# Modelling of Permeation Grouting considering Grout Self-Gravity Effect: Theoretical and Experimental Study

Authors: Yanbin Fu; Xiuling Wang; Sizhan Zhang; Yong Yang
Journal: Advances in Materials Science and Engineering
 (2019)

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## Abstract

Self-gravity is one key parameter for behavior characterization of grout permeation and diffusion. This study proposes mathematical models for permeation grouting with consideration of grout self-gravity effect. The models concerning power law, Bingham, and Newtonian grouts are based on the generalized Darcy’s law and spherical diffusion theory. In addition, a prediction model of grout concretion dimension used for Bingham grout was developed. An analysis of the injection pressure distribution law and a comparative evaluation of diffusion radius considering self-gravity effect using established models were conducted subsequently. Moreover, grouting experiments were performed to check and verify the prediction model. The experimental results showed that injection pressure decreases linearly with increase of diffusion radius for the power-law grout, while nonlinear decrease of injection pressure was confirmed in Bingham and Newtonian grouts in this case. Three grouts approximately diffuse in an “ellipsoidal” shape, and it is confirmed that the diffusion radius is closely related to grout self-gravity. The Newtonian grout produces the maximum diffusion radius compared with the other two grouts whether the gravity effects were considered or not. The grout quantity under a smaller water-to-cementw/c ratio exhibits a significant difference and undergoes two increasing stages, whereas the quantity simply tends to be stable after it reaches its maximum in terms of the larger w/c ratios. The constructed dimension prediction model agrees well with the experimental results, which can be helpful for design and assessment of the grouting scheme.


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## Body

## 1. Introduction

Grouting technique has been extensively used in soil reinforcement, water plugging, and deviation rectification of buildings as well as other engineering practices [1–6]. Based on grout diffusion behaviors and mechanism, three methods are commonly used for grouting activities, compaction grouting [7], fracture grouting [8–11], and permeation grouting [12]. Though multiple forms of grout diffusion occur simultaneously in injected soil, one of them often plays a leading role under certain conditions. For example, materials with large porosity in the sandy layer produce the larger internal pore passages that engender less inhibitory effect during grout flow, which is favorable for effective implementation of the permeation grouting technique [13, 14]. In terms of the utilization of the grout materials, the cement slurry now is most commonly used for grouting practices as a result of low cost compared with other grout materials. In reality, the variations of injected soils and grout materials as well as the uncertainty of external environment always yield challenges for estimation works on grout permeation and diffusion.To promote grouting technique in civil engineering, many researchers have released the studies on grouting materials [15, 16], grouting performance [17–19], and grouting device [20]. Mollamahmutoglu and Yilmaz [21] firstly evaluated some basic properties of the superfine cement suspensions such as stability, viscosity, and setting time, and then permeability behaviors under different grouting pressures of the studied suspensions injected into sand specimens with different gradations were examined. Markou and Droudakis [22] evaluated some relevant parameters influencing grout effectiveness by an experimental investigation. The results confirmed that several parameters such as w/c ratio, bleed capacity of suspensions, and permeability coefficient of used sands actually control the grouted sand properties so as to determine grout quality. Huang et al. [23] developed a novel method of vacuum grouting to overcome shortcomings such as insufficient diffusion radius due to low grouting pressure. Then, a vacuum modification model is proposed after conducting a modification for boundary conditions of Maag’s spherical diffusion model. Kim et al. [24] put forward a method and corresponding equipment with oscillatory injection so as to improve depth of the grout permeation. In this study, the clogging theory was utilized to evaluate enhancement effect of the groutability, and the results indicated that usage of the developed equipment increased grout volume and permeation depth by more than 30% and 25%, respectively.A series of studies are concentrated on modelling for grouting behavior, and three grout constitutive equations (i.e., Newtonian fluid, Bingham fluid, and power-law fluid) were most commonly used for grouting theory analysis [25, 26]. Yoon and El Mohtar [27] proposed a filtration model used to estimate maximum permeation distance in bentonite grouting, and the equation based on yield stress can effectively obtain the maximum distance at a high yield stress. Maghous et al. [28] established a macroscopic model where cement grout flows in the porous medium. The model is presented on the basis of the mass balance equations and filtration law, which can be utilized to investigate the behaviors of the radial flow filtration. In view of the impacts of variations of grout viscosity on grouting evaluation, Zhang et al. [29] developed a step-wise calculation method considering uneven distribution of grout viscosity, in which circular grouting area was divided into minimal annular units. They further studied the distribution law of grout viscosity and pressure in the grouting zone. Some necessary simplifications and assumptions have to be made for compromising with difficulties on accurate definition of complicated process of slurry diffusion [30, 31], resulting in considerable deviation in modelling of actual practices. Though a great deal of efforts from scholars and researchers have been made in model modification [32–34], few investigations have been made to model the behaviors of the permeation grouting by taking into account the self-gravity effect of the grouts. The grout diffusion in rock or soil is generally affected by injection pressure, physical properties of grout (e.g., grout self-gravity), and injected medium. The grout gravity is an important factor for mechanical analysis on grouting engineering, as it plays an increasingly dominant role with the attenuation of injection pressure. Taking into account the self-gravity effect in modelling can provide a more accurate prediction on slurry diffusion. The effect is more significant when grout diffuses at a vertical direction. This consideration may also eliminate uncertainty in practices [35, 36]. For example, the slurry gravity in backfill grouting of shield segments leads to a larger pressure at the top grouting hole than that at the bottom hole [37, 38]. Therefore, it is urgent to focus on this issue, which could provide more useful information to relevant research and engineering practice.In this paper, the generalized Darcy’s law has been utilized to derive grout diffusion models considering impacts of the self-gravity on permeation behaviors for power law, Bingham, and Newtonian grouts and followed by a dimension prediction model of grout concretion. To validate the proposed dimension prediction model, novel experimental equipment including the model box system, the grouting system, the pressure system, and the measurement system has been developed. The proposed model is well checked by the grouting experiments that allow grout flowing through a sand medium as the modeled and measured results are in good agreement with each other.


