Pore Network Modeling: The "Tight Dual Model"

In the following we motivate our image-based approach to represent 3D shapes from the perspective of somebody who wants to simulate transport processes in pore spaces. Our representation for 3D shapes also supports simulations of non-isothermal processes in pore spaces including heat flow, evaporation and condensation, the characterization of structural changes in snow and sea ice as a function of temperature, the description of bone structure, ...

Realistic network simulations of transport processes in the subsurface require realistic pore networks - but what makes a pore network realistic? Obviously, the network must represent a realistic or real pore space, and it must contain the pore space's topological and geometrical information that controls the transport process. These requirements can be met by (1) constructing the pore network from a synthetic porous medium or from a 3D gray scale image of a real pore space and by (2) letting the nodes and the bonds of the pore network represent the pore bodies and pore throats of the pore space. Despite the difficulty of actually defining the pore throats and the pore bodies, this concept of a pore network is implicit in many approaches to represent pore spaces by networks. Dullien, for example, assumes on page 193 of his textbook "Porous Media. Fluid Transport and Pore Structure" that the words "bond" and "pore throat" "have the same meaning", and that the same holds true for "nodes" and "pore bodies. In our approach we distinguish between bonds and pore throats [nodes and pore bodies] since they differ in dimension, i.e., one and two [zero and three]. We follow Dullien, however, in that we establish one-to-one correspondences, i.e., duality, between the words with ``the same meaning''.

Specifically, the pore space and the pore network take the form of cellular complexes, the former consisting of one- to three-dimensional cells and the letter consisting of zero- to two-dimensional cells. Duality now means that there exists a one-to-one correspondence between the cells that make up the pore space and those that make up the pore network such that
(1) the dimensions of a cell and its dual cell always add up to three, and
(2) cell c1 bounds cell c2 if and only if the dual of c2 bounds the dual of c1.

In particular, the zero-dimensional network nodes are the duals of the three-dimensional pore bodies, and the one dimensional network bonds are the duals of the two-dimensional pore throats. Thus, the shapes of the pore bodies [pore throats] can be taken into account when simulating imbibition [drainage] on the network. We have performed drainage simulations on networks derived from simulated sphere packings consisting of up to 13000 spheres and found very good agreement of the capillary pressure -- saturation curves from the simulations and those from real drainage experiments.

Pore network from simulated sphere packing with 86 spheres.
The faces are shown in red and the edges are shown in black.

You can download the latest version of the software package we developed in order to construct the dual model from synthetic sphere packings and from 3D gray scale images: Latest implementation of tight dual model

If you are interested in more details with regards to the approach, please check out the references below.

Personnel

Roland Glantz, Markus Hilpert

Sponsor

This work was funded by the National Science Foundation Grant "CMG Research: a graph based approach for generating pore networks to represent the uncertainty of the subsurface's pore structure."

Reference

Glantz, R. and M. Hilpert (2011). Capillary displacement in totally wetting and infinitely long right prisms. Multiscale Modeling and Simulation. DOI. 10.1137/100786472.
Glantz, R., and M. Hilpert (2008). Tight dual models of pore spaces. Advances in Water Resources 31: 787-806.
Glantz, R., and M. Hilpert (2006). Dual models of two- and three-dimensional porous media. Advances in Water Resources 30: 227-248.
Glantz, R., and M. Hilpert (2004). Generation of two-dimensional pore networks for drainage simulations. Proceedings of the 15th International Conference on Computational Methods in Water Resources in Chapel Hill, North Carolina, USA, June 13-17, 2004, Vol. 1, pp. 3-13.
Hilpert, M., R. Glantz, and C.T. Miller (2003). Calibration of a pore-network model by a pore-morphological analysis. Transport in Porous Media. Transport in Porous Media 51: 267-287.