Global existence of weak solutions to unsaturated poroelasticity
Global existence of weak solutions to unsaturated poroelasticity
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DOI：
10.1051/m2an/2021063
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发表时间：
2021-10
期刊：
影响因子：
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通讯作者：
J. Both;Iuliu Sorin Pop;I. Yotov
中科院分区：
文献类型：
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作者：
J. Both;Iuliu Sorin Pop;I. Yotov
我们研究了非饱和孔弹性，即变饱和多孔介质中的水-力学耦合过程，这里用Biot著名的准静态固结模型的非线性扩展来建模。椭圆型-抛物型耦合偏微分方程组是可变形多孔介质中多相流一般模型的简化形式，在与Richards方程相似的假设下得到。在这项工作中，弱解的存在性是分几步建立的，包括使用物理激励正则化和有限元/有限体积离散的数值逼近。最后，通过Rothe方法和Galerkin方法的结合，以及进一步的紧性论证，证明了原问题的可解性。这种方法特别提供了非饱和孔弹性正则化模型的数值离散化的收敛。在非退化条件和本构关系的自然连续性条件下，最终存在结果成立。从岩土工程应用的角度来看，这些假设是合理的。
We study unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in variably saturated porous media, here modeled by a non-linear extension of Biot's well-known quasi-static consolidation model. The coupled elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in deformable porous media, obtained under similar assumptions as usually considered for Richards' equation. In this work, existence of weak solutions is established in several steps involving a numerical approximation of the problem using a physically-motivated regularization and a finite element/finite volume discretization. Eventually, solvability of the original problem is proved by a combination of the Rothe and Galerkin methods, and further compactness arguments. This approach in particular provides the convergence of the numerical discretization to a regularized model for unsaturated poroelasticity. The final existence result holds under non-degeneracy conditions and natural continuity properties for the constitutive relations. The assumptions are demonstrated to be reasonable in view of geotechnical applications.