## 2. Mathematical Modelling for Slurry Permeation and Diffusion

### 2.1. General and Basic Assumptions

The grout diffusion in the sandy layer is considered as permeation predominates. As shown in Figure1, the point-source-based grout generally diffuses in an approximate spherical shape [39]. The grout flows through the grouting pipe to injected soil, and then diffuses around from bottom of the grouting pipe through pore passages of the soils, along with attenuation of grouting pressure. The grout actually diffuses in a curving path as a result of inhomogeneity of soil pores. However, assuming grout diffuses in a straight path is always required for convenient modelling and analysis.Figure 1
Diagram of the (a) grout permeation diffusion and (b) direction indication of grout concretion.
(a)
(b)The flowing velocity is regarded as a key parameter to judge grout states; however, it is difficult to be determined. Some previous findings suggested that grout permeates in the laminar flow state when it is injected into the ground. Regarding the spherical diffusion model, the pores of soil layers can be considered as radial capillary tubes from center of sphere outward [35, 40, 41]. The size of capillary channels in the sandy layer varies significantly, and it is extremely difficult to reveal the real state through a mathematical model. For convenience, we assume the capillary channels into the tubes with the same radius in modelling. The Dupuit–Forchheimer relation was introduced to transform the permeation model in single tubes into the macroscopic state, and then the generalized Darcy’s law (applicable to laminar flow) is applied to derive the mathematical model on the basis of the capillary group theory [42]. Therefore, consider a laminar incompressible grout of finite length dl and radius r in a circular tube of radius r0 (Figure 2). Further, we assume that microelement shares the same axis with the tube, and α is the angle between the grout flow direction and the x-axis, which varies from 0 to 2π anticlockwise.Figure 2
Geometry of the physical model. (a) Power-law grout and (b) Bingham grout.
(a)
(b)Before formulation of the model, the following basic assumptions are introduced: (1) the sand and soil that grout being injected into are isotropic and homogeneous; (2) the grout is in power law, Bingham, and Newtonian fluids, respectively; (3) the grout is grouted into soil at the bottom of grouting pipes; and (4) the grout diffuses in a spherical shape.


