We present a Parametrization of the Physics Informed Neural Network (P-PINN) approach to tackle the problem of uncertainty quantification in reservoir engineering problems. We demonstrate the approach with the immiscible two phase flow displacement (Buckley-Leverett problem) in heterogeneous porous medium. The reservoir properties (porosity, permeability) are treated as random variables. The distribution of these properties can affect dynamic properties such as the fluids saturation, front propagation speed or breakthrough time. We explore and use to our advantage the ability of networks to interpolate complex high dimensional functions. We observe that the additional dimensions resulting from a stochastic treatment of the partial differential equations tend to produce smoother solutions on quantities of interest (distributions parameters) which is shown to improve the performance of PINNS. We show that provided a proper parameterization of the uncertainty space, PINN can produce solutions that match closely both the ensemble realizations and the stochastic moments. We demonstrate applications for both homogeneous and heterogeneous fields of properties. We are able to solve problems that can be challenging for classical methods. This approach gives rise to trained models that are both more robust to variations in the input space and can compete in performance with traditional stochastic sampling methods.
翻译:我们提出了物理、知情神经网络(P-PINN)的平衡化方法,以解决储油层工程问题中的不确定性量化问题。我们展示了在多孔多孔介质中不强迫的两阶段流动转移(Buckley-Leverett问题)的方法。储油层特性(渗透性、渗透性)被作为随机变量处理。这些特性的分布可以影响液体饱和、前传播速度或突破时间等动态特性。我们探索并利用网络的能力,将复杂的高维功能相互调和。我们观察到,部分差异方程式的随机处理所产生的额外层面往往能产生更顺利的利息量解决方案(分配参数),这证明可以改善PINNS的性能。我们表明,为不确定性空间提供了适当的参数,PINN可以产生与共振成就和共和异性时刻密切匹配的解决方案。我们展示了对等同性和多元性特性领域的应用。我们发现,我们能够解决对古典方法具有挑战性的问题。这一方法使得经过培训的传统空间模型具有更牢固的可变性。