There is a recent surge in research activities on modeling the flow of fluids in porous media with complex pore-networks. A prominent mathematical model, which describes the flow of incompressible fluids in porous media with two dominant pore-networks allowing mass transfer across them, is the double porosity/permeability (DPP) model. However, we currently do not have a complete understanding of unsteady solutions under the DPP model. Also, because of the complex nature of the mathematical model, it is not possible to find analytical solutions, and one has to resort to numerical solutions. It is therefore desirable to have a procedure that can serve as a measure to assess the veracity of numerical solutions. In this paper, we establish that unsteady solutions under the transient DPP model are stable in the sense of Lyapunov. We also show that the unsteady solutions grow at most linear with time. These results not only have a theoretical value but also serve as valuable a posteriori measures to verify numerical solutions in the transient setting and under anisotropic medium properties, as analytical solutions are scarce for these scenarios under the DPP model. This figure shows that the evolution of an unsteady solution under the DPP model satisfies the theoretical bound derived in this paper. $$\Vert \varvec{\Upsilon }\Vert _{{\mathcal {V}}}$$ ‖ Υ ‖ V denotes a norm defined in terms of the velocities in the 2 pore-networks

中文翻译：

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双孔渗模型下非定常解的有界性和增长性

最近，对具有复杂孔隙网络的多孔介质中的流体流动进行建模的研究活动激增。双孔隙率/渗透率 (DPP) 模型是一个突出的数学模型，它描述了多孔介质中不可压缩流体的流动，其中两个主要孔隙网络允许传质穿过它们。但是，我们目前对 DPP 模型下的非定常解还没有一个完整的了解。此外，由于数学模型的复杂性，不可能找到解析解，只能求助于数值解。因此，希望有一个程序可以作为评估数值解的准确性的措施。在本文中，我们建立了瞬态 DPP 模型下的非定常解在 Lyapunov 意义上是稳定的。我们还表明，不稳定的解决方案最多随时间线性增长。这些结果不仅具有理论价值，而且可以作为有价值的后验措施来验证瞬态设置和各向异性介质属性下的数值解，因为在 DPP 模型下这些场景的解析解很少。该图表明，DPP 模型下非定常解的演化满足本文推导出的理论界。$$\Vert \varvec{\Upsilon }\Vert _{{\mathcal {V}}}$$ ‖ Υ ‖ V 表示根据 2 个孔隙网络中的速度定义的范数 因为在 DPP 模型下，这些场景的分析解决方案很少。该图表明，DPP 模型下非定常解的演化满足本文推导出的理论界。$$\Vert \varvec{\Upsilon }\Vert _{{\mathcal {V}}}$$ ‖ Υ ‖ V 表示根据 2 个孔隙网络中的速度定义的范数 因为在 DPP 模型下，这些场景的分析解决方案很少。该图表明，DPP 模型下非定常解的演化满足本文推导出的理论界。$$\Vert \varvec{\Upsilon }\Vert _{{\mathcal {V}}}$$ ‖ Υ ‖ V 表示根据 2 个孔隙网络中的速度定义的范数