### 2.2. Mathematical Model Development

#### 2.2.1. Power-Law Grout

The mechanical equilibrium of the grout microelement for different angles ofα is written mathematically as(1)pπr2−p+dpπr2−2πrτdl−ρgsinαπr2dl=0,where p is the injection pressure; τ is the shear stress on surface of the grout column; ρ is the grout density; and g is the gravitational acceleration.Equation (1) is simplified into the following equation:(2)τ=−r2dpdl+ρgsinα.The rheological equation for power-law fluid is defined as follows [43]:(3)τ=Cγn=C−dvdrn,where C is the viscosity coefficient; γ is the shear rate; and n is the rheological index.Together with equations (2) and (3), it yields(4)dvdr=−τC1/n=−−12Cdpdl+ρgsinα1/nr1/n.Then, the flow velocityv at any position of the grout fluid yields(5)v=−12Cdpdl+ρgsinα1/nnn+1r0n+1/n−rn+1/n.Thus, the grout quantityq per unit time is given as(6)q=∫0r02πrvdr=nπ3n+1−12Cdpdl+ρgsinα1/nr03n+1/n.The average flow velocityv¯ can be given as(7)v¯=qπr02=n3n+1−12Cdpdl+ρgsinα1/nr0n+1/n.Solving the Dupuit–Forchheimer equation that defining flow velocityV=ϕv¯ [43], it yields(8)V=nϕ3n+1−12Cdpdl+ρgsinα1/nr0n+1/n,where ϕ is the porosity of porous media.Here, introducing the effective permeabilityKe and effective viscosity μe, which are defined as [43](9)Ke=ϕr022n3n+1,μe=C3n+1ϕr0nn−1,and equation (8) is rearranged into(10)V=Keμe1/n−dpdl+ρgsinα1/n.For a representative sphere case, its total surface area isA=4πl2. Thus, the total grout quantity Q is given as(11)Q=VAt,where t is the grouting time.Equation (11) is rearranged into(12)V=Q4πtl2=Keμe1/n−dpdl+ρgsinα1/n.Separating variables from equation (12) yields(13)p=−Q4πtnμeKe1−2nl1−2n−ρgsinαl+B,with the boundary conditions p=p0 and l = l0 and yields(14)p0=−Q4πtnμeKe1−2nl1−2n−ρgsinαl0+B,where p0 is the injection pressure and l0 is the radius of the grouting pipe.Solving equation (14) yields(15)B=p0+Q4πtnμeKe1−2nl1−2n+ρgsinαl0.Together with equations (13) and (15), it yields(16)p=p0+Q4πtnμeKe1−2nl01−2n−l1−2n+ρgsinαl0−l.Thus, the total grout quantity can be expressed as(17)Q=43πl3ϕ.Together with equations (16) and (17), it leads to(18)p=p0+ϕ3tnμeKe1−2nl01−2nl3n−l1+n+ρgsinαl0−l.


#### 2.2.2. Bingham Grout

In this case, substituting equation (2) into the rheological equation for Bingham grout that is defined as τ=τ0+μ−dv/dr [43] yields(19)dvdr=τ0−τμ=1μτ0+r2dpdl+ρgsinα,where τ0 is the yield stress.When−r/2dp/dl+ρgsinα≤τ0, the grout is not subject to shear action and there exists a flow core with a radius of rp in the tube; when 0≤r≤rp, no relative motion exists in the grout fluid, and when rp<r≤r0, the relative motion in the fluid produce shear action. Thus, the radius of flow core can be given as(20)rp=−2τ0dp/dl+ρgsinα.Integrating equation (19) yields(21)v=−1μ14dpdl+ρgsinαr02−r2+τ0r0−r,rp≤r≤r0,v=−1μ14dpdl+ρgsinαr02−rp2+τ0r0−rp,0≤r≤rp.The grout quantity per unit time in the circular tube is given by(22)q=∫rpr02πrvdr+πrp2vp.Equations (21) and (22) lead to the following equation:(23)q=−πr048μdpdl+ρgsinα1−43−2τ0/r0dp/dl+ρgsinα+13−2τ0/r0dp/dl+ρgsinα4.The average velocity of the grout in the tube can be expressed as follows:(24)v¯=qπr02=dpdl+ρgsinα1−43−2τ0/r0dp/dl+ρgsinα+13−2τ0/r0dp/dl+ρgsinα4.When there is no grout flow in the tube, it leads to(25)−2τ0/r0dp/dl+ρgsinα=1.Thus, the starting pressure gradientλ is(26)λ=−dpdl=2τ0r0+ρgsinα.Here, introducing the variableK is defined as(27)K=ϕr028.Thus, the grout flow velocityV is(28)V=−Kμdpdl+ρgsinα1−43ρgsinα−λdp/dl+ρgsinα+13ρgsinα−λdp/dl+ρgsinα4.Together with equations (11) and (28), it yields(29)V=Q4πl2t=−Kμdpdl+ρgsinα1−43ρgsinα−λdp/dl+ρgsinα+13ρgsinα−λdp/dl+ρgsinα4.Considering the pressure gradient−dp/dl≫λ, equation (29) is simplified as(30)V=Q4πl2t=−Kμdpdl+ρgsinα1−43ρgsinα−λdp/dl+ρgsinα.Separating variables and integrating equation (30) yield(31)p=Qμ4πtKlt+13ρgsinα−43λl+C,with the boundary conditions p=p0 and l = l0, and equation (31) leads to(32)p0=Qμ4πtKl0t+13ρgsinα−43λl0+C,(33)C=p0−Qμ4πtKl0t+13ρgsinα−43λl0.Together with equations (17), (31), and (33), the injection pressure p at any position is given as(34)p=p0−ϕμ3tKl0l3+ϕμ3tKl2+43λ−13ρgsinαl0−l.


