See the tutorials geodesic distance, geodesic tortuosity and geodesic paths, geodesic channels for basic introductions to the term used in this post. A short summary:
- The geodesic path, GeoPath(v), relative to a set (a box or inlet in Mist) is the shortest path connecting the source point v to the closest pore voxel in the set.
- The geodesic distance, GeoDist(v), relative to a set, is the length of the geodesic path GeoPath(v) relative to the same set.
- The geodesic path, GeoPath(v), computed in direction dir = x, y, or z, is the shortest path connecting the source point v to both starting faces in direction dir (starting face = edge of the 3D volume).
- The (full volume) geodesic tortuosity, τ(v), in direction dir = x, y, or z, is the length of the geodesic path GeoPath(v) computed in direction dir divided by the length of the 3D volume in direction dir.
The geodesic path GeoPath(v), computed in direction dir = x, y or z, is the union of the two geodesic paths GeoPath(v) relative to the two inlets in that direction (inlet <z low> and <z high> in direction z, for instance). The lengths of the geodesic path in direction dir is therefore equal to the sum of the geodesic distance relative to the two inlets. The (full volume) tortuosity in a voxel v can therefore be computed from the geodesic distance, inlet computed relative the two inlets. For direction z, the tortuosity is
![\[\tau(v) = \frac{GeodDist_{\text{\it{z low}}}(v) + GeodDist_{\text{\it{z high}}}(v)}{L}\]](https://mist.math.chalmers.se/wp-content/ql-cache/quicklatex.com-ac833b7e1ae90a589b7f92d495084480_l3.png "Rendered by QuickLaTeX.com")
where L is the length of the volume in direction z and v is any pore voxel connected to both inlets. See this video showing the two inlets <z low> (plane z) and <z high> (plane Z).
The geodesic distance is in Mist computed using a wavefront planner, and the (full volume) geodesic tortuosity and geodesic paths are computed from the corresponding geodesic distance, see the reference manual and Barman et al, 2021 for details.
The inlet tortuosity, which is more commonly reported than the full volume tortuosity in the literature, is defined only for pore voxels at one inlet. The inlet tortuosity at inlet <z low>, for instance, gives the relative length of geodesic paths computed relative to the set <z high>. These are thus geodesic paths starting at inlet <z low> and ending at the closest pore voxel at inlet <z high>. For a voxel v at <z low>, the inlet tortuosity is defined as
![\[\tau_{\text{inlet}}(v) = \frac{GeodDist_{\text{\it{z high}}}(v)}{L}.\]](https://mist.math.chalmers.se/wp-content/ql-cache/quicklatex.com-4a2309cd66f15553fe2d026d8b0f83ba_l3.png "Rendered by QuickLaTeX.com")
Note that this gives the same definition as the full volume tortuosity for voxels v at <z low>, since the geodesic distance
![\[GeodDist_{\text{\it{z low}}}(v),\]](https://mist.math.chalmers.se/wp-content/ql-cache/quicklatex.com-bbbc843a95d46ec60a68d1be122557cd_l3.png "Rendered by QuickLaTeX.com")
is zero here (the distance from a voxel v in <z low> to the closest voxel in <z low> is zero).
The geodesic distance relative to the set <z high> is computed by starting at pore voxels in <z high>, and iteratively taking steps further and further away from those pores (in wavefronts). To compute the inlet tortuosity thus requires the geodesic distance to be computed for (most) connected pore voxels. This means that the full volume tortuosity is computed at around twice the computational cost compared to the inlet tortuosity. An advantage with the full volume tortuosity over the inlet tortuosity is that it characterizes dead-ends and inefficient pores within the pore volume, and can consequently give better predictions of material properties than the inlet tortuosity (see the geodesic tortuosity-tutorial) at only a small increase in computational cost.