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Filtration Law for Polymer Flow through porous media

A. Bourgeat, O. Gipouloux, E. Marusic-Paloka

TL;DR

This work analyzes polymer filtration in periodically porous media using quasi-Newtonian viscosity laws (Carreau and power-law) and a homogenization framework to derive macroscopic nonlinear Darcy laws. By formulating coupled micro–macro problems and then decoupling them, the authors obtain a well-posed macroscopic filtration law with an effective permeability ${\cal U}$ defined via cell problems, and they establish rigorous existence and uniqueness results using monotone operator theory. They provide a detailed Taylor expansion of ${\cal U}$ for Carreau’s law and study the extent to which a polynomial (or Darcy-like) approximation preserves well-posedness, as well as analogous results for the power-law case, including homogeneity properties and asymptotic behavior. Numerical experiments on three periodic pore geometries validate the theory, compare full microscale solutions to truncated Taylor expansions, and reveal geometry-dependent accuracy and nonlinearity in the permeability function. The results advance understanding of macroscopic polymer filtration through porous media and inform stable, physically meaningful reduced models for engineering applications.

Abstract

In this paper we study the filtration laws for the polymeric flow in a porous medium. We use the quasi-Newtonian models with share dependent viscosity obeying the power-law and the Carreau's law. Using the method of homogenization the coupled micro-macro homogenized law, governing the quasi-newtonian flow in a periodic model of a porous medium, was found. We decouple that law separating the micro from the macro scale. We write the macroscopic filtration law in the form of non-linear Darcy's law and we prove that the obtained law is well posed. We give the analytical as well as the numerical study of our model.

Filtration Law for Polymer Flow through porous media

TL;DR

This work analyzes polymer filtration in periodically porous media using quasi-Newtonian viscosity laws (Carreau and power-law) and a homogenization framework to derive macroscopic nonlinear Darcy laws. By formulating coupled micro–macro problems and then decoupling them, the authors obtain a well-posed macroscopic filtration law with an effective permeability defined via cell problems, and they establish rigorous existence and uniqueness results using monotone operator theory. They provide a detailed Taylor expansion of for Carreau’s law and study the extent to which a polynomial (or Darcy-like) approximation preserves well-posedness, as well as analogous results for the power-law case, including homogeneity properties and asymptotic behavior. Numerical experiments on three periodic pore geometries validate the theory, compare full microscale solutions to truncated Taylor expansions, and reveal geometry-dependent accuracy and nonlinearity in the permeability function. The results advance understanding of macroscopic polymer filtration through porous media and inform stable, physically meaningful reduced models for engineering applications.

Abstract

In this paper we study the filtration laws for the polymeric flow in a porous medium. We use the quasi-Newtonian models with share dependent viscosity obeying the power-law and the Carreau's law. Using the method of homogenization the coupled micro-macro homogenized law, governing the quasi-newtonian flow in a periodic model of a porous medium, was found. We decouple that law separating the micro from the macro scale. We write the macroscopic filtration law in the form of non-linear Darcy's law and we prove that the obtained law is well posed. We give the analytical as well as the numerical study of our model.

Paper Structure

This paper contains 14 sections, 11 theorems, 128 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Carreau's coupled homogenized problem
  3. Statement of the main result
  4. Taylor's expansion of the permeability function
  5. Proofs of existence theorems
  6. The power-law coupled homogenized problem
  7. Statement of the main result
  8. Study of the permeability function
  9. Numerical Experiments
  10. Numerical methods used
  11. The Porous media considered
  12. Numerical Simulations for Carreau's viscosity law
  13. Numerical simulation for Power viscosity law
  14. Numerical results

Key Result

Theorem 1

Let ${\cal U}:{\bf R}^n\to{\bf R}^n$ be defined by (jup,1). Then the macroscopic problem has a unique (up to a constant) solution $p^0\in W^{1,r'} (\Omega )$, for any $f\in L^{r'}(\Omega)^n$.

Figures (7)

  • Figure 1: Periodic unitary cell: GEOM1 on the left, GEOM2 in the middle and finally GEOM3 on the right of the figure.
  • Figure 2: Gap between ${\cal U}$ and its Taylor's expansions of order 1, 3 and 5 for GEOM1 on the left, and for GEOM2 on the right (for $\lambda_2 = 100$).
  • Figure 3: Gap between ${\cal U}$ and its Taylor's expansion for GEOM1:$|\xi | \leq 1$, $({\bf e}_1 ,\xi ) = 45^\circ,\;\lambda=1$ on the left, $\lambda=100$ on the right.
  • Figure 4: Gap between ${\cal U}$ and its Taylor's expansion for GEOM2:$|\xi| \leq 1$, $({\bf e}_1 ,\xi ) = 45^\circ$, $\lambda=1$ on the left, $\lambda=100$ on the right.
  • Figure 5: Gap between ${\cal U}$ and its Taylor's development of order 1, 3 and 5 for $|\xi| \le 10$, for GEOM1 on the left, and for GEOM2 in the middle and GEOM3 on the right.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Remark 1
  • Lemma 3
  • Theorem 4
  • Lemma 4
  • ...and 4 more