#### 2.2.3. Newtonian Grout

Considering the constitutive equation, the Newtonian grout can be regarded as a special form of Bingham grout whenτ0 is zero, so the starting pressure gradient λ is given as(35)λ=ρgsinα.Then, the injection pressure of Newtonian grout is(36)p=p0−ϕμ3tKl0l3+ϕμ3tKl2+ρgsinαl0−l.


### 2.3. Prediction Model of Grout Concretion Dimension

Considering the Bingham and Newtonian grouts are the most commonly used types for grouting practices, thereby the Bingham grout was chosen to establish the prediction model for grout concretion dimension in this section.Together with equation (34) and boundary conditions p=0 and l = lmax, which yield(37)p0−ϕμ3tKl0lmax3+ϕμ3tKlmax3+43λ−13ρgsinαl0−lmax=0,where lmax is the maximum diffusion radius.Whenα is 0, π/2, and 3π/2, respectively, the diffusion radius is lmax,0, lmax,π/2, and lmax,3π/2. Thus, the following equation can be obtained:(38a)p0−ϕμ3tKl0lmax,03+ϕμ3tKlmax,03+43λl0−lmax,0=0,(38b)p0−ϕμ3tKl0lmax,π/23+ϕμ3tKlmax,π/23+43λ−13ρgl0−lmax,π/2=0,(38c)p0−ϕμ3tKl0lmax,3π/23+ϕμ3tKlmax,3π/23+43λ+13ρgl0−lmax,3π/2=0,where lmax,0, lmax,π/2, and lmax,3π/2 is the maximum diffusion radius at horizontal direction, vertical upward direction, and vertical downward direction, respectively.l
max
,
0, lmax,π/2, and lmax,3π/2 can be obtained by solving equations (38a), (38b), and (38c), and then the grout concretion dimension at different directions are(39a)Dx=Dy=2lmax,0,(39b)Dz=lmax,π/2+lmax,3π/2,where Dx, Dy, and Dz indicated in Figure 1 are the dimensions of grout concretion at transverse direction, longitudinal direction, and vertical direction, respectively.


### 2.4. Analysis of the Injection Pressure Distribution Law

Some findings of the previous study by Ruan [44] have been utilized to explore the spatial distribution law of injection pressure with diffusion radius variations. In this study, the cement grout was identified as three flow patterns, i.e., power law (w/c ratio = 0.5–0.7), Bingham (w/c ratio = 0.8–1.0), and Newtonian grouts (w/c = 2.0–10.0) based on the variations of the w/c ratios. The cement grout with the w/c ratios of 0.5, 1.0, and 2.0 together with the aforementioned established models have been used to discuss the relationship between diffusion radius at different directions under different injection pressures. The grout parameters are listed in Table 1, and Figure 3 presents the experimental results.Table 1
Parameters of three grouts used for the study.
Grout type
μ
e (Pa·s)
τ
0 (Pa)
ρ (g/cm3)
K
e (cm/s)
n
l
0 (cm)
t (s)
p (kPa)
Power law
—
—
1.86
0.03
0.4
2.5
600
200
Bingham
0.0096
1.563
1.49
0.03
0.4
2.5
600
200
Newtonian
0.0047
0
1.30
0.03
0.4
2.5
600
200
Note. n is porosity.Figure 3
Injection pressure distribution of grouts at different directions. (a) Power-law grout, (b) Bingham grout, and (c) Newtonian grout.
(a)
(b)
(c)Figure3(a) indicates that injection pressure decreases linearly with increase of diffusion radius. Also, attenuation rates of injection pressure at different directions vary marginally in terms of a smaller rheological coefficient. From Figures 3(b) and 3(c), variations of injection pressure along with diffusion radius of Bingham and Newtonian grouts significantly differ from the power-law grout. Nonlinear decreases of injection pressure are found in both two grouts with the increase of diffusion radius. Particularly, compared with the Bingham grout, marginal attenuation in injection pressure was observed within 10 cm of the radius for Newtonian grout, while the slight difference emerges when radius is beyond 10 cm. The injection pressure distributes diminishingly at directions of 3π/2, 0, and π/2 in terms of the same diffusion radius.


