Hydraulic and Electrical Conductivity: Conductivity Exponents and Critical Path Analysis

Abstract

The saturation dependences of the electrokinetic current, the electrical conductivity, and the hydraulic conductivity are investigated by applying concepts from critical path analysis to the pore-size variability and universal scaling from percolation theory to the topological inputs from phase continuity. This division makes it possible to define ranges of saturation for which the pore-size or the topological effects on the given property dominate. We find that in natural media the hydraulic conductivity is predominantly influenced by the pore-size distribution, the electrical conductivity is usually controlled by topology, and there is no input from the pore-size distribution to the electrokinetic current. These calculations are based on a particular model of the medium as well as scaling of flow appropriate from Poiseuille’s law. Qualitatively similar, but quantitatively different conclusions result from different model inputs. It is shown that previously derived non-universal scaling results for the conductivity are obtained from critical path analysis in a limiting case, proving that the treatment is a generalization of known physics.

Appendices

Problems

• 6.1 Prove that r _c is the same for electrical and hydraulic conductivities and draw an analogy to Problem 3.1 in Chap. 3, which asked to show that the value obtained for R _c was independent of whether one obtained r _c first and substituted into the equation for R(r), or whether one integrated over R directly.

• 6.2 Consider a log-normal distribution of pore sizes, but assume as in a fractal model that all pores have the same shapes (this is a necessary assumption in a fractal model, but only an assumption of convenience, otherwise). Derive equivalent expressions for the saturation dependence of the hydraulic and electrical conductivity using critical path analysis and find the moisture contents at which the critical path analysis must be replaced by percolation scaling. These exercises may be performed numerically. Compare the ranges of parameter space (D _p, ϕ), for which Archie’s law may be reasonably derived from percolation theory determined from the log-normal and the power-law distributions. Does a log-normal distribution tend to make Archie’s law more or less widely applicable than is the case for a power-law distribution of pore sizes?

• 6.3 Graphically represent the apparent power μ ^∗(ϕ) of the porosity in Archie’s law when the pore size distribution modifies its value from 2 in the cases that (a) the ratio r ₀/r _m is held constant and (b) the fractal dimensionality is held constant. You will need to keep in mind that in either case μ ^∗ is dln(σ)/dln(ϕ)=(ϕ/σ)dσ/dϕ.

Appendix: Calculation of Water Retention and Unsaturated Hydraulic Conductivity for Pore-Solid Fractal Model with Two Fractal Regimes

6.2.1 Water Retention Curve

We assume that the pore-size distribution of soils follows the pore-solid fractal (PSF) approach proposed by Perrier et al. [55]. This model combined with percolation theory has been successfully applied to model unsaturated hydraulic conductivity of soils with different textures [23]. The continuous probability density function, W(r), of pores consistent with the Hunt and Gee [33] analogy would be:

$$ W ( r ) = \beta \frac{3 - D}{r_{\max}^{3 - D}}r^{ - 1 - D},\quad r_{\min} < r < r_{\max} $$

(6.48)

where β=p/(p+s) in which p and s are the pore and solid fractions [55], respectively, D is the pore-solid interface fractal dimension, r is the pore radius (r∝1/h where h is the tension head), and r _min and r _max are the smallest and largest pore radii, respectively.

Figure 6.28 shows a schematic of a soil water retention curve with two fractal regimes. The first regime covers mostly the large (frequently structural) pores, and the second regime includes the small (textural) pores. The water content at the cross-over point is denoted by θ _x which is equal to the porosity of the second regime ϕ ₂.

