A cut finite element method for elliptic bulk problems with embedded surfaces.

Area of Science:

• Computational fluid dynamics
• Geosciences
• Numerical analysis

Background:

• Simulating fluid flow in fractured porous media is crucial for understanding subsurface processes.
• Existing methods often struggle with accurately capturing the complex interactions at fracture interfaces.
• Developing robust numerical techniques is essential for reliable subsurface flow modeling.

Purpose of the Study:

• To introduce an unfitted finite element method for modeling flow in fractured porous media.
• To accurately represent the coupling and jumps in solution gradients across fractures.
• To provide a framework for simulating flow in both simple and bifurcating fracture networks.

Main Methods:

• Utilizing an unfitted finite element method with Nitsche-type mortaring for fracture coupling.
• Modeling the fracture flow field using averaged traces of bulk variables.
• Incorporating the Laplace-Beltrami operator for fracture transport via averaged gradient projections.
• Proving optimal order error estimates under regularity assumptions.

Main Results:

• The proposed method accurately represents the jump in the normal component of the gradient across fractures.
• Simultaneous modeling of bulk and fracture flow fields is achieved.
• The method is shown to be effective through numerical examples, including bifurcating fractures.

Conclusions:

• The unfitted finite element method provides an accurate and robust approach for flow in fractured porous media.
• The Nitsche-type mortaring effectively handles fracture coupling.
• The method shows promise for applications in various subsurface flow simulations.