### 2.5. Comparison of Diffusion Radius with and without considering Self-Gravity Effect

The proposed models were further used to discuss diffusion radius in aspects of considering and not considering self-gravity effects of the three grouts, and results are shown in Figure4.Figure 4
Diffusion radius of different cement grouts. (a) Power-law grout, (b) Bingham grout, and (c) Newtonian grout.
(a)
(b)
(c)It can be seen from Figure4 that three grouts diffuse approximately in an “ellipsoidal” shape. Though values of the diffusion radius considering gravity effects in horizontal direction exhibit little or even no differences compared with not considering gravity effects, the remarkable differences of the diffusion radius in vertical direction have proved the necessity to take into account the grout gravity effects. The emergence of gravity leads to the decrease of the diffusion radius in the upper part while that increase in the lower part. The Newtonian grout produces the maximum diffusion radius compared with the other two grouts whether considering the gravity effects or not.


## 2.1. General and Basic Assumptions

The grout diffusion in the sandy layer is considered as permeation predominates. As shown in Figure1, the point-source-based grout generally diffuses in an approximate spherical shape [39]. The grout flows through the grouting pipe to injected soil, and then diffuses around from bottom of the grouting pipe through pore passages of the soils, along with attenuation of grouting pressure. The grout actually diffuses in a curving path as a result of inhomogeneity of soil pores. However, assuming grout diffuses in a straight path is always required for convenient modelling and analysis.Figure 1
Diagram of the (a) grout permeation diffusion and (b) direction indication of grout concretion.
(a)
(b)The flowing velocity is regarded as a key parameter to judge grout states; however, it is difficult to be determined. Some previous findings suggested that grout permeates in the laminar flow state when it is injected into the ground. Regarding the spherical diffusion model, the pores of soil layers can be considered as radial capillary tubes from center of sphere outward [35, 40, 41]. The size of capillary channels in the sandy layer varies significantly, and it is extremely difficult to reveal the real state through a mathematical model. For convenience, we assume the capillary channels into the tubes with the same radius in modelling. The Dupuit–Forchheimer relation was introduced to transform the permeation model in single tubes into the macroscopic state, and then the generalized Darcy’s law (applicable to laminar flow) is applied to derive the mathematical model on the basis of the capillary group theory [42]. Therefore, consider a laminar incompressible grout of finite length dl and radius r in a circular tube of radius r0 (Figure 2). Further, we assume that microelement shares the same axis with the tube, and α is the angle between the grout flow direction and the x-axis, which varies from 0 to 2π anticlockwise.Figure 2
Geometry of the physical model. (a) Power-law grout and (b) Bingham grout.
(a)
(b)Before formulation of the model, the following basic assumptions are introduced: (1) the sand and soil that grout being injected into are isotropic and homogeneous; (2) the grout is in power law, Bingham, and Newtonian fluids, respectively; (3) the grout is grouted into soil at the bottom of grouting pipes; and (4) the grout diffuses in a spherical shape.


