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Deep learning for solving partial differential equations (PDEs) has been a major research hotspot. Various neural network frameworks have been proposed to solve nonlinear PDEs. However, most deep learning-based methods need labeled data, while traditional numerical solutions do not need any labeled data. Aiming at deep learning-based methods behaving as traditional numerical solutions do, this paper proposed an approximation-correction model to solve unsteady compressible seepage equations with sinks without using any labeled data. The model contains two neural networks, one for approximating the asymptotic solution, which is mathematically correct when time tends to 0 and infinity, and the other for correcting the error of the approximation, where the final solution is physically correct by constructing the loss function based on the boundary conditions, PDE, and mass conservation. Numerical experiments show that the proposed method can solve seepage equations with high accuracy without using any labeled data, as conventional numerical solutions do. This is a significant breakthrough for deep learning-based methods to solve PDE.

An automatic well test interpretation method for radial composite reservoirs based on convolutional neural network (CNN) is proposed, and its effectiveness and accuracy are verified by actual field data. In this paper, based on the data transformed by logarithm function and the loss function of mean square error (MSE), the optimal CNN is obtained by reducing the loss function to optimize the network with “dropout” method to avoid over fitting. The trained optimal network can be directly used to interpret the buildup or drawdown pressure data of the well in the radial composite reservoir, that is, the log-log plot of the given measured pressure variation and its derivative data are input into the network, the outputs are corresponding reservoir parameters (mobility ratio, storativity ratio, dimensionless composite radius, and dimensionless group characterizing well storage and skin effects), which realizes the automatic initial fitting of well test interpretation parameters. The method is verified with field measured data of Daqing Oilfield. The research shows that the method has high interpretation accuracy, and it is superior to the analytical method and the least square method.

Well test analysis is a crucial technique to monitor reservoir performance, which is based on the theory of seepage mechanics, through the study of well test data, to identify reservoir models and estimate reservoir parameters. Reservoir model recognition is the first and essential step of well test analysis. It is usually judged by professionals’ experience, which results in low efficiency and accuracy. This paper is devoted to applying convolutional neural network (CNN) to well test analysis and proposes a new intelligent reservoir model identification method. Eight reservoir models studied in this paper include homogenous reservoirs with different outer boundaries such as infinite acting boundary, circular, single, angular, channel, U-shaped and rectangular sealing fault boundaries, and a radial composite reservoir with infinite acting boundary. Well testing data used in this paper, including actual field data and theoretical data, are generated by analytical solutions. To improve the classification accuracy of actual field data, noise processing was carried out on the data before training. The CNN that is most suitable for model recognition has been obtained through trial-and-error procedures. The availability of proposed CNN is proved with actual field cases of Daqing oil field, China. The method realizes the automatic identification of reservoir model with the total classification accuracy (TCA) of test data set of 98.68% and 95.18% for original data and noisy data, respectively.

Neural networks are increasingly used widely in the solution of partial differential equations (PDEs). This letter proposes 3D-PDE-Net to solve the three-dimensional PDE. We give a mathematical derivation of a three-dimensional convolution kernel that can approximate any order differential operator within the range of expressing ability and then conduct 3D-PDE-Net based on this theory. An optimum network is obtained by minimizing the normalized mean square error (NMSE) of training data, and L-BFGS is the optimized algorithm of second-order precision. Numerical experimental results show that 3D-PDE-Net can achieve the solution with good accuracy using few training samples, and it is of highly significant in solving linear and nonlinear unsteady PDEs.

Image denoising is a common problem during image processing. Salt and pepper noise may contaminate an image by randomly converting some pixel values into 255 or 0. The traditional image denoising algorithm is based on filter design or interpolation algorithm. There exists no work using the convolutional neural network (CNN) to directly remove salt and pepper noise to the authors’ knowledge. In this study, they utilise CNN with the multi-layer structure for the removal of salt and pepper noise, which contains padding, batch normalisation and rectified linear unit. In training, they divide images into three parts: training set, validation set and test set. Experimental results demonstrate that the architecture can effectively remove salt and pepper noise for the various noisy images. In addition, their model can remove high-density noise well due to the extensive local receptive fields of the deep neural networks. Finally, extensive experimental results show that their denoiser is effective for those images with a large number of interference pixels which may cause misjudgement. In a word, they generalise the application of CNN to salt and pepper noise removal and obtain competitive results.

Stochastic reconstruction of digital core images is a vital part of digital core physics analysis, aiming to generate representative microstructure samples for sampling and uncertainty quantification analysis. This paper proposes a novel reconstruction method of the digital core of shale based on generative adversarial networks (GANs) with powerful capabilities of the generation of samples. GANs are a series of unsupervised generative artificial intelligence models that take the noise vector as an input. In this paper, the GANs with a generative and a discriminative network are created respectively, and the shale image with 45 nm/pixel preprocessed by the three-value-segmentation method is used as training samples. The generative network is used to learn the distribution of real training samples, and the discriminative network is used to distinguish real samples from synthetic ones. Finally, realistic digital core samples of shale are successfully reconstructed through the adversarial training process. We used the Fréchet inception distance (FID) and Kernel inception distance (KID) to evaluate the ability of GANs to generate real digital core samples of shale. The comparison of the morphological characteristics between them, such as the ratio of organic matter and specific surface area of organic matter, indicates that real and reconstructed samples are highly close. The results show that deep convolutional generative adversarial networks with full convolution properties can reconstruct digital core samples of shale effectively. Therefore, compared with the classical methods of reconstruction, the new reconstruction method is more promising.

