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 steady-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）均表现出多孔隙网络结构。双重孔隙/渗透率模型提供了一种基于力学的方法来描述上述多孔材料中的流体动力学特性。然而，当前理论成果主要解决稳态响应问题，其在瞬态领域的对应研究仍有待完善。本文的主要目标是填补这一认知空白。我们通过严格的数学论证，揭示了双重孔隙/渗透率模型解在瞬态条件下满足的三个基本性质：时间逆向唯一性、互易性以及变分原理。我们采用“能量法”——通过利用流动流体的物理总动能——来确立第一个性质，并利用柯西-黎曼卷积证明后续两个性质。本文报道的关于双孔隙介质中流体动力学特性的定性研究成果具有（a）理论意义、（b）实际应用价值及（c）重要的教学价值。特别地，这些成果将有助于实践工作者和计算科学家检验数值模拟程序的准确性。时间逆向唯一性为求解逆问题奠定了坚实的理论基础——此类问题需根据后期时刻的解数据反推预设的初始条件。