A physics-informed convolutional neural network is proposed to simulate two phase flow in porous media with time-varying well controls. While most of PICNNs in existing literatures worked on parameter-to-state mapping, our proposed network parameterizes the solution with time-varying controls to establish a control-to-state regression. Firstly, finite volume scheme is adopted to discretize flow equations and formulate loss function that respects mass conservation laws. Neumann boundary conditions are seamlessly incorporated into the semi-discretized equations so no additional loss term is needed. The network architecture comprises two parallel U-Net structures, with network inputs being well controls and outputs being the system states. To capture the time-dependent relationship between inputs and outputs, the network is well designed to mimic discretized state space equations. We train the network progressively for every timestep, enabling it to simultaneously predict oil pressure and water saturation at each timestep. After training the network for one timestep, we leverage transfer learning techniques to expedite the training process for subsequent timestep. The proposed model is used to simulate oil-water porous flow scenarios with varying reservoir gridblocks and aspects including computation efficiency and accuracy are compared against corresponding numerical approaches. The results underscore the potential of PICNN in effectively simulating systems with numerous grid blocks, as computation time does not scale with model dimensionality. We assess the temporal error using 10 different testing controls with variation in magnitude and another 10 with higher alternation frequency with proposed control-to-state architecture. Our observations suggest the need for a more robust and reliable model when dealing with controls that exhibit significant variations in magnitude or frequency.

翻译：本文提出了一种物理信息卷积神经网络，用于模拟具有时变井控的多孔介质两相流。与现有文献中多数物理信息卷积神经网络（PICNN）专注于参数-状态映射不同，本文提出的网络通过时变控制参数化解，建立控制-状态回归模型。首先，采用有限体积法离散流动方程，并构建符合质量守恒定律的损失函数。诺伊曼边界条件被无缝融入半离散化方程中，无需额外损失项。网络架构包含两个并行的U-Net结构，网络输入为井控参数，输出为系统状态。为捕捉输入与输出之间的时间依赖关系，网络被精心设计以模拟离散化状态空间方程。我们逐步训练每个时间步的网络，使其能够同时预测每个时间步的油压和水饱和度。在完成一个时间步的训练后，利用迁移学习技术加速后续时间步的训练过程。所提模型用于模拟不同油藏网格块数的油水多孔流动场景，并与相应数值方法在计算效率和精度方面进行对比。结果表明，由于计算时间不随模型维度扩展，PICNN在有效模拟包含大量网格块的系统方面具有潜力。我们使用10种幅值变化的测试控制与另10种频率变化的测试控制，通过控制-状态架构评估时间误差。观察结果提示，当控制参数在幅值或频率上出现显著变化时，需要更稳健可靠的模型。