Understanding fluid movement in multi-pored materials is vital for energy security and physiology. For instance, shale (a geological material) and bone (a biological material) exhibit multiple pore networks. Double porosity/permeability models provide a mechanics-based approach to describe hydrodynamics in aforesaid porous materials. However, current theoretical results primarily address state-state response, and their counterparts in the transient regime are still wanting. The primary aim of this paper is to fill this knowledge gap. We present three principal properties -- with rigorous mathematical arguments -- that the solutions under the double porosity/permeability model satisfy in the transient regime: backward-in-time uniqueness, reciprocity, and a variational principle. We employ the ``energy method" -- by exploiting the physical total kinetic energy of the flowing fluid -- to establish the first property and Cauchy-Riemann convolutions to prove the next two. The results reported in this paper -- that qualitatively describe the dynamics of fluid flow in double-pored media -- have (a) theoretical significance, (b) practical applications, and (c) considerable pedagogical value. In particular, these results will benefit practitioners and computational scientists in checking the accuracy of numerical simulators. The backward-in-time uniqueness lays a firm theoretical foundation for pursuing inverse problems in which one predicts the prescribed initial conditions based on data available about the solution at a later instance.

翻译：理解多孔材料中的流体运动对能源安全与生理学至关重要。例如，页岩（地质材料）和骨骼（生物材料）均呈现出多孔网络结构。双重孔隙/渗透率模型为描述上述多孔材料中的水动力学提供了基于力学的分析方法。然而，当前理论成果主要针对稳态响应，其在瞬态工况下的对应研究仍存在空白。本文旨在填补这一知识缺口。我们通过严谨的数学论证，提出了双重孔隙/渗透率模型下的解在瞬态工况中满足的三项主要性质：时间逆唯一性、互易性及变分原理。采用“能量法”——通过利用流动物理总动能——建立第一项性质，并运用柯西-黎曼卷积证明后两项性质。本文所报告的定性描述双孔介质中流体动力学特性的结果具有：（a）理论重要性，（b）实际应用价值，以及（c）显著的教学意义。特别地，这些结果将有助于实践者和计算科学家检验数值模拟器的准确性。时间逆唯一性为开展反问题研究奠定了坚实的理论基础——此类问题旨在依据解在后续时刻的已知数据预测预设的初始条件。