基于模型降维和递归神经网络的油藏参数反演

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

摘要/Abstract

摘要：

提出一种基于映射递归神经网络(aNN)的代理模型, 借助于深度学习框架中易于使用的自动微分工具, 有效生成模型降阶伴随算子, 实现高效反演建模。与降阶切线线性模型相似, 基于投影的神经网络(POD-aNN)结构加速降阶子空间的伴随模型的建立。POD-aNN由一个降维单元和一个用于投影的中间非线性过渡单元组成, 二者分别用于将状态系统分解为低维子空间和近似系统状态的时变演化。因此, 伴随模型在缩减的空间中运行, 计算量和内存需求可以忽略不计。结合二维石油模型展开油藏参数反演研究。结果表明: 该方法在显著降低计算量的前提下取得了满意的参数反演结果, 从而证明了该方法的有效性, 并为应用神经网络模型到实际参数反演案例提供理论基础。

关键词: 计算流体力学, 降阶模型, 伴随模型, 递归神经网络, 参数反演

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

肖聪, 张士诚, 马新仿, 周彤, 侯腾飞. 基于模型降维和递归神经网络的油藏参数反演[J]. 计算物理, 2022, 39(5): 564-578.

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.

图/表 14

图1 POD-aNN神经网络模型示意图(a) POD-aNN体系结构; (b) 残差网络结构1; (c) 残差网络结构2

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

图2 POD-aNN神经网络模型时序预测流程

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

图3 参考模型的空间渗透率场图解

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

表1 模型大小、流体特性和注入井/生产井设置以及训练POD-aNN的超参数

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

图4 饱和度本征谱的衰减及特征值的映射基

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

表2 二维石油模型的POD-aNN神经网络结构设置

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)

图5 不同超参数下损失函数的演化

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

图6 不同训练样本数的平均相对误差γSOE1和γSOE2

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

图7 不同POD模式的(a)POD-aNN和(b)HFM预测的时变相位饱和

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

图8 (a) POD-aNN和(b) HFM预测的三个位置的流体饱和度

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

图9 POD-aNN和FD计算的目标函数梯度及目标函数梯度角度

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

图10 不同训练集的目标函数的演化和地质渗透率场反演结果

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

图11 不同POD模式的目标函数的演化和地质渗透率场反演结果

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

图12 多后验渗透率场的平均和标准偏差(STD)

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

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