Fig. 6.28

Depiction of two separate fractal regimes in a soil water retention curve and corresponding definitions

In a porous medium having a probability density function that scales differently in two different regimes, e.g., D ₁ and D ₂ (Fig. 6.28), the total porosity may be found by integrating r ³ W(r) between r _min and r _max to obtain

$$\begin{aligned} \phi =& \phi _{1} + \phi _{2} = \beta_{1}\frac{3 - D_{1}}{r_{\max}^{3 - D_{1}}}\int_{r_{\mathrm{x}}}^{r_{\max}} r^{3}r^{ - 1 - D_{1}}dr + \beta_{2}\frac{3 - D_{2}}{r_{\mathrm{x}}^{3 - D_{2}}}\int_{r_{\min}}^{r_{\mathrm{x}}} r^{3}r^{ - 1 - D_{2}}dr \\ =& \beta_{1} \biggl[ 1 - \biggl( \frac{r_{\mathrm{x}}}{r_{\max}} \biggr)^{3 - D_{1}} \biggr] + \beta_{2} \biggl[ 1 - \biggl( \frac{r_{\min}}{r_{\mathrm{x}}} \biggr)^{3 -D_2}\biggr] \end{aligned}$$

(6.49)

where r _x is the pore radius at the cross-over point where the fractal behavior of the medium changes from regime 1 to regime 2, and D ₁ and D ₂ are the pore-solid interface fractal dimension of the first and second regimes, respectively.

The water content of pores with radii less than or equal to r in the second regime (h _x<h<h _max) is determined as follows (r<r _x):

$$ \theta = \beta_{2}\frac{3 - D_{2}}{r_{\mathrm{x}}^{3 - D_{2}}}\int_{r_{\min}}^{r} r^{3}r^{ - 1 - D_{2}}dr = \beta_{2} \biggl[ \biggl( \frac{r}{r_{\mathrm{x}}} \biggr)^{3 - D_{2}} - \biggl( \frac{r_{\min}}{r_{\mathrm{x}}} \biggr)^{3 - D_{2}} \biggr] $$

(6.50)

Likewise, the water content of pores with radii less than or equal to r in the first regime (h _min<h<h _x) would be (r>r _x):

$$\begin{aligned} \theta =& \beta_{1}\frac{3 - D_{1}}{r_{\max}^{3 - D_{1}}}\int_{r_{\mathrm{x}}}^{r} r^{3}r^{ - 1 - D_{1}}dr + \beta_{2}\frac{3 - D_{2}}{r_{\mathrm{x}}^{3 - D_{2}}}\int_{r_{\min}}^{r_{\mathrm{x}}} r^{3}r^{ - 1 - D_{2}}dr \\ =& \phi _{2} + \beta_{1} \biggl[ \biggl( \frac{r}{r_{\max}} \biggr)^{3 - D_{1}} - \biggl( \frac{r_{\mathrm{x}}}{r_{\max}} \biggr)^{3 - D_{1}} \biggr] \end{aligned}$$

(6.51)

Combining Eqs. (6.50) and (6.51) with the capillary equation gives the following piecewise soil water retention curve model:

$$\begin{aligned} \theta = \left \{ \begin{array}{l} \phi \quad h < h_{\min}\\ \phi _{2} + \beta_{1} \bigl[ \bigl( \frac{h}{h_{\min}} \bigr)^{D_{1} - 3} - \bigl( \frac{h_{\mathrm{x}}}{h_{\min}} \bigr)^{D_{1} - 3} \bigr] = \phi - \beta_{1} \bigl[ 1 - \bigl( \frac{h}{h_{\min}} \bigr)^{D_{1} - 3} \bigr]\\ \quad h_{\min} < h < h_{\mathrm{x}} \\ \beta_{2} \bigl[ \bigl( \frac{h}{h_{\mathrm{x}}} \bigr)^{D_{2} - 3} - \bigl( \frac{h_{\min}}{h_{\mathrm{x}}} \bigr)^{D_{2} - 3} \bigr] = \phi _{2} - \beta_{2} \bigl[ 1 - \bigl( \frac{h}{h_{\mathrm{x}}} \bigr)^{D_{2} - 3} \bigr] \quad h_{\mathrm{x}} < h < h_{\max} \end{array} \right . \end{aligned}$$

(6.52)

6.2.2 Unsaturated Hydraulic Conductivity

Following Ghanbarian-Alavijeh and Hunt [23], we use critical path analysis combined with the PSF model to find the critical pore radius for percolation for saturated and unsaturated conditions. Generally, there are two possibilities.