A dual-porosity and dual-permeability (DPDP) model was developed to describe the fluids flow and transient pressure behaviors of multistage hydraulically fractured horizontal wells in shale-gas reservoirs by incorporating gas adsorption and gas permeability corrections. This 3-dimensional, 1-phase numerical model was developed using finite volume method and solved implicitly with unstructured PEBI (Perpendicular Bisection) gridding. Parametric studies were conducted to understand the pressure transient behaviors with effects of gas adsorption, permeability correction, transmissibility between fracture and matrix system (called inter-porosity flow ability, IPF), and natural fracture spacing. The results showed that pressure transient behaviors were not only controlled by the individual ultimate adsorption capacity (UAC) and IPF but rather the combination of IPF ability and UAC derived herein. The physics was that for a block by omitting the flow in matrix systems and accumulation term, the flow rate of free gas between the matrix cell and the fracture cell was equal to the flow rate caused by adsorbed or desorbed due to a pressure change in the matrix cell. This was the reason that if the IPF ability was relatively small, large amount of free gas would flow into the wellbore, which affected the pressure transient behaviors. This finding was quantified with a new correlation that was a function of IPF ability, UAC and pressure difference between fracture and matrix.

Threshold pressure gradient (TPG) is an important mechanism for fluid flow through low-permeability reservoirs, which would explain the phenomena that pressure derivative curves exhibit a straight line of slope that is larger than zero after radial flow regime. A mathematical single-phase flow model incorporating pseudo TPG is proposed to describe the flow behavior in low permeability reservoirs. Fully implicit numerical simulation based on PEBI grid is developed to study the transient pressure response. Two field data are used to calibrate and validate the proposed model and the code. Based on one of field data, parametric studies are conducted to investigate the effect of minimum TPG, pseudo TPG and nonlinear exponent on the pressure transient response for a vertical well. We find that pseudo TPG can explain the unique and consistent characteristic of the pressure transient response in low permeability reservoirs. The finding is useful for petroleum engineers to interpret the field data to obtain some basic parameters for low permeability reservoirs.

The well testing technique is an important tool in estimating well and reservoir characteristics, such as permeability, skin factor and so on. For a long time, researchers have been searching for automatic well testing interpretation tools, but the results are disappointing. This paper proposes using convolutional neural network (CNN) as an automatic well test interpretation approach for infinite acting reservoirs. The CNN takes pressure change and pressure derivative data of the log-log plot for inputs. The wellbore storage coefficient, skin factor and reservoir permeability are redefined into a dimensionless group C D e 2 S as the output of the CNN. In this method, the best trained CNN structure is obtained by minimizing mean square error (MSE) and mean relative error (MRE). This new method is tested for its effectiveness and accuracy in Daqing oil field, China. It demonstrates that, for wells in infinite reservoir, CNN could be an effective automatic well test interpretation technique. CNN also shows the potential for more complicated scenarios.

Mesh denoising is a classical problem and has made great progress, but it has not been solved perfectly. Our paper presents an adaptive and dynamic mesh denoising method, which can remove noise effectively while preserving sharp features and visually meaningful fine-scale components. Most state-of-the-art approaches still fall short of robustly handling various noisy 3D models, because optimal parameters are usually selected manually, and remain unchanged for the whole model and throughout the whole denoising procedure. Actually, the parameters should be adaptively adjusted according to the feature intensity of different regions in each iteration and dynamically changed in subsequent iterations to avoid over-smooth. In this paper, the parameters are determined automatically and adjusted dynamically, which allows the real shape of the surface to be restored as much as possible, especially in feature regions. Extensive qualitative and quantitative experiments on various noisy meshes have demonstrated the effectiveness and robustness of our approach.

Summary It is common practice that one of the reservoir properties is recognized as a complex function of several interrelated factors in neural-network applications used in oil-reservoir studies. Few methods are based on one reservoir property being recognized as a function of time and the spatial locations, which means that the reservoir-property data can be described as a time vector series. It is a great challenge for the artificial neural network to describe a time vector series because the neural network is unable to approximate a multivariate vector function effectively. By combining the principle of implicit curves and surfaces with the neural network, we present a novel way to process the time vector series. The method includes the following steps: mapping data, constructing an explicit function, training the neural network, extracting the isoline, and inverse mapping. This paper presents an application to predict the isotope-logging data in 2002 with the known data from 2001 and 2003.

Time vector series abound in many fields, which are different from time series. Based on mapping and implicit function interpolation, the method of implicit interpolation over time vector series is developed and studied. Time vector series are mapped into a sequence of closed curves or surfaces, which are seen as the interpolation data set. Based on the interpolation data set, the implicit interpolation function is constructed and from which an isoline is drawn. The inverse mapping of the isoline is the prediction of vector series at the new time. Based on the well logging data an application is given to predict the unknown isotope logging data of year 2004 with the known data of year 2003 and 2005 which shows the performance of the method.