Model-reduced Autoregressive Neural Network for Parameter Inversion

doi:10.19596/j.cnki.1001-246x.8473

Abstract

Abstract:

We present an architecture of projection-based autoregressive neural network (aNN) where model-reduced adjoint is efficiently produced with the help of an easy-to-use auto-differentiation (AD) tool in deep-learning frameworks. Analogy to reduced-order tangent linear model, a projection-based aNN (POD-aNN) structure is proposed to accelerate the construction of adjoint model based on reduced subspace. The POD-aNN consists of a dimensionality reduction and an intermediate non-linear transition unit which is used to project a state system to a low-dimensional subspace and approximate time-varying evolution of system states in low dimension, respectively. The adjoint model is run in reduced space with negligible computational cost and memory requirement. Once the gradient is obtained in reduced space it is projected back in full space and then the inversion modeling is conducted. Characteristics and performance of the method are illustrated with two sets of inverse modeling experiments in a synthetic 2D fluid flow model with random spatially dependent parameters. It shows that the proposed POD-aNN obtains satisfactory results with significantly reduced computational cost and, therefore, demonstrates promising applicability to practical reservoir models.

Key words: computational fluid mechanics, reduced-order modeling, adjoint modeling, autoregressive neural network, parameter inversion

Cong XIAO, Shicheng ZHANG, Xinfang MA, Tong ZHOU, Tengfei HOU. Model-reduced Autoregressive Neural Network for Parameter Inversion[J]. Chinese Journal of Computational Physics, 2022, 39(5): 564-578.

Figures/Tables 14

Fig.1 Schematic illustration of POD-aNN neural network architecture

Fig.2 Consecutive prediction in the model states using a trained POD-aNN model

Fig.3 Illustration of spatial permeability field in the reference model

Table 1 Model size, fluid properties, injection/production well settings and hyperparameters for training POD-aNN

参数	取值	参数	取值
网格数	20 × 20	生产时间/d	1800
水、油密度/(kg·m^-3)	1 014; 859	训练数据集，N_s	400，600，800，1000
水、油黏度/(mP·s)	0.4; 2	测试数据集，N_t	200
地层水、油饱和度	S_w = 0.2; S_o = 0.8	初始学习效率	0.001
初始地层压力/MPa	30	优化算子	Adam
生产井压力/MPa	25	每训练周期样本数	100
注入井排量/(m³·d^-1)	150	训练周期数	500

Fig.4 Decay of eigen-spectrum for saturation and global basis vectors

Table 2 POD-aNN neural network structure setting for two-dimensional oil model

神经网络层	输入维数	输出维数
输入层	(N_ψ+ N_ξ, 1)	(140, 2)
残差块1	(140, 2)	(140, 2)
残差块2	(140, 2)	(130, 2)
残差块1	(130, 2)	(130, 2)
残差块2	(130, 2)	(120, 2)
残差块1	(120, 2)	(120, 2)
残差块2	(120, 2)	(120, 1)
全连接层	(120, 1)	(N_ξ, 1)

Fig.5 Evolution of loss functions with respect to hyper-parameters

Fig.6 Field-average relative errors γSOE1 and γSOE2 with respect to the number of training samples

Fig.7 Saturation predicted with (a) POD-aNN and (b) HFM corresponding to different number of POD modes

Fig.8 Time-varying phase saturation predicted with (a) POD-aNN and (b) HFM at three locations

Fig.9 Objective function gradients and their angles computed with POD-aNN and FD

Fig.10 Evolution of logarithmic objective function and updated geological permeability fields with respect to different number of training samples

Fig.11 Evolution of logarithmic objective function and updated geological permeability fields with respect to different number of POD modes

Fig.12 Average and standard deviation (STD) of an ensemble log-permeability fields

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