[18] The principal interest of section 3 is to interpret the results obtained from binarized 3D images of Fontainebleau sandstone and to provide recommendations and information on the choice of numerical parameters, permeability formulations and microCT resolution. Nevertheless, given the relatively complex structure of the sandstone, we first apply the LBTRT scheme to simpler model structures (straight channel, assembly of straight channels in parallel or series, 2D model porous media with different degrees of heterogeneity). These results will help us interpret the results obtained from the microCT images.
3.1. Flow in a Narrow Straight Channel
[21] We now investigate the influence of the numerical error on the permeability of the channel. For this geometry, the effective permeability calculated using the Darcy's law (see equation (12)) is
with
The effective permeability is thus related to the effective width by
The energy dissipation approach (see equation (13)) gives a permeability
with
or, using equation (17), we have leading to
so that equation (22) reduces to . Both approaches lead to the same value of the effective permeability using equation (18):
[22] The exact permeability is obtained for . However, to compute the permeabilities from the simulation data, integrals in equations (20) and (23) are replaced by sums. To investigate the influence of the numerical integration rule (midpoint rule, summation), we compute the mean averaged values from the effective solution:
and
Note that relation (27) is exact since the firstorder derivatives are computed exactly for the parabolic profile using equation (16). For Darcy's law, the discretization of equations (19) and (26) gives
In this case numerical values are equal to the exact solution .
[25] Thus, although the exact permeability obtained by Darcy and by the energy dissipation approaches are identical using an exact integration rule, the simulation results differ due to the summation over the mesh. This difference plays an important role in a low resolved channel.
[26] However, replacing the midpoint rule by a trapezoidal rule, the “best” value is for Darcy's law and for energy dissipation method (cf. equation (29) and (30) for midpoint rule). At the same time, applying the Simpson's rule, e.g., replacing in the first line in equation (26) by , one obtains the following exact effective solution: which coincides with when , i.e., when . Similarly, Simpson's rule gives , and then for . At this point we stress the Simpson's rule (S), midpoint (M) and trapezoidal (T) rules are related: . This relation is satisfied for using Darcy's law: , because of the linearity of the effective width square with . However, it is not so for the energy dissipation method computed as the ratio of two mean quantities. On this basis, we should expect less linear dependency on for this method.
[27] Although one could suggest that a highorder integration method will also improve the precision in arbitrary geometry, we believe that it would be so only provided that (1) the effective solution imposes noslip velocity at the boundary nodes with the effective local secondorder precision, at least, and (2), one may perform accurate interpolations for the intermediate integration points. In porous media, it is difficult to reach the local secondorder precision because of insufficient pore resolution and imprecise description of solid boundaries. At the same time, when the boundaries are precisely described, the higherorder accurate boundary schemes alone strongly reduce solution dependency on [Ginzburg and d'Humières, 2003; Ginzburg et al., 2008a]. They are the best candidates for most accurate permeability measurements, with or without using highly accurate integration rules.
3.2. Permeability of Independent Channels Placed in Parallel and Series
[29] For the parallel configuration, the permeability varies linearly with . However, for the second configuration, a negative curvature is observed. Also, the dependence on is more pronounced for the distribution with the large standard deviation. This behavior can be explained by the socalled bottleneck effect. If a porous medium is characterized by a topology and heterogeneity in a way that it always allows the fluid to pass through large pores, global permeability and therefore dependency on is determined by those highly resolved pores. Thus, in this case dependency on is minor. However, in the case of a bottleneck system, where fluid is forced to pass through small restrictions, global system behavior depends on those underresolved regions [Talon et al., 2010]. Thus, dependency on becomes important and nonlinear.
3.3. TwoDimensional Model Porous Media: Array of Circles and Truncated Gaussian Distribution
[30] Our attention is now drawn to the influence of viscosity by comparing the TRT to BGK scheme, the choice of driving conditions, the approach to determine permeability (Darcy: K_{D} or energy dissipation: K_{E}) and the convergence time as a function of the model parameters. To this goal, a regular and a random 2D porous medium was generated. The first corresponds to an array of disks whereas the second was generated by assigning random values taken from a Gaussian distribution to a two dimensional grid [Adler et al., 1990]. All grid cells whose value is below a certain value c_{th} correspond to the solid phase and others to the void space. The value of c_{th} sets the porosity of the media to . The two porous structures, displayed in Figure 3, have the same porosity, .
[31] Table 1 gives the permeability values obtained with TRT and BGK scheme for the random porous media. Results are given for both driving conditions (DC1: body force, DC2: pressure gradient). The values of the permeability obtained by TRT scheme are clearly viscosity independent whereas those determined by BGK do depend on . Table 1 also gives the permeability obtained with the different driving conditions (only for TRT scheme): the values are close demonstrating that conditions are equivalent. The small difference between the values can be explained by the size of the porous media, being periodic it is infinite for DC1, whereas for DC2 it has a finite size.
Table 1. Permeability of the 2D Random Porous Media as a Function of the Viscosity and the Two Driving Conditions Calculated by the TRT and the BGK Methods^{a}Viscosity, ν  TRT: DC1  TRT: DC2  BGK: DC2 

(Λ = 0.01)  (Λ = 0.2)  (Λ = 0.01)  (Λ = 0.2) 


0.01  0.5986  0.8177  0.5942  0.8124  0.5308 
0.05  0.5986  0.8177  0.5942  0.8124  0.6319 
0.1  0.5986  0.8177  0.5942  0.8124  0.7282 
0.2  0.5986  0.8177  0.5942  0.8124  0.8953 
0.4  0.5986  0.8177  0.5942  0.8124  1.2002 
[35] Figure 6 gives the convergence time as a function of for the regular and the random porous media determined using the two criteria. As expected, the convergence time is shorter for criteria 2 than for criteria 1. The convergence time also varies with the fluid viscosity with a global minimum that depends on the porous structure and the applied criteria.
[37] As a consequence, the computation time may be optimized by using a proper combination of viscosity and speed of sound c_{S}. This behavior strongly depends on the porous structure. For instance, convergence time decreases with the viscosity in the case of Poiseuille flow (results not shown).
3.4. Application of the TRT to Fontainebleau Sandstone
[43] Table 2 shows the convergence time of the Fontainebleau sandstone as a function of the viscosity for . To smooth data a moving average of N = 50 was applied before differentiation. Convergence criteria 2 was used. In contrast to the 2D porous media, there is no minimum in convergence time. Convergence time decreases with decreasing viscosity. This is in contrast to results of Pan et al. [2006] and Ginzburg and d'Humières [2003] for artificial porous structure where convergence time increases for lower viscosity.
Table 2. Convergence Time for Permeability Measurements of the Fontainebleau Sandstone as a Function of the Viscosity for Λ = 0.2^{a}Viscosity, ν  Convergence Time 


0.01  618 
0.05  1325 
0.1  1747 
0.2  2344 
0.4  3094 