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于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流体动力学提供了基于力学的分析方法。然而，当前的理论成果主要涉及稳态响应，其在瞬态领域的研究尚属空白。本文旨在填补这一认知空白。我们通过严谨的数学论证，提出了双重孔隙/渗透率模型在瞬态条件下解所满足的三个主要性质：时间反演唯一性、互易性及变分原理。利用流体的物理总动能（"能量法"）确立了第一个性质，并通过柯西-黎曼卷积证明了后两个性质。本文报告的结果——定性描述了双孔介质中流体运动的动力学特征——具有（a）理论意义、（b）实际应用价值及（c）重要的教学价值。具体而言，这些结果将有助于实践者和计算科学家验证数值模拟器的准确性。时间反演唯一性为开展反问题研究奠定了坚实的理论基础——此类问题旨在根据后续时刻的解数据反推预设初始条件。