Application of physics-informed neural network in two-phase flow

doi:10.11949/0438-1157.20241241

Abstract

Abstract:

The influx of research on machine learning and data science from the field of computer science into chemical engineering presents transformative opportunities for chemical engineering paradigms. Among them, physics-informed neural network (PINN) has gained wide attention because it embeds physical equations into neural networks so that the network output satisfies physical laws. This work begins by introducing the algorithm ideas and sampling strategies of PINN. It further discuss various treatment of the PINN loss function, mainly including cases with no observational data, equation reduction, equation discretization, and partial embedding of physical equations. Finally, it provides an overview of recent progress in the application of PINN to areas such as gas-liquid two-phase flow, two-phase flow in porous media, liquid-solid two-phase flow, and heat transfer in two-phase flow.

Key words: fluid mechanics, multiphase flow, numerical simulation, physics-informed neural network, sampling strategy, loss function

摘要：

机器学习和数据科学相关研究从计算机科学学科涌向化学工程，将为化学工程领域创造变革范式的机会，其中物理信息神经网络（PINN）因将物理方程嵌入神经网络中使得网络输出满足物理规律而获得广泛关注。首先介绍PINN的算法思想及其采样策略；进一步讨论对PINN的损失函数不同的处理方式，主要包括无观测值、方程降阶、方程离散化和只嵌入部分物理方程等；最后概述了PINN方法在气液两相流、多孔介质两相流、液固两相流、两相流传热等领域最新进展。

关键词: 流体力学, 多相流, 数值模拟, 物理信息神经网络, 采样策略, 损失函数

CLC Number:

TQ 021.9

Cheng ZHANG, Xue LI, Mao YE, Zhongmin LIU. Application of physics-informed neural network in two-phase flow[J]. CIESC Journal, 2024, 75(11): 3835-3856.

张橙, 李雪, 叶茂, 刘中民. 物理信息神经网络在两相流中的应用[J]. 化工学报, 2024, 75(11): 3835-3856.

Figures/Tables 17

Fig.1 Physics-informed neural network for partial differential equation[48]

Fig.2 (a) Illustration of physics-informed neural network for the phase-field method; (b) Different time marching strategies of physics-informed neural networks for the phase-field method: (b1) single network training in the whole time domain, (b2) multiple-networks training in various time sequences[64]

Table 1 Differences between a normal neural network and a deep mixed residual method network in interface evolution equation［67］

\frac{\partial C}{\partial t} + (u \cdot \nabla) C = \nabla \cdot (M_{0} \nabla [C (C^{2} - 1) - ε^{2} \nabla^{2} C])

L = {‖\frac{\partial C}{\partial t} + (u \cdot \nabla) C - \nabla \cdot \{M_{0} \nabla [C (C^{2} - 1) - ε^{2} \nabla^{2} C]\}‖}_{2}^{2}

深度混合残差形式

$\frac{\partial C}{\partial t} + (u \cdot \nabla) C = \nabla \cdot (M_{0} \nabla ϕ)$

$ϕ = C (C^{2} - 1) - ε^{2} \nabla^{2} C$

L = {‖\frac{\partial C}{\partial t} + (u \cdot \nabla) C - \nabla \cdot (M_{0} \nabla ϕ)‖}_{2}^{2} + {‖ϕ - [C (C^{2} - 1) - ε^{2} \nabla^{2} C]‖}_{2}^{2}

Table 1 Differences between a normal neural network and a deep mixed residual method network in interface evolution equation［67］

界面演化方程

显式形式

损失函数

普通形式

\frac{\partial C}{\partial t} + (u \cdot \nabla) C = \nabla \cdot (M_{0} \nabla [C (C^{2} - 1) - ε^{2} \nabla^{2} C])

L = {‖\frac{\partial C}{\partial t} + (u \cdot \nabla) C - \nabla \cdot \{M_{0} \nabla [C (C^{2} - 1) - ε^{2} \nabla^{2} C]\}‖}_{2}^{2}

深度混合残差形式

$\frac{\partial C}{\partial t} + (u \cdot \nabla) C = \nabla \cdot (M_{0} \nabla ϕ)$

$ϕ = C (C^{2} - 1) - ε^{2} \nabla^{2} C$

L = {‖\frac{\partial C}{\partial t} + (u \cdot \nabla) C - \nabla \cdot (M_{0} \nabla ϕ)‖}_{2}^{2} + {‖ϕ - [C (C^{2} - 1) - ε^{2} \nabla^{2} C]‖}_{2}^{2}

Fig.3 (a) The CNN structure as a solver for simulation and data assimilation; (b) The training scheme for PICNN-based reservoir simulation[68]

