We formally derive interface conditions for modeling fractures in Darcy flow problems and, more generally, thin inclusions in heterogeneous diffusion problems expressed as the divergence of a flux. Through a formal integration of the governing equations within the inclusions, we establish that the resulting interface conditions are of Wentzell type for the flux jump and Robin type for the flux average. Notably, the flux jump condition is unconventional, involving a tangential diffusion operator applied to the average of the solution across the interface. The corresponding weak formulation is introduced, offering a framework that is readily applicable to finite element discretizations. Extensive numerical validation highlights the robustness and versatility of the proposed modeling technique. The results demonstrate its effectiveness in accommodating a wide range of material properties, managing networks of inclusions, and naturally handling fractures with varying apertures -- all without requiring an explicit geometric representation of the fractures.

翻译：本文通过形式推导，建立了达西流问题中裂缝的界面条件，更一般地，建立了以通量散度形式表达的异质扩散问题中薄层夹杂物的界面条件。通过对控制方程在夹杂物内部进行形式积分，我们证实所得界面条件对于通量跳跃为Wentzell型，对于通量平均则为Robin型。值得注意的是，通量跳跃条件具有非传统形式，其中涉及沿界面切向的扩散算子作用于解在界面上的平均值。文中引入了相应的弱形式表述，为有限元离散化提供了可直接应用的框架。大量数值验证突显了所提建模方法的鲁棒性与普适性。结果表明，该方法能有效适应广泛的材料属性、处理夹杂物网络、并自然应对不同开度的裂缝——所有这些均无需对裂缝进行显式的几何表征。