6.2.2.1 Possibility 1: θ _t<ϕ ₁

Using the pore-solid fractal approach, we define the critical volume content of percolation θ _t from critical path analysis for saturated conditions (θ=ϕ) as follows:

$$ \theta_{\mathrm{t}} = \frac{p_{1}}{p_{1} + s_{1}}\frac{3 - D_{1}}{r_{\max}^{3 - D_{1}}}\int _{r_{\mathrm{c}} ( \theta = \phi )}^{r_{\max}} r^{3}r^{ - 1 - D_{1}}dr = \beta_{1} \biggl[ 1 - \biggl( \frac{r_{\mathrm{c}} ( \theta = \phi )}{r_{\max}} \biggr)^{3 - D_{1}} \biggr] $$

(6.53)

Solution of Eq. (6.53) combined with Eq. (6.52) for r _c(θ=ϕ) gives

$$ r_{\mathrm{c}} ( \theta = \phi ) = r_{\max} \biggl[ \frac{\beta_{1} - \theta_{\mathrm{t}}}{\beta_{1}} \biggr]^{\frac{1}{3 - D_{1}}},\quad \theta_{\mathrm{t}} < \phi _{1} $$

(6.54)

For unsaturated condition as we show in Fig. 6.29, three cases are possible:

• (I) θ−θ _t>ϕ ₂ and θ>ϕ ₂ (see Fig. 6.29)

  Fig. 6.29

  

  Three distinct ranges of moisture content that control the determination of r _c when θ _t is less than either partial porosity

  The critical water content for percolation θ _t would be

  $$ \theta_{\mathrm{t}} = \beta_{1}\frac{3 - D_{1}}{r_{\max}^{3 - D_{1}}}\int _{r_{\mathrm{c}} ( \theta )}^{r} r^{3}r^{ - 1 - D_{1}}dr = \beta_{1} \biggl[ \biggl( \frac{r}{r_{\max}} \biggr)^{3 - D_{1}} - \biggl( \frac{r_{\mathrm{c}} ( \theta )}{r_{\max}} \biggr)^{3 - D_{1}} \biggr] $$

  (6.55)

  Rewriting Eq. (6.55) combined with Eq. (6.52) for r _c(θ) yields

  $$ r_{\mathrm{c}} ( \theta ) = r_{\max} \biggl[ \frac{\beta_{1} - \phi + \theta - \theta_{\mathrm{t}}}{\beta_{1}} \biggr]^{\frac{1}{3 - D_{1}}} $$

  (6.56)

• (II) θ−θ _t<ϕ ₂ and θ>ϕ ₂ (see Fig. 6.29)

  $$ \theta_{\mathrm{t}} = \beta_{2} \biggl[ 1 - \biggl( \frac{r_{\mathrm{c}} ( \theta )}{r_{\mathrm{x}}} \biggr)^{3 - D_{2}} \biggr] + \beta_{1} \biggl[ \biggl( \frac{r}{r_{\max}} \biggr)^{3 - D_{1}} - \biggl( \frac{r_{\mathrm{x}}}{r_{\max}} \biggr)^{3 - D_{1}} \biggr] $$

  (6.57)

  Solving Eq. (6.57) combined with Eq. (6.52) for r _c(θ) results

  $$ r_{\mathrm{c}} ( \theta ) = r_{\mathrm{x}} \biggl[ \frac{\beta_{2} - \phi _{2} + \theta - \theta_{\mathrm{t}}}{\beta_{2}} \biggr]^{\frac{1}{3 -D_{2}}} $$

  (6.58)

  in which $r_{\mathrm{x}} = r_{\max} [ \frac{\beta_{1} - \phi _{1}}{\beta_{1}} ]^{1 / ( 3 - D_{1} )}$.

