Influence of the Topology on the Flow Properties of Porous Media Explored from 3D Simulations

Microscopy Microanalysis Microstructures, Jul 2018

Patricia Jouannot, Jean-Paul Jernot

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Influence of the Topology on the Flow Properties of Porous Media Explored from 3D Simulations

Microsc. Microanal. Microstruct. Influence of the Topology on the Flow Properties of Porous Media Explored from 3D Simulations Patricia Jouannot 0 Jean-Paul Jernot 0 0 LERMAT , URA CNRS 1317, ISMRA, 14050 Caen Cedex , France 2014 A 3D-simulated random packing of spheres is digitized and densified with two basic tools of mathematical morphology: dilation and closing. Using these transformations separately, the densification of the initial structure follows two different topological paths. As a consequence, a fluid flow simulated by geodesic dilations inside the porous networks shows two different behaviours depending on the topological paths of the densification. - space on the initial structure digitized on the FCC grid. In this case, the elementary structuring element, B, is a cuboctahedron (one central point with its twelve equidistant neighbours). Successive dilations or closings carried out on the initial structure give the 3D dilated and closed images. The dilated structures (respectively closed) named Di (Fi) are the result of a transformation of size i realized on the initial structure DO (FO, identical to DO) using the cuboctahedron. An example of these structures is given in Figure 1 where plane cuts through the structures Dl and F3 are visualized. The topology of each structure is quantitatively described by the Euler-Poincaré characteristic, N3, measured in 3D space. This parameter is defined by with: s, the number of disconnected surfaces, G, the genus of each surface (the genus of a surface being the maximum number of cuts that can be made through this surface without disconnecting it in several parts) [ 4 ]. In the case of a local analysis, the value of N3 must be referred to a unit volume of structure (Nv = N3 /V ). For all the simulations, the field size is constant and is chosen as the unit volume. This parameter is followed as a function of the densification for dilated and closed structures. At the beginning of the densification, there are no major changes in the topological state of the stacking (DO, Dl, FO, FI). The slight decrease of Nv is only due to the increase of the compacity [ 2 ]. A further densification induces a modification of the porQus network: some branches are removed, the genus decreases and Nv increases (D2, D3, F2, F3, F4). Then, the pores become isolated (D4, D5, ..., F5 ... ) and finally disappear (Nv = 0). The two processes of densification can be compared in Figure 2. For the same porosity, two different topological states are obtained. Thus, two topological paths are followed all along the densification process, different connectivity properties being associated with each path. Geodesic Propagations Through the Porous Networks The topology of the structures has just been described by the global parameter Nv. More information can be obtained from an exploration of the porous network taking into account its local connectivity. EXPLORATION OF THE POROUS PHASE. - The different porous networks are explored by a simple but powerful transformation of mathematical morphology: the geodesic dilation [ 5 ]. For each structure, the porous phase of the upper plane is taken as a marker and the geodesic dilations start from this plane. A kind of flow is generated from the upper plane (z = 0) to the lower one (z = 163). Each elementary step of the propagation is called the time, t. The progressive invasion of the porous phase is described by two parameters: the percentage of the porous surface invaded on the lower plane, % A(P); the percentage of the porous volume invaded in the whole structure, % V(P). The curves corresponding to the structures Dl and F3 are given in Figure 3. Following the topological evolution of the structures (Fig. 2), four steps can be distinguished according to the differences of behaviours with respect to the propagation: i) A complete filling of the porous phase is obtained for the initial structures DO, Dl, FI. The time necessary to reach the bottom plane or to invade the whole volume is nearly the minimum one (163). This is due to an almost plane filling of the porous phase and a regular invasion illustrated by the linear variation of the porous volume invaded with the propagation time. ii) For the structures D2 and F2, a still complete filling of the pores is observed: the porous network is thus entirely interconnected. Nevertheless, the propagation time is longer than for the initial structures, the paths being