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

Pushpi Paranamana, Eugenio Aulisa, Akif Ibragimov, Magdalena Toda

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.

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

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 . 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.

Paper Structure

This paper contains 19 sections, 14 theorems, 105 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Formulation of the problem and preliminary results
  3. Reservoir modeling
  4. Diffusive Capacity and Pseudo-Steady State regime (PSS)
  5. PSS solution for the Initial Boundary Value Problem.
  6. Steady state auxiliary BVP
  7. Fractured reservoir modeling
  8. Geometric modeling of the fracture
  9. Geometric model of the fracture as a 3-D manifold
  10. Flow equation inside the fracture
  11. Reduced model of the flow inside the fracture
  12. Reduced model of the fracture: when the metric $G$ does not depend on $\lambda$
  13. Estimates for the difference between the solutions of the original model and the reduced models
  14. Estimates for the difference between the solutions inside the fracture $\Omega_f$.
  15. Estimates for the difference between the solutions in coupled fracture-porous media domain with linear isotropic flows.
  16. ...and 4 more sections

Key Result

Lemma \oldthetheorem

For $f_{\beta}(\|\eta \|)$ defined as above, the function $f_{\beta}(\|\eta \|)\|\eta \|$ is strictly monotonic on bounded sets. More precisely,

Figures (12)

  • Figure 1: Fractured-Reservoir Domain
  • Figure 2: Schematic of the fracture domain $\Omega_f$ as a manifold.
  • Figure 3: Fracture domain $\Omega_f$.
  • Figure 4: Domain of the reduced model.
  • Figure 5: Fracture geometry 1: pressure distribution inside a thin fracture-$2h(u)=0.025$ (left) and a thick fracture-$2h(u)=1$ (right), obtained using the original model.
  • ...and 7 more figures

Theorems & Definitions (38)

  • Remark 1
  • Remark 2
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Corollary 1
  • Proposition 1
  • proof
  • Definition \oldthetheorem
  • ...and 28 more