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Joint filtration of compressible immiscible liquids

Anvarbek Meirmanov

TL;DR

The work addresses a rigorous derivation of macroscopic Biot-type models for the joint filtration of two compressible immiscible fluids in an elastic porous medium, starting from a detailed microscopic description $\textbf{A}^{\varepsilon}$ that couples stationary Stokes flow in the fluids with stationary Lame dynamics in the solid. Using homogenization as $\varepsilon\to 0$ and introducing smooth viscosity approximations $\mu_{j,h}^{\varepsilon}$ via models $\textbf{B}^{\varepsilon}_{k,h}$, the authors aim to obtain homogenized macroscopic models $\textbf{H}_{k,h}$ that preserve the two-fluid dynamics and the elastic response. The manuscript develops a comprehensive toolkit—tensor conventions, Poincaré and embedding inequalities, mollification, extension lemmas, and Hölder estimates—to enable a rigorous two-scale convergence analysis and to bridge microscopic and macroscopic descriptions. This framework yields a principled way to move beyond phenomenological macroscopic models (e.g., Buckley-Leverett) by grounding macroscopic behavior in a detailed pore-scale description, with potential impact on oil slurry displacement and radioactive slurry filtration modeling.

Abstract

We consider initial boundary value problems arising in mathematical models for joint motion of two immiscible viscous liquids in a pore space of the solid skeleton. First, we consider this physical process at the microscopic level (the pore size $\varepsilon$ is approximately 5-20 microns) governed by the model $\textbf{A}^{\varepsilon}$, consisting of Lame equations for the solid skeleton and the Stokes equations for the liquid components. Next, assuming the existence of a weak solution to the corresponding initial boundary value problem at the microscopic level for the model the $\textbf{B}^{\varepsilon}$, that approximate the model $\textbf{A}^{\varepsilon}$, and using the homogenization procedure we derive the Biot's model describing the physical process in consideration at the macroscopic level for two weakly viscous immiscible fluids in the elastic solid skeleton at the macroscopic level.

Joint filtration of compressible immiscible liquids

TL;DR

The work addresses a rigorous derivation of macroscopic Biot-type models for the joint filtration of two compressible immiscible fluids in an elastic porous medium, starting from a detailed microscopic description that couples stationary Stokes flow in the fluids with stationary Lame dynamics in the solid. Using homogenization as and introducing smooth viscosity approximations via models , the authors aim to obtain homogenized macroscopic models that preserve the two-fluid dynamics and the elastic response. The manuscript develops a comprehensive toolkit—tensor conventions, Poincaré and embedding inequalities, mollification, extension lemmas, and Hölder estimates—to enable a rigorous two-scale convergence analysis and to bridge microscopic and macroscopic descriptions. This framework yields a principled way to move beyond phenomenological macroscopic models (e.g., Buckley-Leverett) by grounding macroscopic behavior in a detailed pore-scale description, with potential impact on oil slurry displacement and radioactive slurry filtration modeling.

Abstract

We consider initial boundary value problems arising in mathematical models for joint motion of two immiscible viscous liquids in a pore space of the solid skeleton. First, we consider this physical process at the microscopic level (the pore size is approximately 5-20 microns) governed by the model , consisting of Lame equations for the solid skeleton and the Stokes equations for the liquid components. Next, assuming the existence of a weak solution to the corresponding initial boundary value problem at the microscopic level for the model the , that approximate the model , and using the homogenization procedure we derive the Biot's model describing the physical process in consideration at the macroscopic level for two weakly viscous immiscible fluids in the elastic solid skeleton at the macroscopic level.
Paper Structure (9 sections, 5 theorems, 39 equations)

This paper contains 9 sections, 5 theorems, 39 equations.

Table of Contents

  1. Introduction.
  2. The problem statement.
  3. Some auxiliary statements.
  4. Matrices and differential operators.
  5. Poincaré inequality.
  6. The simplest embedding theorem.
  7. Mollifiers.
  8. Extension Lemma
  9. Holder's inequality.

Key Result

Lemma 3.1

Let $Q\subset\mathbb{R}^{3}$ be bounded domain with Lipschitz piecewise smooth boundary. Then for any function $\boldsymbol{w}\in\stackrel{\!\!\circ}{W}^{1}_{2}(Q)$ holds true the inequality [11]: where $M(Q)<\infty$ for bounded $Q$. If $\,\Omega\subset\bigcup_{|\boldsymbol{k}|=1}^{n^{3}}\Omega^{\,\boldsymbol{k},\varepsilon}$ and $\boldsymbol{w}\in\stackrel{\!\!\circ}{W}^{1}_{2}(\Omega^{\,\boldsy

Theorems & Definitions (8)

  • Lemma 3.1
  • Lemma 3.2
  • Definition 1
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • Definition 2