# Water Retention Curve of Biocemented Sands Using MIP Results
## 2 Model Proposed  
The water retention curve can be derived from pore-size-distribution data. The model proposed by [PERSON] et al. [15] for clayey soils was adopted as a starting point; however, it was adapted to discard the adsorbed water at the level of the microstructure and its volume change, and, instead, it considered the deformation of the soil skeleton resulting from the intrusion process [35]. In this study, it was assumed that, although the intruded mercury would fill both the void's volume of the sample (inner volume) and the space between the sample and the sample holder (outer volume), there would be differences because some air can be trapped in the sample voids, and some of those cannot be accessed due to soil structure. Because it is difficult to characterize this differential, it is assumed that the exterior mercury was applying pressure in a material with some compressible air trapped in closed and inaccessible pores, and that the material behaves elastically under isotropic compression. Only the drying path of the WRC was simulated, because the intrusion of non-wetting mercury in MIP tests is equivalent to air entry during drying.  
The suction, \(s\), can be computed directly from mercury pressure, \(p_{Hg}\), using the Laplace equation (Equation (3)):  
\[s=\frac{T_{w}cos\theta_{w}}{T_{Hg}cos\theta_{Hg}}p_{Hg} \tag{3}\]  
where \(T_{w}\) and \(T_{Hg}\) are the surface tensions of water and mercury, respectively (\(T_{w}\) = 0.073 N/m and \(T_{Hg}\) = 0.489 N/m); and \(\theta_{w}\) and \(\theta_{Hg}\) are the contact angles of water and mercury, respectively (\(\theta_{w}\) = 0\({}^{\circ}\) and \(\theta_{Hg}\) = 120\({}^{\circ}\)).  
As defined by [PERSON] et al. [15], the degree of saturation, \(Sr\), can be computed by using Equation (4), and the water content can be computed by using Equation (5):  
\[S_{r}=(1-S_{rm})+\frac{w_{res}}{w_{sat}}S_{rm} \tag{4}\]  
\[w=(1-S_{rm})(w_{sat}-w_{res})+w_{res} \tag{5}\]  
In both equations, \(S_{rm}\) is the degree of saturation computed by considering the volume of mercury intruded until that step in relation to the maximum volume of intruded mercury (Equation (6)), and \(w_{res}\) represents the voids that cannot be filled with mercury (Equation (7)). Both parameters are determined by knowing the \(e_{MIP}\), the void ratio filled with mercury until the current stage (volume of mercury intruded divided by the volume of solids of the sample), and \(e_{MIP,max}\), the maximum void ratio that was filled in the intrusion phase (maximum volume of mercury intruded divided by the volume of solids of the sample). Finally, \(w_{sat}\) is the water content of the soil when fully saturated. The void ratio, \(e\), at preparation is determined as usual (Equation (8), where \(G_{s}\) is the density of the solid particles):  
\[S_{rm}=\frac{e_{MIP}}{e_{MIP,max}}=\frac{V_{intruded}}{V_{max,intrud}} \tag{6}\]  
\[w_{res}=\frac{e-e_{MIP,max}}{G_{s}} \tag{7}\]  
\[w_{sat}=\frac{e}{G_{s}} \tag{8}\]  
Parameter \(w_{res}\) in the model proposed here is updated during the intrusion process to consider changes in the void ratio, \(\Delta e\) (Equation (9)), of the soil when compressed inside the intrusion penetrometer. These changes are assumed to be isotropic and elastic during intrusion, and therefore the elastic compressibility index, \(\kappa\), is adopted to compute the reduction in void ratio during compression caused by increasing mercury pressure (Equation (10)). From a physical point of view, it must be ensured that the void ratio computed while considering volume changes must be larger than \(e_{MIP,max}\):  
\[w_{res}=\frac{e-\Delta e-e_{MIP,max}}{G_{s}} \tag{9}\]  
\[\Delta e=-\kappa\Delta lnp_{Hg} \tag{10}\]