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Saturation-suction relation

At this point it should be clear that the suction \(s=p^a-p^w\) and degree of saturation \(S\) play an important role in describing the behaviour of partially saturated soils. The variation of saturation with respect to suction is defined by soil–water retention curve (SWRC). It is commonly presented in a graph of either gravimetric water content (\(w\)), volumetric water content (\(\theta_w\)) or degree of saturation (\(S\)) in the vertical axis against matric suction \(s\) in the horizontal axis in a logarithmic scale. Different methods exist to determine the SWRC for soils. In most methods, the suction in a soil sample is controlled/changed and the degree of saturation is measured. The methods differ in particular in the maximum suction that can be applied. A typical SWRC for granular soils is represented in Figure 1.

Figure 1. Different stages of a typical Soil-Water Retention Curve (SWRC).

The SWRC in Figure 1 was obtained by draining water from an initially saturated soil sample (\(S=1\) and \(s=0\) kPa) by increasing the suction.

  • During the first stage of the experiment, no change in saturation is observed although the suction was increased (\(s>0\) kPa).
  • After further increasing the suction, water starts to drain from the soil sample and the degree od saturation decreases. The suction at which the water starts to drain out from the biggest soil pores is referred to as the air-entry value (AEV). The AEV has been found to depend on the grain size distribution of the soil. A larger proportion of fine particles implies smaller intra-particle pore spaces between soil particles, resulting in a higher AEV.
  • With increasing suction, more water drains from the soil sample and the degree of saturation decreases (Phase 3). During drainage, the large pore channels drain first and, as the suction continues to increase, the pore channels with a narrower diameter drain as well - analogous to the observations in the Jamin-tube in section Capillary rise.
  • At some point (Phase 4), further increase in suction does not result in any significant change in saturation. This point on the SWRC is referred to as the residual degree of saturation \(S^{res}\) or residual suction.
  • The last slide of Figure 1 shows the observed behaviour for a wetting of the previously drained soil sample. As can be seen from the illustration a hysteresis exists between the drainage and the wetting curve. At the same suction, the soil contains more water during drainage than during wetting. The reason for this is that drainage is determined by the small pores, as these can retain the water at the passive capillary rise height. Wetting (rise of the water to the active capillary rise height), on the other hand, is determined by the large pores, as these cannot be overcome. See the observations for the Jamin-tube in section Capillary rise.

Measuring the SWRC in the laboratory

... to come ...

Mathematical description

Many different mathematical models exist linking the degree of saturation \(S\) to the suction \(s\). Alternatively, these models are sometimes formulated in terms of volumetric water content \(\theta_w\) instead of the degree of saturation \(S\). A widely used model is the one proposed by Van Genuchten3:

\[ S = S^{res} + (S^{sat}-S^{res}) \left( \dfrac{1}{1+(\alpha s)^n} \right)^{m} ~~~\text{with}~~~ m=1-\dfrac{1}{n} \]

Therein, \(S^{sat}\) and \(S^{res}\) are the degree of saturation after a wetting process at zero suction and the residual degree of saturation at very high suctions, respectively. \(\alpha\), \(n\) and \(m\) are parameters controlling the shape of the SWRC. \(m=1-(1/n)\) is often assumed. \(\alpha\) controls the AEV, the smaller the value of \(\alpha\) the larger the AEV is. The parameter \(n\) controls the steepness of the transition zone. The influence of the models parameter are illustrated in Figure 2.

Figure 2. Influence of the parameters of the van Genuchten model on the shape of the SWRC.

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. ↩