The penetration grouting is very widely used in geotechnical engineering nowadays, but the slurry diffusion radius is not long enough because of low grouting pressure. The vacuum grouting method is proposed to solve this problem. However, there is no diffusion theory model of vacuum grouting, which makes the practical application lack scientific basis. In this paper, the distribution law of vacuum negative pressure in soil is deduced. Then, the boundary conditions of Maag’s spherical diffusion model are modified by the vacuum pressure distribution law. After that, the vacuum modification model is deduced. Finally, Maag’s model and modified model are analyzed according to a published experiment, which proves that the vacuum modification model is suitable for predicting the slurry diffusion of vacuum grouting. The proposed model provides a reference for the theoretical study of vacuum grouting.
The sand layer which has the characteristics of loose structure, no cohesion, and poor selfstability is often encountered in geotechnical engineering. The sandy layer can cause severe damage to underground projects [
Therefore, researchers have been studying the mechanism of grouting pressure of slurry diffusion and analyzed the mechanical equilibrium mechanism of diffusion termination [
Researchers in recent years have analyzed a lot about the grouting combined with vacuum in concrete structure field. Assaad and Daou [
Therefore, this paper takes Newtonian fluid as the research object to deduce the diffusion model of vacuum penetration grouting on the basis of Maag’s spherical diffusion model. The new diffusion model explains how the diffusion distance be enlarged in mathematical way and predicts the diffusion distance of the slurry by analysis formula by considering grouting coefficients. The vacuum modification model provides a reference to the scientific application of vacuum penetration grouting.
Maag derived the diffusion formula of the grout in sand layer and proposed a spherical diffusion model of Newtonian fluid [
Maag’s spherical diffusion model is shown in Figure
Schematic diagram of derivation of Maag’s model.
According to Darcy’s law:
According to the boundary conditions:
Equation (
Considering
In order to study the influence of vacuum on the diffusion radius, the distribution of vacuum pressure in soil is analyzed firstly. The plane seepage model is used to derive the distribution formula of vacuum pressure. It is assumed that the seepage of air in soil is steady and follows Darcy’s law. The vacuum pressure is converted into the corresponding water head through the derivation of the model, which can be coupled with the grouting pressure head. Finally, the new boundary conditions are submitted into Maag’s spherical diffusion model and the expression of the diffusion radius of the slurry in vacuum condition is derived.
Figure
Soil particle and pore model.
The soil porosity ratio
It can be found that
The velocity of the air passing through a certain cross section of soil can be expressed as
According to Darcy’s law:
Suppose
Then the following result is obtained:
The boundary conditions are assumed as
In the conditions, the pressure
Finally, the distribution of the pressure
It is known that the 101 kPa can be converted into the water head of about 10.3 m, so the pressure difference
The vacuum distribution is shown in Figure
Schematic diagram of vacuum pressure distribution.
Because the seepage radius of slurry is comparatively much smaller than the range of vacuum pressure distribution, it is assumed that the vacuum pressure on the left and right side of the distance of
Assume
Then Maag’s spherical diffusion radius can be modified based on the boundary conditions as
The new boundary conditions are submitted into Darcy’s law and the equations are integrated to obtain the diffusion radius under the vacuum effect:
It can be seen from (
According to the analysis above, the final diffusion radius of vacuum modification model will be influenced by many factors, like vacuum degree, soil conditions, and so on. In this paper, the vacuum modification formula is verified by the published vacuum grouting experiment by Shen [
The vacuum grouting experiment system is presented in Figure
The experimental system [
The medium sand, fine sand, and silty sand were used as the injected stratum. The physical properties of experimental sand soil are shown in Table
Physical properties of sand samples.
Sand soil 






Medium  1.61  1.45 × 10^{−2}  37.8  10.7  2.59 
Fine  1.71  5.59 × 10^{−3}  34.5  12.5  2.61 
Silty  1.79  9.75 × 10^{−4}  31.0  15.3  2.60 
Where
Acid sodium silicate solution was used in the experiment. The sodium silicate solution is 30° Be’ and the concentration of dilute sulphuric acid solution is 10%. The volume ratio of sodium silicate solution to sulphuric acid is 0.7. The gel time of the slurry is about 350 s. The slurry is Newtonian fluid and the initial viscosity is 5 mPa·s.
In the experiment,
Diffusion radii of Maag’s spherical model and vacuum modification model.

Sand soil 





60  Medium  251.5  136  85.9  50.1 
Fine  134.0  91  64.4  26.6  
Silty  60.6  48  36  12  


40  Medium  138.5  122  85.9  36.1 
Fine  82.7  86  64.4  21.6  
Silty  38.5  46  36  10  


20  Medium  58.7  110  85.9  24.1 
Finer  37.0  79  64.4  14.6  
Silty  18.3  44  36  8 
Where,
The comparison of diffusion radii in Maag’s and vacuum modification models in vacuum condition is shown in Figures
Diffusion radii under vacuum degree of 60 kPa.
Diffusion radii under vacuum degree of 40 kPa.
Diffusion radii under vacuum degree of 20 kPa.
It is revealed from Figures
The radii differences under different vacuum degrees are shown in Figure
Radii differences under different vacuum degrees.
Figure
In Shen’s [
Theoretical and experimental values of the diffusion radius under different vacuum degrees and soil conditions.
Sand stratum 





Medium  60  145  136  −6.2 
40  90  122  35.5  
20  70  110  57.1  


Fine  60  110  91  −17.2 
40  80  86  7.5  
20  55  79  43.6  


Silty  60  75  48  −36 
40  60  46  −23.3  
20  45  44  −2.2 
Where,
The variation of experimental values and theoretical values is shown in Figures
The variation of theoretical and experimental values in medium sand layer.
The variation of theoretical and experimental values in fine sand layer.
The variation of theoretical and experimental values in silt layer.
It can be obtained from Figures
(1) Vacuum grouting method can enlarge the grout diffusion distance compared to the traditional method without vacuum pressure.
(2) It can be concluded from the theoretical analysis and experiment that the modified Maag’s spherical diffusion model is more reasonable to predict the diffusion of grout than the traditional Maag’s model. At the same time, it is shown in the vacuum modification model that the higher the absolute value of vacuum pressure, the larger the diffusion radius.
The authors declare that they have no conflicts of interest.
This study is supported by “The Fundamental Research Funds for the Central Universities of China” (no. 292015082).