Figure 1: Schematic of (a ) vacuum and surcharge combined preloading; (b ) surcharge preloading; and (c ) vacuum preloading (Mohamedelhassan and Shang 2002)

(Gibson, Schiffman et al. 1989) defined the excess pore water pressure in the consolidation process in two ways as: excess over the hydrostatic pressure (the pressure distribution when the pore water is stationary) and the pore pressure in excess of a steady-state flow condition. These definitions explain best the two distinct mechanisms which govern in a combined vacuum and surcharge preloading. Applying surcharge preloading would increase the excess pore pressure and because of the very low permeability of clays it can’t be dissipated which is under the first definition. On the other hand applying vacuum pressure through PVDs creates a water head between soil and PVD which accelerates the flow in clay soils and eases the discharge of water which is in accordance with the second definition. Although the effect of vacuum preloading is somehow the same as surcharge preloading in the acceleration of the consolidation process, they shouldn’t be mistaken with each other as they have two different mechanisms. The vacuum pressure effect is often demonstrated by negative pore pressure. Care should be taken to not mistake the negative algebraic term with its real mechanism as it is the pore pressure in excess of a steady-state flow condition in soil in the vicinity of PVDs under vacuum preloading. (Lu, Likos et al. 2021) has discussed in detail the inefficiency of common definition of pore pressure in soil mechanic and emphasized the necessity for developing better theories and seeking better engineering solutions for problems in geotechnical engineering.

To clarify the difference assume that at a given soil depth of z under combined vacuum and surcharge treatment system (+P) is the quantity of the excess pore pressure as a result of embankment surcharge and (-P) is the quantity of the excess pore pressure as a result of the vacuum pump. Assume the superposition law is valid and as result, there should be no settlement because of consolidation as the quantity of excess pore pressure is equal to zero i.e. (+P – P) but in contrast, the consolidation settlement would occur. Of course, this is not the case and it is an ideal situation in the real-world where considerable settlement takes place. It shows the superficial way of using superposition law. But now the question arises about if the absolute value of (│-P│) adopted in the analysis would be considering the case of (2P) a reasonable approach in dealing with such a situation. This example demonstrates the actual performance of two distinct mechanisms of surcharge and vacuum preloading. In reality, none of the stated situations would occur. As it would be seen the superposition law is not valid and moreover, another phenomena exists which is the interaction between PVDs and vacuum and surcharge or the hydro-mechanical coupling (in brief coupling). In coupled consolidation analysis, the excess pore pressure and deformations would be calculated simultaneously while considering compressibility of soil particles and pore fluid (Biot 1941) to observe the stated coupling effect where it can be increasing or decreasing based on different situations.

(Mohamedelhassan and Shang 2002) reported a minor disagreement between analytical solutions and consolidation results which might be attributed to laboratory deficiencies or the coupling effect of vacuum and surcharge. Refer to formulas (1), (2) ,and (3) and assuming constant permeability under various preloading (not a valid assumption) the simplified 1D excess pore pressure might be written as:

\(u\left(z\ ,\ t\ \right)=\ u_{v}\left(z\ ,\ t\ \right)+\ u_{s}\ \left(\ z\ ,\ t\ \right)+\ u_{\text{vs}}(z\ ,\ t)\)(4)

Where u_vs is the coefficient of consolidation for a combined surcharge and vacuum preloading that considers the hydro-mechanical coupling effect in analytical solutions.

(Bergado, Chai et al. 1998) has reported the monitoring results of two trial embankments in Second Bangkok International Airport and (Indraratna and Rujikiatkamjorn 2006) modeled these two embankments using FEM modeling in plane-strain condition. The second trial embankment is used for primary model verification. This case history was specifically selected because of variable vacuum pressure that was applied. The related data concerning the history of preloading and material properties can be accessed through these articles. GEOSTUDIO 2018 SIGMA/W coupled analysis in plane-strain condition was used for modeling. It should be noted that the effect of well resistance and clogging were considered in the model by boundary conditions and the smear effect was considered by the approach proposed by (Indraratna and Redana 2000). As stated by (Cai 2021) in order to consider nonlinearity of the consolidation arising from evolving permeability and compressibility of the soil due to change in void ratio during consolidation and non-Darcian flow regime for low permeability soil and large strain elasto-plastic behavior of the soil, a permeability modifier was applied in FEM analyses (geostudio 2018).

(a)