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Dynamic properties of double porosity/permeability model

K. B. Nakshatrala

TL;DR

The paper addresses transient dynamics in double porosity/permeability (DPP) models for materials with two interconnected pore networks, a regime previously less understood compared to steady-state analyses. It develops and proves three core properties for the transient DPP system: backward-in-time uniqueness, dynamic reciprocity, and a variational principle, using an energy-based approach and time-domain Cauchy-Riemann convolutions. These results offer rigorous theoretical insights and practical tools for verifying numerical solvers and enabling inverse problems, by treating the coupled macro- and micro- pore-network flows with a coherent variational framework. The findings have broad relevance for porous media applications such as shale and bone, and provide a foundation for future numerical verification and inversion methodologies in transient DPP models.

Abstract

Understanding fluid movement in multi-pored materials is vital for energy security and physiology. For instance, shale (a geological material) and bone (a biological material) exhibit multiple pore networks. Double porosity/permeability models provide a mechanics-based approach to describe hydrodynamics in aforesaid porous materials. However, current theoretical results primarily address steady-state response, and their counterparts in the transient regime are still wanting. The primary aim of this paper is to fill this knowledge gap. We present three principal properties -- with rigorous mathematical arguments -- that the solutions under the double porosity/permeability model satisfy in the transient regime: backward-in-time uniqueness, reciprocity, and a variational principle. We employ the ``energy method'' -- by exploiting the physical total kinetic energy of the flowing fluid -- to establish the first property and Cauchy-Riemann convolutions to prove the next two. The results reported in this paper -- that qualitatively describe the dynamics of fluid flow in double-pored media -- have (a) theoretical significance, (b) practical applications, and (c) considerable pedagogical value. In particular, these results will benefit practitioners and computational scientists in checking the accuracy of numerical simulators. The backward-in-time uniqueness lays a firm theoretical foundation for pursuing inverse problems in which one predicts the prescribed initial conditions based on data available about the solution at a later instance.

Dynamic properties of double porosity/permeability model

TL;DR

The paper addresses transient dynamics in double porosity/permeability (DPP) models for materials with two interconnected pore networks, a regime previously less understood compared to steady-state analyses. It develops and proves three core properties for the transient DPP system: backward-in-time uniqueness, dynamic reciprocity, and a variational principle, using an energy-based approach and time-domain Cauchy-Riemann convolutions. These results offer rigorous theoretical insights and practical tools for verifying numerical solvers and enabling inverse problems, by treating the coupled macro- and micro- pore-network flows with a coherent variational framework. The findings have broad relevance for porous media applications such as shale and bone, and provide a foundation for future numerical verification and inversion methodologies in transient DPP models.

Abstract

Understanding fluid movement in multi-pored materials is vital for energy security and physiology. For instance, shale (a geological material) and bone (a biological material) exhibit multiple pore networks. Double porosity/permeability models provide a mechanics-based approach to describe hydrodynamics in aforesaid porous materials. However, current theoretical results primarily address steady-state response, and their counterparts in the transient regime are still wanting. The primary aim of this paper is to fill this knowledge gap. We present three principal properties -- with rigorous mathematical arguments -- that the solutions under the double porosity/permeability model satisfy in the transient regime: backward-in-time uniqueness, reciprocity, and a variational principle. We employ the ``energy method'' -- by exploiting the physical total kinetic energy of the flowing fluid -- to establish the first property and Cauchy-Riemann convolutions to prove the next two. The results reported in this paper -- that qualitatively describe the dynamics of fluid flow in double-pored media -- have (a) theoretical significance, (b) practical applications, and (c) considerable pedagogical value. In particular, these results will benefit practitioners and computational scientists in checking the accuracy of numerical simulators. The backward-in-time uniqueness lays a firm theoretical foundation for pursuing inverse problems in which one predicts the prescribed initial conditions based on data available about the solution at a later instance.
Paper Structure (8 sections, 7 theorems, 109 equations, 2 figures)

This paper contains 8 sections, 7 theorems, 109 equations, 2 figures.

Table of Contents

  1. INTRODUCTION AND MOTIVATION
  2. TRANSIENT DOUBLE POROSITY/PERMEABILITY (DPP) MODEL
  3. Mathematical preliminaries
  4. BACKWARD-IN-TIME UNIQUENESS
  5. A remark on $\mathcal{E}(t)$
  6. A RECIPROCAL RELATION
  7. A VARIATIONAL PRINCIPLE
  8. CLOSURE

Key Result

Theorem 1

Let denote two solutions that satisfy Eqs. Eqn:Backwards_BoLM_1--Eqn:Backwards_pBC_2 and not necessarily the initial velocity conditions Eqn:Backwards_vIC_1--Eqn:Backwards_vIC_2. If, at some instance of time $\mathcal{T} > 0$, we have then the velocities under the two solutions coincide at all prior times: Moreover, the macro- and micro-pressures under the two solutions match, albeit up to an a

Figures (2)

  • Figure 1: Solutions to the forward and backward problems. The backward-in-time solution, one of the central topics of this paper, is not necessarily a solution to the backward problem.
  • Figure 2: Illustrating the non-uniqueness of backward-in-time solutions.

Theorems & Definitions (15)

  • Theorem 1: Backward-in-time uniqueness
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 1
  • Proposition 4
  • proof
  • proof : Proof of Theorem \ref{['Thm:Backwards_Backward_uniqueness_theorem']}
  • Theorem 5: Dynamic reciprocity
  • ...and 5 more