• (III) θ−θ _t<ϕ ₂ and θ<ϕ ₂ (see Fig. 6.29)

  $$ \theta_{\mathrm{t}} = \beta_{2} \biggl[ \biggl( \frac{r}{r_{\mathrm{x}}} \biggr)^{3 - D_{2}} - \biggl( \frac{r_{\mathrm{c}} ( \theta )}{r_{\mathrm{x}}} \biggr)^{3 - D_{2}} \biggr] $$

  (6.59)

  Solution of Eq. (6.59) combined with Eq. (6.52) for r _c(θ) gives

  $$ r_{\mathrm{c}} ( \theta ) = r_{\mathrm{x}} \biggl[ \frac{\beta_{2} - \phi _{2} + \theta - \theta_{\mathrm{t}}}{\beta_{2}} \biggr]^{\frac{1}{3 - D_{2}}} $$

  (6.60)

6.2.2.2 Possibility 2: θ _t>ϕ ₁

In this case, the critical water content for percolation θ _t at saturation (θ=ϕ) is

$$ \theta_{\mathrm{t}} = \beta_{2} \biggl[ 1 - \biggl( \frac{r_{\mathrm{c}} ( \theta = \phi )}{r_{\mathrm{x}}} \biggr)^{3 - D_{2}} \biggr] + \phi _{1} $$

(6.61)

Rewriting Eq. (6.61) combined with Eq. (5.5) for r _c(θ=ϕ) gives

$$ r_{\mathrm{c}} ( \theta = \phi ) = r_{\mathrm{x}} \biggl[ \frac{\beta_{2} - \theta_{\mathrm{t}} + \phi _{1}}{\beta_{2}} \biggr]^{\frac{1}{3 - D_{2}}},\quad \theta_{\mathrm{t}} > \phi _{1} $$

(6.62)

As we show in Fig. 6.30, there are two cases possible for the unsaturated condition:

• (I) θ−θ _t<ϕ ₂ and θ>ϕ ₂ (see Fig. 6.30)

  $$ \theta_{\mathrm{t}} = \beta_{2} \biggl[ 1 - \biggl( \frac{r_{\mathrm{c}} ( \theta )}{r_{\mathrm{x}}} \biggr)^{3 - D_{2}} \biggr] + \theta - \phi _{2} $$

  (6.63)

  Solution of Eq. (6.63) for r _c(θ) gives

  $$ r_{\mathrm{c}} ( \theta ) = r_{\mathrm{x}} \biggl[ \frac{\beta_{2} - \phi _{2} + \theta - \theta_{\mathrm{t}}}{\beta_{2}} \biggr]^{\frac{1}{3 - D_{2}}} $$

  (6.64)

  Fig. 6.30

  

  Two separate distinct ranges of moisture contents that guide determination of r _c when θ _t is larger than the porosity associated with the upper fractal regime

• (II) θ−θ _t<ϕ ₂ and θ<ϕ ₂ (see Fig. 6.30)

  $$ \theta_{\mathrm{t}} = \beta_{2} \biggl[ \biggl( \frac{r}{r_{\mathrm{x}}} \biggr)^{3 - D_{2}} - \biggl( \frac{r_{\mathrm{c}} ( \theta )}{r_{\mathrm{x}}} \biggr)^{3 - D_{2}} \biggr] $$

  (6.65)

  Solution of Eq. (6.65) for r _c(θ) yields

  $$ r_{\mathrm{c}} ( \theta ) = r_{\mathrm{x}} \biggl[ \frac{\beta_{2} - \phi _{2} + \theta - \theta_{\mathrm{t}}}{\beta_{2}} \biggr]^{\frac{1}{3 -D_{2}}} $$

  (6.66)

  In fact, when θ _t>ϕ ₁, r _c(θ) follows the same function of water content (Eqs. (6.64) and (6.66)) for both θ>ϕ ₂ and θ<ϕ ₂.

