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Geometric model of the fracture as a manifold immersed in porous media
Journal of Mathematical Physics ( IF 1.3 ) Pub Date : 2021-05-11 , DOI: 10.1063/1.5109730
Pushpi Paranamana 1 , Eugenio Aulisa 2 , Magdalena Toda 2
Affiliation  

In this work, we analyze the flow filtration process of slightly compressible fluids in porous media containing fractures with complex geometries. We model the coupled fracture-porous media system where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. We develop a model to examine the flow inside fractures with complex geometries and variable thickness on a Riemannian manifold. The fracture is represented as the normal variation of a surface immersed in R3. Using operators of Laplace–Beltrami type and geometric identities, we model an equation that describes the flow in the fracture. A reduced model is obtained as a low dimensional boundary value problem. We then couple the model with the porous media. Theoretical and numerical analyses have been performed to compare the solutions between the original geometric model and the reduced model in reservoirs containing fractures with complex geometries. We prove that the two solutions are close and, therefore, the reduced model can be effectively used in large scale simulators for long and thin fractures with complicated geometry.

中文翻译:

作为浸入多孔介质中的歧管的断裂几何模型

在这项工作中,我们分析了微可压缩流体在包含具有复杂几何形状的裂缝的多孔介质中的流动过滤过程。我们对耦合的裂缝-多孔介质系统进行建模,其中在多孔介质中考虑线性 Darcy 流,并在裂缝内使用非线性 Forchheimer 方程。我们开发了一个模型来检查在黎曼流形上具有复杂几何形状和可变厚度的裂缝内的流动。断裂被表示为浸入其中的表面的法向变化电阻3. 使用 Laplace-Beltrami 类型和几何恒等式的算子,我们对描述裂缝中的流动的方程进行建模。作为低维边界值问题获得简化模型。然后我们将模型与多孔介质耦合。已经进行了理论和数值分析,以比较原始几何模型和包含具有复杂几何形状的裂缝的储层的简化模型之间的解。我们证明这两种解是接近的,因此,简化的模型可以有效地用于具有复杂几何形状的细长裂缝的大型模拟器。
更新日期:2021-05-28
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