p1 |

Dconv3 | (1, 16, 47, 47) | k5s2p1 |

Dconv4 | (1, 16, 51, 51) | k5s1p0 |

In this section, we evaluate the performance of the criss-cross physics-informed convolutional neural network by assessing it on two heterogeneous reservoir problems: a single-phase problem and an oil-water two-phase problem.

Initially, we examine a 2D heterogeneous reservoir problem, which is derived from an actual block, and labeled data is generated through numerical simulations. The geological model features closed boundaries on all sides. The initial pressure is 300 bar. The dimension of the reservoir model is ^{3}/d respectively located in (510, 510), (210, 210), (770, 310).

The CC-PINN takes a single-channel input, represented by the time matrix, and produces a single-channel output in the form of the pressure field image. The network architecture is illustrated in _{2} error, as described by

where the

In this study, the data of the initial 65-time steps were extracted as the training dataset to develop a predictive model for the pressure response over the subsequent 35-time steps response. Subsequently, we established the geological and mathematical models, along with the requisite network configurations, to train and evaluate the CC-PINN model. The predicted pressure of CC-PINN at time step 75 and 100 are illustrated in

The histogram of relative L_{2} error (a) obtained by CC-PICNN and PINN on the test dataset is shown in _{2} error and R^{2} score metric reach the same level, the CC-PICNN is more precise and smoother, since the spatial heterogeneity of permeability is better satisfied and expressed by a predefined convolutional layer. However, for PINN, the error tends to increase as time evolves, probably due to the cumulative effect of the error.

The results show the satisfactory accuracy of the CC-PINN and superior performance compared with the PINN model. The accuracy of CC-PINN is further validated by comparing the bottom-hole pressure (BHP) curves to reference numerical solutions and PINN. As shown in

Oil well-1 | Oil well-2 | Oil well-3 | |
---|---|---|---|

CC-PICNN | 0.9938 | 0.9999 | 0.9996 |

PINN | 0.9275 | <0 | 0.5324 |

In this sub-section, the CC-PINN model is demonstrated in the context of a two-phase heterogeneous reservoir scenario. Unlike the single-phase problem, the governing equations (

The geological model to be solved is a domain covering 510 × 510 × 10 m^{3} with four no-flow boundary conditions. The coordinates of the injection and production wells are (5, 5) and (505, 505), respectively. In this work, we assume that the rock and fluid are incompressible. The porosity is 0.2 and the permeability is shown in ^{3}/day and a constant well bottom pressure of 150 bar, respectively. The parameter configuration utilized in CC-PINN remains consistent with the single-phase problem in

The predicted pressure and water saturation of CC-PINN at time step 75 and 100 are illustrated in

Based on the predicted data, it is observed that the PINN model exhibits an evolution over time, as evidenced by the histograms of R^{2} scores and relative L_{2} errors for the 35 predicted test time steps by CC-PINN and PINN in _{2} errors and a corresponding decrease in R^{2} scores, which could potentially be attributed to error accumulation. The aforementioned findings are consistent with the established literature on error propagation in machine learning models.

The accuracy of CC-PINN is further validated by comparing the bottom-hole pressure (BHP) of the injection well and liquid production rate of oil well curves to the PINN model and reference numerical solutions benchmark, as shown in

Model | Oil well | Injection well | |
---|---|---|---|

BHP | CC-PINN | \ | 0.857070452 |

PINN | \ | <0 | |

Oil production rate | CC-PINN | 0.982053324 | \ |

PINN | <0 | \ | |

Water production rate | CC-PINN | 0.85433558 | \ |

PINN | <0 | \ |

The oil production rate and water production rate curves obtained by CC-PICNN exhibit a remarkable degree of congruence with the reference numerical solutions, as opposed to those obtained by PINN. Specifically, the non-convergence of the predicted curve obtained by PINN can be attributed to the inadequacy of convergence in the well-containing grid pressure and water saturation. The non-convergence can be ascribed to the inadequate prediction of well grid pressure and non-adherence of two-phase saturation to the conservation condition of summing up to unity. This leads to inaccuracies in production pressure difference and relative permeability, ultimately culminating in substantial errors in the resultant production curve.

Although showing good ability in addressing spatially heterogeneous problems, the current framework has several limitations, and many technical challenges are still present. Firstly, the data used in this paper are derived from numerical simulations, which may not always be applicable to scientific problems. Furthermore, the acquisition of numerical simulation data is resource-intensive, which may limit the practical application of the model. Therefore, it is crucial to explore methods that use little or no labeled data in future research. Secondly, the CNN-based physics-informed model in this paper can only handle regular boundaries at present and cannot solve complex boundary conditions. This is a technical challenge that necessitates further research and development.

In this work, a criss-cross physics-informed convolutional neural network (CC-PINN) is proposed for predicting porous media fluid flow with spatial heterogeneity. A 2D convolutional layer with a criss-cross convolutional kernel is introduced to achieve the seamless enforcement of flux continuity and integration of physical meaning into CNN. The layer is designed to enable the direct use of powerful classic CNN backbones for expressing transmissibility between adjacent cells, discretized residuals of PDEs, harmonic means of permeability, upstream-weighted differencing schemes, and boundary conditions. The introduction of this layer is motivated by the need to ensure accurate representation and conservation of information in the discretized domain. The criss-cross convolutional kernel utilized in this layer facilitates the computation of the relevant physical quantities and guarantees the preservation of their inherent properties in the CNN model. The initial conditions, and discretized PDEs residual expressed by the criss-cross convolutional kernel are imposed in the loss function as physical penalty terms for data match loss, additionally, the boundary conditions are seamlessly integrated into the training process by padding operation in CNN. In essence, the solving process of PDEs in numerical simulators is replaced by the training procedure of deep learning, and the solving scheme can be learned by the network. Once trained, the CC-PINN can be used to predict future pressure and saturation distributions. The effectiveness and merit of the proposed CC-PINN have been demonstrated by solving several dynamic subsurface flow instances in reservoir porous media, encompassing a spectrum of heterogeneous problems, ranging from single-phase to two-phase complexities. Through numerical analysis as a benchmark, we compared the proposed CC-PICNN model with the state-of-the-art PINN model. The findings demonstrate that the CC-PINN exhibits a faster convergence rate than the PINN model, and the accuracy of the CC-PINN model surpasses that of the PINN model, provided that the total training dataset and iteration number remain constant. These results highlight the superiority of the proposed CC-PINN model over the existing state-of-the-art PINN model for the prediction of porous media fluid flow with spatial heterogeneity.

None.

This work was supported by the National Natural Science Foundation of China (No. 52274048), Beijing Natural Science Foundation (No. 3222037), the CNPC 14th Five-Year Perspective Fundamental Research Project (No. 2021DJ2104), and the Science Foundation of China University of Petroleum, Beijing (No. 2462021YXZZ010).

The authors confirm contribution to the paper as follows: study conception and design: Liang Xue, Jiangxia Han; data collection: Ying Jia, Mpoki Sam Mwasamwasa, Felix Nangukar; analysis and interpretation of results: Charles Sangweni, Hailong Liu, Qian Li; draft manuscript preparation: Jiangxia Han, Liang Xue. All authors reviewed the results and approved the final version of the manuscript.

The raw data supporting the findings of this study are available from the

The authors declare that they have no conflicts of interest to report regarding the present study.

_{2}minimum miscibility pressure based on artificial neural network