To model unsaturated hydraulic conductivity, we invoke Poiseuille’s law for self-similar fractal porous media in which the hydraulic conductance g of a given pore is proportional to r ⁴ and the inverse of the pore length (l), also assumed to be proportional to r(l∝r). Therefore, g is a function of pore radius cubed, r ³ [23, 27]. Since the hydraulic conductivity K(θ) is directly proportional to the critical hydraulic conductance g _c, the unsaturated hydraulic conductivity normalized with the saturated hydraulic conductivity (as the reference point) is given by

$$ \frac{K ( \theta )}{K_{\mathrm{s}}} = \frac{g_{\mathrm{c}} ( \theta )}{g_{\mathrm{c}} ( \theta = \phi )} = \frac{r_{\mathrm{c}}^{3} ( \theta )}{r_{\mathrm{c}}^{3} ( \theta = \phi )} $$

(6.67)

Thus, the new piecewise unsaturated hydraulic conductivity model for the two possibilities outlines above would be

6.2.2.3 Possibility 1: θ _t<ϕ ₁

$$ \frac{K ( \theta )}{K_{\mathrm{s}}} = \left \{ \begin{array}{l} \bigl[\frac{\beta_{1} - \phi + \theta - \theta_{\mathrm{t}}}{\beta_{1} - \theta_{\mathrm{t}}} \bigr]^{3 / ( 3 - D_{1} )}, \quad \theta - \theta_{\mathrm{t}} > \phi _{2}, \theta > \phi _{2} \\ \bigl( \frac{\beta_{1}^{3 / ( 3 - D_{1} )}}{\beta_{2}^{3 / ( 3 - D_{2} )}} \bigr) \bigl( \frac{\beta_{1} - \phi _{1}}{\beta_{1}} \bigr)^{3 / ( 3 - D_{1} )}\frac{ [ \beta_{2} - \phi _{2} + \theta - \theta_{\mathrm{t}} ]}{ [ \beta_{1} - \theta_{\mathrm{t}} ]^{3 / ( 3 - D_{1} )}}^{3 / ( 3 - D_{2} )}, \\ \quad \theta - \theta_{\mathrm{t}} < \phi _{2},\theta > \phi _{2},\theta < \phi _{2} \end{array} \right . $$

(6.68)

6.2.2.4 Possibility 2: θ _t>ϕ ₁

$$ \frac{K ( \theta )}{K_{\mathrm{s}}} = \biggl[ \frac{\beta_{2} - \phi _{2} + \theta - \theta_{\mathrm{t}}}{\beta_{2} + \phi _{1} - \theta_{\mathrm{t}}} \biggr]^{3 / ( 3 - D_{2} )} $$

(6.69)

If one assumes that the percolation threshold is equal to 0, the unsaturated hydraulic conductivity model proposed above (Eqs. (6.68) and (6.69)) is simplified to

$$ \frac{K ( \theta )}{K_{\mathrm{s}}} = \left \{ \begin{array}{l@{\quad}l} \bigl[\frac{\beta_{1} - \phi + \theta}{\beta_{1}} \bigr]^{3 / ( 3 - D_{1} )}, & \theta > \phi _{2} \\ \bigl( \frac{\beta_{1}^{3 / ( 3 - D_{1} )}}{\beta_{2}^{3 / ( 3 - D_{2} )}} \bigr) \bigl( \frac{\beta_{1} - \phi _{1}}{\beta_{1}} \bigr)^{3 / ( 3 - D_{1} )}\frac{ [ \beta_{2} - \phi _{2} + \theta ]}{\beta_{1}^{3 / ( 3 - D_{1} )}}^{3 / ( 3 - D_{2} )}, & \theta < \phi _{2} \end{array} \right . $$

(6.70)

In all comparisons that we have made with experiment, we have found that θ _t=0 was appropriate, for reasons that we explain in the text. Nonetheless, in principle one should refer to Eqs. (6.68) and (6.69) for more general results. As long as both regimes are associated with textural pores, the simplest assumption should be used, namely that θ _t is independent of moisture content. In this case one would normally expect θ _t to be finite. But in the case that structural and textural pores are present, it must be considered possible that θ _t takes on a different value for the two types of pores. If θ _t is large, it is important that the results we obtain here would not extend to low saturations, where universal scaling of the hydraulic conductivity would be expected. But when the percolation threshold occurs at zero moisture content, the range of moisture contents at which universal scaling should be observed will typically be 0<θ<0.1, and is often even more restricted. In the case that θ _t=0 we could neglect this complication, since the experimenters never explored moisture contents sufficiently low that universal scaling of the hydraulic conductivity would be encountered.