Friction modifies the quasistatic mechanical response of a confined, poroelastic medium

TL;DR

The paper develops a continuum framework that couples Coulomb wall friction with uniaxial poroelasticity in confined media, addressing quasi-static loading by a piston or by fluid pressure and unloading. It derives a diffusion–advection equation for the effective stress and introduces a friction number $\mathcal{F}= \dfrac{2\mu K L}{R}$ that governs the strength of frictional effects, producing Janssen-like attenuation in piston-driven cases and energy-augmenting gradients in fluid-driven cases. During compression, friction can stiffen the apparent response and, upon decompression, generate slip fronts and hysteresis with distinct energy partitioning between the two loading modes. The findings clarify how wall friction can bias measurements of permeability and stiffness in confined poroelastic systems and offer diagnostic signatures, such as slip-front propagation, to distinguish friction from particle rearrangements in experiments.

Abstract

The mechanical response of elastic porous media confined within rigid geometries is central to a wide range of industrial, geological, and biomedical systems. However, current models for these problems typically overlook the role of wall friction, and particularly its interaction with confinement. Here, we develop a theoretical framework to describe the interplay between the mechanics of the medium and Coulomb friction at the confining walls for slow, quasistatic deformations in response to two canonical uniaxial forcings: piston-driven loading and fluid-driven loading, followed by unloading. We find that, during compression, the stress field evolves according to a quasistatic advection-diffusion equation, extending classical poroelasticity results. The magnitude of friction is controlled by a single dimensionless number proportional to the friction coefficient and the aspect ratio of the confining geometry. During decompression, a portion of the solid matrix remains stuck due to friction, leading to hysteresis and to the propagation of a slip front. In piston-driven loading, the frictional stress is directly coupled to the solid effective stress, leading to exponential damping of the loading and striking changes to the displacement field. However, this coupling limits the energy dissipated by friction. In fluid-driven loading, the pressure gradient locally adds energy, decoupling the frictional stress from the effective stress. The displacement remains qualitatively unchanged but is quantitatively reduced due to large energy dissipation. In both cases, friction can have a substantial impact on the apparent mechanical properties of the medium.

Figures (13)

• Figure 1: A confined cylindrical porous medium is initially at rest (left) and is then compressed either by a permeable piston (centre) or a fluid flow (right). The shading illustrates the level of stress experienced by the solid matrix in the absence of friction, in the steady state: the stress level is uniform in piston-driven compression, and increases linearly from top to bottom in the fluid-driven compression (see section \ref{['subsub_poroelast']}).
• Figure 2: Magnitude of the effective stress, as a function of relative position ($\tilde{z}$), due to forcing by a piston (a, equation \ref{['eqn_SigmaPiston']}) or by a fluid flow (b, equation \ref{['eqn_StressFluidComp']}), with $\mathcal{F} \in \{0,1, 2, 3, 4\}.$ The dotted curve corresponds to the frictionless case ($\mathcal{F}=0$, section \ref{['subsub_poroelast']}), and the coloured arrow points toward increasing friction number. The imposed stress and fluid pressure are fixed at $|\tilde{\sigma}^{\prime \star}| = \Delta \tilde{P}^\star = 0.05$. (c): magnitude of the stress at the bottom of the medium ($\tilde{z}=0$) as a function of the friction number (equations \ref{['eqn_usPiston']} and \ref{['eqn_usfluide']}).
• Figure 3: Magnitude of the relative displacements, as a function of relative position ($\tilde{z}$), due to forcing by a piston (a) or by a fluid flow (b), with $\mathcal{F} \in \{0,1, 2, 3, 4\}.$ The dashed curves correspond to the frictionless case ($\mathcal{F}=0$, section \ref{['subsub_poroelast']}). The imposed stress and fluid pressure are fixed at $|\tilde{\sigma}^{\prime \star}| = \Delta \tilde{P}^\star = 0.05$. (c): magnitude of displacement of the top of the medium as a function of the friction number.
• Figure 4: Effective stress as a function of relative position ($\tilde{z}$) for a porous medium relaxing from a compressed state (with an applied forcing $|\tilde{\sigma}^{\prime \star}_c|=|\Delta \tilde{P}^\star_c|=0.05$) toward a fully decompressed state ($\tilde{\sigma}^{\prime \star}=\Delta \tilde{P}^\star=0$) in 11 steps. The loading is imposed by a piston (left half, blue curves) or by a fluid flow (right half, red curves), and the stress field is evaluated from the analytical solutions (continuous coloured curves) and from a full numerical resolution (dotted black). For each forcing, two typical cases are presented: one with relatively low friction ($\mathcal{F} = 0.5$) and one with relatively high friction ($\mathcal{F} = 5$). Arrows show the evolution with decreasing load.
• Figure 5: Position of the slip front ($\tilde{z}_\mathrm{slip}$) as a function of the loading intensity, scaled by the intensity of the initial compression ($\tilde{\sigma}^{\prime \star}_c$, $\Delta \tilde{P}^\star_c$), for piston-driven decompression and fluid-driven decompression. In both cases, $\mathcal{F}$ takes the following values: ${0.25, 0.5, 1, 2, 3, 4, 5}$. Theoretical predictions from the analytical model are displayed as solid coloured lines, while numerical results appear as black dotted lines.