Fig.4 Schematic diagram of the semi-physics-informed neural network BubbleNet[69]

Fig.5 (a) Predicted results of phase-field variable C at a reverse single vortex based on physics-informed neural network for the phase-field method: (a1) t=0 s, (a2) t=0.5 s, (a3) t=1.0 s, (a4) t=1.5 s, (a5) t=2.0 s; (b) At bubble-rising problem: temporal evolution of bubble center of mass (b1) and rising velocity (b2) (the red solid lines correspond to the present result, while the blue dashed lines correspond to the reference result obtained from Aland and Voigt)[64]

Fig.6 (a) Computational domain and initial condition of RT instability; (b) Comparison of evolution of volume fraction of RT instability : (b1) results from modified PF-PINNs, (b2) results from reference[67]

Fig.7 (a) Computational domain and initial condition of single bubble flow case and multiple bubble flow case; (b) At single flow case: comparison of physical quantity u, v, p, φ between CFD, DNN, BubbleNet (without Poisson equation) and BubbleNet (with Poisson equation); (c) At multiple bubble flow case: comparison of physical quantity u, v, p, φ between CFD, DNN, BubbleNet (without Poisson equation) and BubbleNet (with Poisson equation)[69]

Fig.8 (a) Initial condition of one-dimensional computational domain; (b) Initial condition of two-dimensional computational domain; (c) Evolution of collocation points with new adaptivity (from left to right represent different governing equations and new collocation points after each adaptation step are shown in red)[58]

Fig.9 (a) An illustration of the reservoir model with injector, producer and boundary conditions; (b) The permeability field for the heterogeneous test case; (c) The pressure (c1) and water saturation fields (c2) at the end of simulation for the heterogeneous test case; (d) The change of well-block and bottom-hole pressures of the producer obtained by PICNN for the heterogeneous reservoir, compared to reference solution by the conventional FV approach[68]

Fig.10 (a) Physics-informed neural network with observed values for prediction of pressure, gas saturation and water production;A comparison between the ground truth and predicted value together with the corresponding relative error for (b) gas saturation and (c) pressure field: multi-output data-driven model (top), physics-informed model without interpolated points (middle) and physics-informed model with interpolated points (bottom)[71]

Table 2 Recorded MSE values （original un-normalized scale） for STT （single-output training）， MTT （multi-output training）， PI w/o interp（physics-informed without interpolated points）， PI with interp（physics-informed with interpolated points） for water production rate， pressure field and gas saturation field［71］

场	产水率	压力	气体饱和度
STT	1.05	0.13	$4.45 \times 10^{- 5}$
MTT	$1.40 \times 10^{- 2}$	$2.09 \times 10^{- 2}$	$2.43 \times 10^{- 5}$
PI w/o interp	$4.14 \times 10^{- 4}$	$2.35 \times 10^{- 2}$	$2.11 \times 10^{- 5}$
PI with interp	$2.52 \times 10^{- 4}$	$5.83 \times 10^{- 7}$	$6.03 \times 10^{- 6}$

场	产水率	压力	气体饱和度
STT	1.05	0.13	$4.45 \times 10^{- 5}$
MTT	$1.40 \times 10^{- 2}$	$2.09 \times 10^{- 2}$	$2.43 \times 10^{- 5}$
PI w/o interp	$4.14 \times 10^{- 4}$	$2.35 \times 10^{- 2}$	$2.11 \times 10^{- 5}$
PI with interp	$2.52 \times 10^{- 4}$	$5.83 \times 10^{- 7}$	$6.03 \times 10^{- 6}$

Fig.11 (a) One-dimensional spontaneous imbibition configuration; (b) PINN configuration for self-similar spontaneous imbibition solution[73]

Fig.12 (a) PINN predicted average normalized water flux for transient spontaneous imbibition; (b) PINN predicted capillary dimensionless group for transient spontaneous imbibition; (c) Absolute difference in average normalized water flux between PINN and FD results for transient spontaneous imbibition; (d) Absolute difference in capillary dimensionless group between PINN and FD results for transient spontaneous imbibition [73]

Fig.13 Schematic diagram of training the PINN system for solving two-phase steady state flow with capillary heterogeneity at various flow conditions[74]

Fig.14 1D saturation profiles of the actual core-flooding experiment (red crosses) and PINNs predictions (dotted black lines) at fractional flows of 0.50 (a), 0.80 (b), 0.90 (c), and 0.95 (d)[74]

Fig.15 Configuration and training strategy of the PINN algorithm for heat transfer in two-phase flow: (a) network architecture; (b) training strategy[75]

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