Evolving Microstructure

Homogenization & multiscale media

Overview

Porous media can change their geometry, for instance through sedimentation or dissolution processes. I investigate how this evolution influences effective macroscopic equations, including porosity, fluid flow, and transport. More advanced models couple the evolving geometry to the unknown concentration or velocity, so that the motion of the pores becomes part of the solution and leads to free-boundary problems. Mathematically, a central challenge is that the PDEs are posed on time-dependent domains, rather than merely having time-dependent coefficients on a fixed domain.

Schematic homogenization of an evolving microstructure

We are interested in local changes of the microstructure. For example, sedimentation of material transported through a porous medium can induce a local growth wave. The animation illustrates how such evolution appears at different microscopic scales and in the effective medium.

Time t/T = 0.00 Speed

Microscopic domains \(\Omega_\varepsilon(t)\)

ε = 1/4

ε = 1/8

ε = 1/16

ε → 0

Effective medium

Reference cell \(Y_x(t)\)

\(Y_x(t)\)

Click a microscopic panel to select a local cell. The zoomed cell shows the same shape at the same time. Shading in the effective domain illustrates local porosity, not a computed homogenized solution.

Illustrative coupled transport and flow model

Let \(\Omega_\varepsilon(t)\subset\Omega\) be the evolving fluid region. Its internal pore boundary is \(\Gamma_\varepsilon(t)=\partial\Omega_\varepsilon(t)\setminus\partial\Omega\), and the part of its boundary on the exterior of the sample is \(\Sigma_\varepsilon(t)=\partial\Omega_\varepsilon(t)\cap\partial\Omega\). An illustrative model couples advection and diffusion of a concentration \(c_\varepsilon\) to the quasi-stationary Stokes velocity \(\mathbf v_\varepsilon\). We denote by \(\mathbf n_\varepsilon\) the outward unit normal to the fluid region.

Microscopic model in \(\Omega_\varepsilon(t)\)

Advection–diffusion with nonlinear Robin boundary condition

\[ \partial_t c_\varepsilon -\nabla\!\cdot(D\nabla c_\varepsilon) +\nabla\!\cdot(c_\varepsilon\mathbf v_\varepsilon) =f_\varepsilon \quad\text{in }\Omega_\varepsilon(t). \] \[ -D\nabla c_\varepsilon\cdot\mathbf n_\varepsilon =\varepsilon\,r(c_\varepsilon) \quad\text{on }\Gamma_\varepsilon(t). \]

\(r\) is a (possibly nonlinear and signed) surface reaction rate. The factor \(\varepsilon\) compensates for the increasing pore surface area per unit volume.

Quasi-stationary Stokes system with boundary conditions

\[ \begin{aligned} -\varepsilon^2\mu\Delta\mathbf v_\varepsilon +\nabla p_\varepsilon &=\mathbf g_\varepsilon,\\ \nabla\cdot\mathbf v_\varepsilon&=0 \end{aligned} \quad\text{in }\Omega_\varepsilon(t). \] \[ \begin{aligned} \mathbf v_\varepsilon &=\mathbf V_{\Gamma,\varepsilon} &&\text{on }\Gamma_\varepsilon(t),\\ \boldsymbol\sigma_\varepsilon\mathbf n_\varepsilon &=\mathbf h_\varepsilon &&\text{on }\Sigma_\varepsilon(t). \end{aligned} \]

No-slip means that the fluid velocity equals the moving pore-wall velocity \(\mathbf V_{\Gamma,\varepsilon}\) on \(\Gamma_\varepsilon(t)\); for a stationary wall, \(\mathbf V_{\Gamma,\varepsilon}=0\). On the exterior boundary, \(\mathbf h_\varepsilon\) prescribes Neumann-type traction data. Here \(\mu>0\) is the viscosity and \(\boldsymbol\sigma_\varepsilon=\varepsilon^2\mu\nabla\mathbf v_\varepsilon-p_\varepsilon I\) is the stress-like flux consistent with the Laplacian formulation (rather than the symmetric physical Cauchy stress). The \(\varepsilon^2\)-scaling is illustrative of a Darcy-type limit.

Possible homogenized model in \(\Omega\)

Effective advection–reaction–diffusion equation for the concentration

\[ \partial_t(\theta c_0)-\nabla\!\cdot(\mathbb D_{\mathrm{eff}}\nabla c_0)+\nabla\!\cdot(c_0\mathbf v_0)=\theta f_0-\mathcal R_{\mathrm{eff}}(c_0). \]

\(\theta=\theta(t,x)\) is the local porosity, \(\mathbb D_{\mathrm{eff}}\) the effective diffusion tensor, \(\mathbf v_0\) the macroscopic volume flux (Darcy velocity), and \(\mathcal R_{\mathrm{eff}}\) the effective surface-reaction term.

Darcy-type effective flow

\[ \mathbf v_0=-\mathbb K_{\mathrm{eff}}(\nabla p_0-\mathbf g_0). \] \[ \partial_t\theta+\nabla\cdot\mathbf v_0=0. \]

In the first equation, the permeability tensor \(\mathbb K_{\mathrm{eff}}\) depends on the evolving pore geometry. For incompressible fluid flow with impermeable moving pore walls and no volumetric sources, the second equation expresses macroscopic fluid-volume conservation: the change in pore volume fraction \(\theta\) is balanced by the divergence of the averaged volumetric flux \(\mathbf v_0\).

Geometry-dependent coefficients

\[ \theta(t,x)=\frac{|Y^*(t,x)|}{|Y|}, \qquad \mathbb D_{\mathrm{eff}}=\mathbb D_{\mathrm{eff}}(t,x),\qquad \mathbb K_{\mathrm{eff}}=\mathbb K_{\mathrm{eff}}(t,x). \]

Here \(Y^*(t,x)\) denotes the instantaneous fluid part of the reference cell \(Y\). These are schematic examples of possible limit terms, not a general homogenization theorem for moving-boundary models. The precise form, including transport and memory contributions, depends on the scaling and cell problems.

Coupled geometry and free boundaries. In more advanced descriptions, the pore geometry is not prescribed. Sedimentation or dissolution can be driven by the very same surface rate \(r(c_\varepsilon)\) as in the nonlinear Robin condition. For example, the following kinetic law, with \(\kappa>0\), couples the moving boundary to the concentration:

\[ \mathbf V_{\Gamma,\varepsilon}\cdot\mathbf n_\varepsilon =-\varepsilon\kappa\,r(c_\varepsilon). \]

The microscopic problem then becomes a free-boundary problem, and effective transport, flow, and porosity must be determined together. The sign of the normal speed depends on the convention for the normal and on whether the reaction deposits or removes material.

Publications on this topic

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