# Geometric model of the fracture as a manifold immersed in porous media

## TL;DR

This paper introduces a geometric model for fluid flow in fractured porous media, using Riemannian manifolds to accurately represent complex fracture geometries and develop reduced models for efficient large-scale simulations.

## Contribution

It develops a novel geometric framework modeling fractures as manifolds immersed in space, coupling it with porous media flow, and validates reduced models for complex fracture geometries.

## Key findings

- The reduced model closely approximates the original geometric model.
- The approach effectively handles complex, variable-thickness fractures.
- The model facilitates large-scale reservoir simulations with complex fracture networks.

## Abstract

In this work, we analyze the flow filtration process of slightly compressible fluids in porous media containing man made 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 $\mathbb{R}^3$. 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 BVP. We then couple the model with the porous media. Theoretical and numerical analysis 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.