Deviation from Archie’s law in partially saturated porous media: Wetting film versus disconnectedness of the conducting phase

We experimentally study the electrical transport in partially water-saturated pore network. The porous medium under investigation is a Fontainebleau sandstone, characterized by x-ray tomography. We show the existence of two electrical conductivity regimes. At high water saturation, the electric resistivity follows a well-known Archie law. Below a water saturation $S_w\sim 0.2$, a strong negative deviation from this Archie law is observed. We attribute this transition to the existence of “a thick liquid film,” assuring the ionic conduction in the low saturation regime. A numerical simulation is proposed to confirm this scenario. Two possible protocols are used to distribute the brine phase in the pore network of a three-dimensional microtomography image. The first one is based on a minimization of the interfacial energy. The second takes into account a quasistatic capillary displacement. The classical random-walk algorithm is used to compute the electric conductivity at various water saturations. Without the “thick film,” both of the two fluid-placing protocols show a disconnectedness transition of the brine phase when $S_w<0.2$. Adding this “film” to solid surface, the electrical continuity is maintained. The bending down trend is correctly reproduced, showing that in this range, the electric response cannot be described by a power law as usual.

Figure 1

Scanning electron microscopy (SEM) image of the Fontainebleau sandstone.

Figure 2

$R_{\text{ind}}\text{−}{S}_w$ curves of the Fontainebleau sandstone in drainage and imbibition modes. Black square: measurement in drainage mode. Circles: measurement in imbibition. Dotted line: Archie law [Eq. (2)] with $n=2$.

Figure 3

3D microtomography of the dry Fontainebleau sandstone pore network. The size of the cube is 1.5 mm ($500\times {10}^3$ voxels).

Figure 4

Computation of the capillary condensation in a cubic structure composed of eight spheres using simulated annealing algorithm: (a) initial state and (b) final state.

Figure 5

3D reconstruction of the water distribution inside the Fontainebleau sandstone pore network at $S_w=0.3$ using a simulated annealing algorithm. The size of the cube is 1.5 mm ($500\times {10}^3$ voxels).

Figure 6

(Color online) On the upper part: a section of a 3D microtomography of a partially water-saturated pore network $\left(S_w=0.3\right)$. The water phase appears in medium gray level. The part below: water distribution inside the pore network at $S_w=0.3$ using a simulated annealing algorithm (see text). The solid is in black, the water phase is in medium gray, and the empty pore network is in white.

Figure 7

Box-counting data for wetting fluid distribution obtained by a simulated annealing. Full line: $S_w=1$. Full triangle: $S_w=0.40$. Open square: $S_w=0.20$. Full square: $S_w=0.10$. Open circle: $S_w=0.05$.

Figure 8

3D visualization of the brine configuration at $S_w=0.28$. The brine is invaded by air along the $X$ axis. We can find the small brine aggregates isolated by air. The largest size of the box is 1.5 mm ($500\times 500\times 400$ voxels).

Figure 9

Box-counting data for the nonwetting fluid distribution obtained by a quasistatic capillary displacement. Open triangle: $S_w=0.65$. Cross: $S_w=0.26$. Open square: $S_w=0.06$. The full line provides a guideline following an algebraic law with $\overline{D}=2.5$.

Figure 10

Box-counting data for the wetting fluid distribution obtained by a quasistatic capillary displacement. Full line: $S_w=1$. Full triangle: $S_w=0.65$. Open square: $S_w=0.26$. Full square: $S_w=0.13$. Open circle: $S_w=0.06$.

Figure 11

3D visualization of the graph of retraction of the water phase computed using simulated annealing algorithm at $S_w=0.30$. The size of the cube is 1.5 mm ($500\times {10}^3$ voxels).

Figure 12

Evolution of the distribution of the number of connections (the degree distribution) of the retraction graphs associated to the fluid distribution obtained either by a simulated annealing or by a quasistatic capillary displacement. Open circles: saturated pore network. Full triangles: Pore network saturated at $S_w=0.28$ under static capillary displacement. Open square: capillary condensation of the pore network at $S_w=0.3$ computed with a simulated annealing algorithm.

Figure 13

Evolution of the topological constant $C$ [Eq. (4)] with the degree of water saturation. Open circles: static capillary displacement. Full circles: capillary condensation of the pore network computed with a simulated annealing algorithm.

Figure 14

Computation of the formation factor for a homologous series of saturated pore networks generated by a succession of mathematical erosion or dilation. The seed pore network is the 3D microtomography shown in Fig. 3. $-1$ stands for a first erosion. $+1$ stands for a first dilation. $+2$ stands for two consecutive dilations of the pore network. The straight line in log-log scale is the best fit according to the Archie law.

Figure 15

Computed $R_{\text{ind}}\text{−}{S}_w$ curves with brine phase configuration obtained either by simulated annealing or by capillary invasion and comparison with the experimental measurements. Black square: experimental measurements. White square: simulated annealing configuration. Circle: capillary displacement configuration.

Figure 16

Scanning electron microscopy image of grain surface roughness using backscattered electrons.

Figure 17

Brine film configuration: the film elements cover the pore surface when the pore is emptied.

Figure 18

Computed $R_{\text{ind}}\text{−}{S}_w$ curves involving a thick film with a thickness equal to $0.6\mu \text{m}$. The last points (at small $S_w$) of the computed curves depend only on the resistivity of the film. Black square: experimental measurement. Open square: simulated annealing. Open circle: capillary displacement.