Table 3

Tortuosities of microstructures. Average tortuosities with the corresponding standard deviation std, computed on 40 realizations of each model. Models (1), (2) and (3) assess the impact of volume fraction. Models (4) and (5) assess the impact of heterogeneity. Models (1’), (2’) and (3’) assess the impact of anisotropy; all propagating direction are evaluated, x (compression direction) and y (perpendicular direction) are displayed as well as the mean value (mean) over the three spatial directions.

	Tortuosity
	τg	τm	τB	τP	τG	${\tau }_{G_r}$
Model	lσ	lσ	lσ	lσ	lσ	lσ
(1) Mean	1.0178	1.0444	1.0444	1.0471	1.0511	1.0500
	0.0006	0.0001	0.0001	0.0001	0.0002	0.0003
(2) Mean	1.0525	1.0855	1.0855	1.0892	1.0936	1.0945
	0.0007	0.0002	0.0002	0.0002	0.0004	0.0006
(3) Mean	1.1334	1.1872	1.1875	1.1891	1.1951	1.2049
	0.0013	0.0009	0.0009	0.0005	0.0013	0.0018
(4) Mean	1.0161	1.0460	1.0461	1.0491	1.0539	1.0525
	0.0006	0.0001	0.0001	0.0001	0.0003	0.0006
(5) Mean	1.0130	1.0478	1.0479	1.0513	1.0575	1.0563
	0.0007	0.0002	0.0002	0.0002	0.0005	0.0008
(1’) x	1.0906	1.0847	1.0849	1.0906	1.0985	1.0973
	0.0014	0.0004	0.0004	0.0003	0.0006	0.0010
(1’) y	1.0128	1.0323	1.0323	1.0342	1.0378	1.0366
	0.0009	0.0001	0.0001	0.0001	0.0003	0.0003
(1’) Mean	1.0194	1.0497	1.0497	1.0529	1.0580	1.0567
	0.0018	0.0045	0.0045	0.0049	0.0052	0.0053
(2’) x	1.1754	1.1660	1.1664	1.1754	1.1840	1.1890
	0.0021	0.0007	0.0007	0.0006	0.0014	0.0021
(2’) y	1.0404	1.0638	1.0638	1.0663	1.0705	1.0705
	0.0009	0.0002	0.0002	0.0002	0.0005	0.0006
(2’) Mean	1.0580	1.0977	1.0979	1.1026	1.1084	1.1102
	0.0048	0.0088	0.0088	0.0094	0.0098	0.0102
(3’) x	1.4068	1.3967	1.3983	1.4068	1.4187	1.4703
	0.0052	0.0020	0.0021	0.0017	0.0040	0.0067
(3’) y	1.1012	1.1430	1.1432	1.1420	1.1528	1.1587
	0.0019	0.0007	0.0007	0.0005	0.0017	0.0028
(3’) Mean	1.1565	1.2269	1.2276	1.2300	1.2421	1.2626
	0.0142	0.0219	0.0221	0.0228	0.0229	0.0269