Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parameter Estimation for the Reduced Fracture Model via a Direct Filter Method

Phuoc Toan Huynh, Feng Bao, Thi-Thao-Phuong Hoang

TL;DR

This work addresses the inverse problem of estimating fracture widths $d$ in fractured porous media from real-time flow observations. It couples a reduced fracture diffusion model with an online direct filter method, within a Bayesian data-assimilation framework, to estimate the width parameter $d$ by treating $\theta=1/d$ as a stochastic process and updating its distribution as data arrive. Numerical experiments on one single fracture, two parallel fractures, and two intersecting fractures demonstrate accurate, rapid convergence of the width estimates, with convergence rates influenced by the chosen exploration noise. The approach integrates a mass-conservative forward model based on mixed finite elements with an efficient online parameter-estimation strategy, offering a practical tool for real-time fracture characterization in subsurface systems.

Abstract

In this work, we present a numerical method that provides accurate real-time detection for the widths of the fractures in a fractured porous medium based on observational data on porous medium fluid mass and velocity. To achieve this task, an inverse problem is formulated by first constructing a forward formulation based on the reduced fracture model of the diffusion equation. A parameter estimation problem is then performed online by utilizing a direct filter method. Numerical experiments are carried out to demonstrate the accuracy of our method in approximating the target parameters.

Parameter Estimation for the Reduced Fracture Model via a Direct Filter Method

TL;DR

This work addresses the inverse problem of estimating fracture widths in fractured porous media from real-time flow observations. It couples a reduced fracture diffusion model with an online direct filter method, within a Bayesian data-assimilation framework, to estimate the width parameter by treating as a stochastic process and updating its distribution as data arrive. Numerical experiments on one single fracture, two parallel fractures, and two intersecting fractures demonstrate accurate, rapid convergence of the width estimates, with convergence rates influenced by the chosen exploration noise. The approach integrates a mass-conservative forward model based on mixed finite elements with an efficient online parameter-estimation strategy, offering a practical tool for real-time fracture characterization in subsurface systems.

Abstract

In this work, we present a numerical method that provides accurate real-time detection for the widths of the fractures in a fractured porous medium based on observational data on porous medium fluid mass and velocity. To achieve this task, an inverse problem is formulated by first constructing a forward formulation based on the reduced fracture model of the diffusion equation. A parameter estimation problem is then performed online by utilizing a direct filter method. Numerical experiments are carried out to demonstrate the accuracy of our method in approximating the target parameters.
Paper Structure (11 sections, 37 equations, 11 figures)

This paper contains 11 sections, 37 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Reduced fracture model for the diffusion equations
  3. Data assimilation
  4. General framework
  5. Direct filter for parameter estimation
  6. Numerical results
  7. Problem setup
  8. Test case 1: One Single Fracture
  9. Test case 2: Two Parallel Fractures
  10. Test case 3: Two intersecting fractures
  11. Conclusion

Figures (11)

  • Figure 1: The domain $\Omega$ with the fracture $\Omega_{f}$ (left) and the fracture-interface $\gamma$ (right).
  • Figure 2: [Test case 1] (Left) Geometry and boundary conditions of the test case. (Right) Example of an uniform triangular mesh for spatial discretization.
  • Figure 3: [Test case 1] Pressure field (left) and velocity field (right) at time $T=5$.
  • Figure 4: [Test case 1] The estimations for $d$.
  • Figure 5: [Test case 2] Geometry and boundary conditions of the test case.
  • ...and 6 more figures