Concepts

What OpenImpala computes

OpenImpala takes a segmented 3D voxel image (where each voxel is labelled with a phase ID) and computes effective transport properties by solving partial differential equations directly on the voxel grid.

Phase data

Images are segmented into integer phase IDs stored in an AMReX iMultiFab. Typically:

• Phase 0 = pore / void

• Phase 1 = solid matrix

This is configurable via the phase parameter. Multi-phase transport is supported: each phase can be assigned a different transport coefficient.

Volume fraction

The simplest metric: the fraction of voxels belonging to a given phase.

$$\varepsilon = \frac{N_{\text{phase}}}{N_{\text{total}}}$$

Percolation

Before solving transport equations, OpenImpala checks whether the target phase forms a connected path from inlet to outlet using a GPU-accelerated flood-fill algorithm. If the phase does not percolate, transport is zero.

Tortuosity

Tortuosity quantifies how much a winding pore structure impedes transport compared to a straight channel. OpenImpala solves the steady-state diffusion equation:

$$\nabla \cdot (D \nabla \phi) = 0$$

with Dirichlet boundary conditions at inlet ($\phi = 0$) and outlet ($\phi = 1$), and zero-flux Neumann conditions on lateral faces.

The effective diffusivity is computed from the resulting flux:

$$D_{\text{eff}} = \frac{|\text{average flux}|}{\text{cross-section area} \times |\nabla\phi_{\text{imposed}}|}$$

Tortuosity is then:

$$\tau = \frac{\varepsilon_{\text{active}}}{D_{\text{eff}}}$$

where $\varepsilon_{\text{active}}$ is the volume fraction of the percolating (connected) phase.

For a uniform medium on an $N$-cell grid, the discrete solution gives $D_{\text{eff}} = N/(N-1)$, so $\tau = (N-1)/N$.

Effective diffusivity tensor

For anisotropic microstructures, the full effective diffusivity tensor $\mathbf{D}_{\text{eff}}$ is computed by solving the cell problem from homogenisation theory:

$$\nabla_\xi \cdot \left( D \nabla_\xi \chi_k \right) = -\nabla_\xi \cdot \left( D \hat{e}_k \right)$$

for corrector functions $\chi_k$ in each direction $k \in {x, y, z}$, with periodic boundary conditions. The tensor components are:

$$D_{\text{eff},ij} = \frac{1}{|Y|} \int_Y D(\mathbf{x}) \left( \delta_{ij} + \frac{\partial \chi_j}{\partial x_i} \right) , d\mathbf{x}$$

Face coefficients

Inter-cell diffusivities use the harmonic mean of adjacent cell values:

$$D_{\text{face}} = \frac{2 D_L D_R}{D_L + D_R}$$

This is physically correct for resistances in series and ensures that a solid cell ($D = 0$) adjacent to a pore cell correctly blocks transport.