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Hydraulic conductivity in partially saturated soils

The rate of flow of water through a porous medium is regulated by the hydraulic conductivity or coefficient of permeability of the soil. The hydraulic conductivity is the primary soil property required when analyzing steady-state and transient (or unsteady-state) flow of an incompressible fluid through a porous medium.

For saturated soils, the hydraulic conductivity is generally assumed to be a constant. However, the hydraulic conductivity for an unsaturated soil can change by several orders of magnitude depending on the degree of saturation of the soil.

An experiment illustrating this behaviour was performed by Mualem (1976)2. The experiment involved flowing water through glass beads subjected to different negative pore-water pressures, i.e. different suctions. The glass beads were initially saturated and then subjected to a series of steps with decreasing water pressures, i.e. increasing suction. At the different steps, the hydraulic conductivities of the glass beads and the degree of saturation was measured starting at saturated conditions. The results of this experiment are presented in Figure 1.

Figure 1. Flow of water through glass beads. Left: Saturation-suction relation, Right: Evolution of hydraulic conductivity with changing suction. Experimental data from Mualem (1977)2.

The results showed that the hydraulic conductivity of the glass beads started to decrease when the suction was approx. 3 kPa. A further increase in suction caused the hydraulic conductivity to rapidly decrease by several orders of magnitude from \(k^w\approx 2\cdot 10^{-2}\) m/s to \(k^w\approx 10^{-5}\) m/s.

The experiments of Mualem (1976) also show that the hydraulic conductivity is different for wetting and drying processes and that similar hysteretic effects are observed similar to the ones of the saturation-surction relation.

The reason for the decrease of the hydraulic conductivity is that water can only flow through that portion of a porous medium that consists of water. As the amount of water (i.e. the degree of saturation) in a soil decreases, the hydraulic conductivity decreases as well because there is less cross-sectional area through which water can flow. However, there is not a one-to-one relationship between the amount of water in the soil and the hydraulic conductivity. The hydraulic conductivity decreases at a much faster rate than the degree of saturation. The reason for this non-linear relation is that a reduction in the amount of water in the soil also increases the tortuosity of the flow path. As a consequence, an arithmetic decrease in the degree of saturation generally results in a logarithmic decrease in the hydraulic conductivity. A dry soil has a much lower hydraulic conductivity than a wet soil.

Mathematical description

Different methods for the estimation of the partially saturated hydraulic conductivity exist. Most methods rely on the concept of "relative permeability" where the hydraulic conductivity for various degrees of saturation \(k^{ps}\) is expressed as the product of the saturated hydraulic conductivity \(k^s\) and a relative permeability \(k^{rw}\):

\[ k^{ps} = k^{rw}k^s \]

where \(k^{rw}\) is most often assumed to be a function of the degree of saturation \(S\) and thus \(k(S)\). \(k^s\) is often assumed to be constant or depending on the porosity \(n\). Compared to the relative permeability \(k^{rw}\), the saturated hydraulic conductivity \(k^s\) is easier to measure. For the estimation of \(k^{rw}\) one often relies on empirical relations, two of which are discussed in more detail in the following subsections.

Cozeny/Carman

The famous Cozeny/carman model linking the saturated hydraulic conductivity \(k^s\) to the porosity \(n\) reads

\[ k^s = \dfrac{1}{C} \dfrac{n^3}{(1-n)^2} \dfrac{\gamma_w}{\mu_w} d_e^2 \]

Therein, \(C\) is a fitting parameter usually in the range of \(180-270\). \(\gamma_w\), \(\mu_w\) and \(d_e\) are the unit weight of water, the dynamic viscosity of water and the effective grain diameter, respectively. In a partially saturated soil, the pore space \(n\) is only partly occupied by water. The pore space that is occupied by water can be calculated as \(n_w = Sn\). In the saturated case (\(S=1\)) \(n_w=n\) and in the dry case (\(S=0\)) \(n_w=0\) hold. Considering \(n_w\) instead of \(n\) allows to apply the Cozeny/Carman relation to unsaturated soils:

\[ k^{ps} = \dfrac{1}{C} \dfrac{n_w^3}{(1-n_w)^2} \dfrac{\gamma_w}{\mu_w} d_e^2 \]

With the definition of the relative permeability \(k^{rw}=k^{ps}/k^s\) we obtain:

\[ k^{rw} = \dfrac{k^{ps}}{k^s} = \dfrac{n_w^3(1-n)^2}{n^3(1-n_w)^2} = \dfrac{(Sn)^3(1-n)^2}{n^3(1-Sn)^2} \]

Van Genuchten - Mualem

... to come ...

References

  1. Kaye, G. W. C., & Laby, T. H. (1928). Tables of physical and chemical constants and some mathematical functions. Longmans, Green and Company Limited. ↩

  2. Y. Mualem, ‘A new model for predicting the hydraulic conductivity of unsaturated porous media’, Water Resources Research, vol. 12, no. 3, pp. 513–522, Jun. 1976, doi: 10.1029/wr012i003p00513. ↩

  3. M. Th. van Genuchten, ‘A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils’, Soil Science Society of America Journal, vol. 44, no. 5, pp. 892–898, Sep. 1980, doi: 10.2136/sssaj1980.03615995004400050002x. ↩