more and more tortuous. iii) From now on (D3, F3, F4) some parts of the porous network become isolated. A whole filling of the porous phase is no more possible. The partial invasion of the bottom plane is irregular and the invasion time is very long. iv) For the last structures, the propagation flow does not reach the bottom plane. The porous phase contains a large proportion of isolated pores and the invaded volume is small. SIMULATION OF FLOW. - The exploration of the porous network by means of geodesic propagations looks like a simulation of flow [ 6, 7 ]. The volumetric rate of flow per unit volume of structure, Qv, can be defined as the porous volume invaded during the time interval Ot: The evolution of this parameter as a function of the densification is reported in Figure 4 and the results are collected in Table I (for simplification, Ot = 100 steps is taken as the unit time). In the figure, the dotted line represents a linear decrease of Qv with the compacity. The results concerning DO (FO), Dl and Fl are placed on this dotted line. For these structures, the value of N3 reported per unit volume of the solid phase, i.e. N3jV(S) = NV/VV(S), is almost the same. This means that they are topologically equivalent and the flow only depends on the quantity of the porous phase. For the more densified structures, the linear behaviour is no longer observed: topological changes occur inside the porous networks. They take place at lower densities for the structures Fi than for the structures Di (Fig. 2) and Qv decreases more rapidly for the closed structures than for the dilated ones. However, the decrease of Qv could also be attributed to a decrease of the size of the pores along the densification. Then, for each structure (Di, Fi), a granulometry of the porous phase has been carried out by openings [8] in 3D space. For comparable values of Nv (e.g. D2 and F2), as expected, the corresponding results of Qv are higher for the closed structure F2 possessing a larger mean size of pores and a larger porosity. For comparable values of the porosity (e.g. D1 and F3), the mean pore size is larger for the porous network of F3 than for Dl (Fig. 5). With almost the same porosity and a mean pore size roughly 50% larger for F3 than for Dl, one can expect that the flow would be easier through the F3 structure. Surprisingly, the reverse is observed: the value of Qv is lower for F3 than for Dl. This indicates that, in this case, the flow is more influenced by the topology of the porous network than by its size. Conclusion Two transformations of mathematical morphology, dilation and closing, have been used to densify a random stacking of spheres. The resulting structures follow two different topological paths: the dilation promotes the densification while the closing favours the topological modifications of the porous medium. The two topological paths of densification induce two different behaviours with respect to the same simulation of flow. At the beginning, the flow only depends on the porosity because the topological characteristics are equivalent. Then, for comparable values of the porosity, the flow is mainly disrupted by the topological modifications of the porous network, whatever the pore size. This emphasizes qualitatively the role played by the topology on the flow properties of a porous medium. Acknowledgements This work was performed in the frame of the "Pôle Traitement et Analyse d’Image" de BasseNormandie. [1] Visscher W.M. and Bolsterli M. , Random packing of equal and unequal spheres in two and three dimensions , Nature 239 ( 1972 ) 504 - 507 . [2] Bhanu Prasad P. and Jernot J.P , Topological description of the densification of a granular medium , J. Microsc . 163 ( 1991 ) 211 - 220 . [3] Serra J. , Image Analysis and Mathematical Morphology , Vol. 1 (London Academic Press, 1982 ). [4] DeHoff R.T , Aigeltinger E.H. and Craig K.R. , Experimental determination of the topological properties of three dimensional microstructures , J. Microsc . 95 ( 1972 ) 69 - 91 . [5] Lantuéjoul C. and Beucher S. , On the use of the geodesic metric in image analysis , J. Microsc . 121 ( 1981 ) 39 - 49 . [8] Matheron G. , Éléments pour une théorie des milieux poreux , (Masson, Paris, 1967 ). [6] Jernot J.P. , Bhanu Prasad P. and Demaleprade P , Three-dimensional simulation of flow through a porous medium , J. Microsc . 167 ( 1992 ) 9 - 21 . [7] Jouannot P. , Étude topologique de structures aléatoires biphasées, Thèse de Doctorat de l' Université de Caen ( 1994 ).

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Patricia Jouannot, Jean-Paul Jernot. Influence of the Topology on the Flow Properties of Porous Media Explored from 3D Simulations, Microscopy Microanalysis Microstructures, 485-490, DOI: doi:10.1051/mmm:1996148