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An Efficient Discrete Fracture Model with Minimal Degrees of Freedom for Fractured Porous Media with Low-Permeable Barriers

Core Concepts
A simple yet effective extension of the Box-DFM method to include low-permeable barriers, requiring only minimal additional degrees of freedom and maintaining the symmetric positive-definite property of the stiffness matrix.
The paper presents an extension of the Box-DFM (Discrete Fracture Model) method to include the effects of low-permeable barriers in fractured porous media. The key highlights are: The extended Box-DFM method remains identical to the traditional Box-DFM in the absence of barriers, requiring only minimal additional degrees of freedom to accommodate pressure discontinuities across barriers. The method adopts a broken Lagrange finite element space as the trial functions and uses the low-dimensional Darcy's law to compute the flux across cell interfaces aligned with low-permeable barriers. The resulting stiffness matrix of the extended Box-DFM is proven to be symmetric positive-definite, inheriting the favorable properties of the original Box-DFM. Extensive numerical tests on published benchmark problems demonstrate the validity and performance of the extended method, with comparisons to existing finite volume discrete fracture models. The range of validity is also studied, showing the method performs well when the tangential permeability of barriers is no greater than the porous matrix, but may not be suitable for barriers with high tangential permeability.
The permeability tensor of the porous matrix is denoted as Km. The permeability of high-permeable fractures is denoted as kf. The permeability of low-permeable barriers is denoted as kb. The aperture of fractures and barriers is denoted as a.
"The box method discrete fracture model (Box-DFM) is an important finite volume-based discrete fracture model (DFM) to simulate flows in fractured porous media." "The inclusion of barriers requires only minimal additional degrees of freedom to accommodate pressure discontinuities and necessitates minor modifications to the original coding framework of the Box-DFM." "We use extensive numerical tests on published benchmark problems and comparison with existing finite volume DFMs to demonstrate the validity and performance of the method."

Deeper Inquiries

How can the extended Box-DFM method be further improved to handle barriers with high tangential permeability

To improve the handling of barriers with high tangential permeability in the extended Box-DFM method, several enhancements can be considered: Incorporating Tangential Flow: One approach could be to modify the method to account for tangential flow along barriers with high permeability. This could involve introducing additional degrees of freedom to capture the tangential pressure gradients and fluxes accurately. Hybrid Approach: A hybrid approach could be adopted where the method switches between neglecting tangential flow for low-permeability barriers and incorporating it for high-permeability barriers. This adaptive strategy would ensure accurate representation of flow behavior based on the barrier properties. Advanced Interface Conditions: Developing more sophisticated interface conditions at the intersections of high-permeability barriers could help in better capturing the flow dynamics. These conditions could involve a combination of pressure continuity, flux balance, and permeability adjustments to handle the complexities of the barrier network effectively. Numerical Stability: Ensuring numerical stability when incorporating high tangential permeability is crucial. Techniques such as stabilization methods or adaptive mesh refinement can be employed to maintain accuracy and convergence in the presence of varying permeability values.

What are the potential limitations of the interface model assumption used in the extended Box-DFM, and how could it be relaxed

The interface model assumption used in the extended Box-DFM method simplifies the treatment of barriers by neglecting tangential flow and assuming pressure continuity across barriers. However, this assumption has potential limitations: Accuracy Concerns: Neglecting tangential flow may lead to inaccuracies in scenarios where barriers with high tangential permeability significantly impact the flow behavior. The assumption of pressure continuity may not hold true in all cases, especially for barriers with complex geometries or varying permeability distributions. Intersection Handling: The interface model may struggle to accurately represent the behavior at intersections of high-permeability barriers, where flow patterns can be intricate. Relaxing this assumption would require a more detailed approach to account for the interactions at these intersections. Complex Barrier Networks: In cases of complex barrier networks with varying permeabilities and orientations, the interface model may oversimplify the flow dynamics. Relaxing this assumption would involve developing more sophisticated models to capture the diverse effects of barriers on fluid flow. To relax the interface model assumption, the method could be enhanced by: Introducing more advanced interface conditions that consider the specific characteristics of barriers, such as varying permeabilities and orientations. Incorporating additional degrees of freedom to account for pressure and flux variations along barriers. Implementing adaptive strategies to switch between different modeling approaches based on the properties of the barriers in the network.

Can the extended Box-DFM be coupled with other numerical methods, such as mixed finite element or discontinuous Galerkin, to handle more complex fracture-barrier networks

The extended Box-DFM can be coupled with other numerical methods to handle more complex fracture-barrier networks effectively. Some potential approaches include: Mixed Finite Element Methods: Coupling the extended Box-DFM with mixed finite element methods can offer advantages in handling heterogeneous media with varying permeabilities. This combination can provide a more robust framework for modeling flow in fractured porous media with barriers. Discontinuous Galerkin Methods: Integrating the extended Box-DFM with discontinuous Galerkin methods can enhance the accuracy and flexibility of the model. This coupling can enable the treatment of discontinuities across fractures and barriers more effectively, especially in cases of complex geometries and material properties. Adaptive Mesh Refinement: Combining the extended Box-DFM with adaptive mesh refinement techniques can improve the resolution of the model in regions of interest, such as around barriers with high permeability. This coupling can enhance the overall accuracy and efficiency of the simulation in capturing the intricate flow patterns in the fractured porous media. By integrating the extended Box-DFM with these advanced numerical methods, researchers can address the challenges posed by complex fracture-barrier networks and achieve more comprehensive and accurate simulations of fluid flow in such systems.