标题： 多孔介质中的流动：一种用于模拟压力依赖粘度的霍普夫-科尔变换方法 显示英文标题

标题： Flow Through Porous Media: A Hopf-Cole Transformation Approach for Modeling Pressure-Dependent Viscosity

摘要： 大多数有机液体表现出粘度与压力相关，这使得在压力显著高于环境条件的应用中（例如，地质碳封存）必须考虑这种行为。描述通过多孔介质流动并考虑粘度-压力依赖性的数学模型是非线性的（例如，Barus模型）。这种非线性使数学分析复杂化，并使数值解更加耗时且容易出现收敛问题。在本文中，我们证明了Hopf-Cole变换，最初是为Burgers方程开发的，可以将描述具有压力依赖粘度的多孔介质流动的控制方程重新表述为线性形式。转换后的方程在转换变量中类似于Darcy方程，使得（a）能够进行系统数学分析以建立唯一性和最大值原理，（b）推导出基于力学的原则，以及（c）使用针对Darcy方程优化的求解器开发高效的数值解。值得注意的是，线性Darcy方程的许多性质自然地扩展到依赖于压力的非线性模型。例如，这些非线性模型的解遵循类似于Darcy方程中观察到的倒数关系。 显示英文摘要

摘要： Most organic liquids exhibit a pressure-dependent viscosity, making it crucial to consider this behavior in applications where pressures significantly exceed ambient conditions (e.g., geological carbon sequestration). Mathematical models describing flow through porous media while accounting for viscosity-pressure dependence are nonlinear (e.g., the Barus model). This nonlinearity complicates mathematical analysis and makes numerical solutions more time-intensive and prone to convergence issues. In this paper, we demonstrate that the Hopf-Cole transformation, originally developed for Burgers' equation, can recast the governing equations -- describing flow through porous media with pressure-dependent viscosity -- into a linear form. The transformed equations, resembling Darcy's equations in the transformed variables, enable (a) systematic mathematical analysis to establish uniqueness and maximum principles, (b) the derivation of a mechanics-based principle, and (c) the development of efficient numerical solutions using solvers optimized for Darcy equations. Notably, many properties of the linear Darcy equations naturally extend to nonlinear models that depend on pressure. For example, solutions to these nonlinear models adhere to a reciprocal relation analogous to that observed in Darcy's equations.