## 2.2. Mathematical Model Development

### 2.2.1. Power-Law Grout

The mechanical equilibrium of the grout microelement for different angles ofα is written mathematically as(1)pπr2−p+dpπr2−2πrτdl−ρgsinαπr2dl=0,where p is the injection pressure; τ is the shear stress on surface of the grout column; ρ is the grout density; and g is the gravitational acceleration.Equation (1) is simplified into the following equation:(2)τ=−r2dpdl+ρgsinα.The rheological equation for power-law fluid is defined as follows [43]:(3)τ=Cγn=C−dvdrn,where C is the viscosity coefficient; γ is the shear rate; and n is the rheological index.Together with equations (2) and (3), it yields(4)dvdr=−τC1/n=−−12Cdpdl+ρgsinα1/nr1/n.Then, the flow velocityv at any position of the grout fluid yields(5)v=−12Cdpdl+ρgsinα1/nnn+1r0n+1/n−rn+1/n.Thus, the grout quantityq per unit time is given as(6)q=∫0r02πrvdr=nπ3n+1−12Cdpdl+ρgsinα1/nr03n+1/n.The average flow velocityv¯ can be given as(7)v¯=qπr02=n3n+1−12Cdpdl+ρgsinα1/nr0n+1/n.Solving the Dupuit–Forchheimer equation that defining flow velocityV=ϕv¯ [43], it yields(8)V=nϕ3n+1−12Cdpdl+ρgsinα1/nr0n+1/n,where ϕ is the porosity of porous media.Here, introducing the effective permeabilityKe and effective viscosity μe, which are defined as [43](9)Ke=ϕr022n3n+1,μe=C3n+1ϕr0nn−1,and equation (8) is rearranged into(10)V=Keμe1/n−dpdl+ρgsinα1/n.For a representative sphere case, its total surface area isA=4πl2. Thus, the total grout quantity Q is given as(11)Q=VAt,where t is the grouting time.Equation (11) is rearranged into(12)V=Q4πtl2=Keμe1/n−dpdl+ρgsinα1/n.Separating variables from equation (12) yields(13)p=−Q4πtnμeKe1−2nl1−2n−ρgsinαl+B,with the boundary conditions p=p0 and l = l0 and yields(14)p0=−Q4πtnμeKe1−2nl1−2n−ρgsinαl0+B,where p0 is the injection pressure and l0 is the radius of the grouting pipe.Solving equation (14) yields(15)B=p0+Q4πtnμeKe1−2nl1−2n+ρgsinαl0.Together with equations (13) and (15), it yields(16)p=p0+Q4πtnμeKe1−2nl01−2n−l1−2n+ρgsinαl0−l.Thus, the total grout quantity can be expressed as(17)Q=43πl3ϕ.Together with equations (16) and (17), it leads to(18)p=p0+ϕ3tnμeKe1−2nl01−2nl3n−l1+n+ρgsinαl0−l.


### 2.2.2. Bingham Grout

In this case, substituting equation (2) into the rheological equation for Bingham grout that is defined as τ=τ0+μ−dv/dr [43] yields(19)dvdr=τ0−τμ=1μτ0+r2dpdl+ρgsinα,where τ0 is the yield stress.When−r/2dp/dl+ρgsinα≤τ0, the grout is not subject to shear action and there exists a flow core with a radius of rp in the tube; when 0≤r≤rp, no relative motion exists in the grout fluid, and when rp<r≤r0, the relative motion in the fluid produce shear action. Thus, the radius of flow core can be given as(20)rp=−2τ0dp/dl+ρgsinα.Integrating equation (19) yields(21)v=−1μ14dpdl+ρgsinαr02−r2+τ0r0−r,rp≤r≤r0,v=−1μ14dpdl+ρgsinαr02−rp2+τ0r0−rp,0≤r≤rp.The grout quantity per unit time in the circular tube is given by(22)q=∫rpr02πrvdr+πrp2vp.Equations (21) and (22) lead to the following equation:(23)q=−πr048μdpdl+ρgsinα1−43−2τ0/r0dp/dl+ρgsinα+13−2τ0/r0dp/dl+ρgsinα4.The average velocity of the grout in the tube can be expressed as follows:(24)v¯=qπr02=dpdl+ρgsinα1−43−2τ0/r0dp/dl+ρgsinα+13−2τ0/r0dp/dl+ρgsinα4.When there is no grout flow in the tube, it leads to(25)−2τ0/r0dp/dl+ρgsinα=1.Thus, the starting pressure gradientλ is(26)λ=−dpdl=2τ0r0+ρgsinα.Here, introducing the variableK is defined as(27)K=ϕr028.Thus, the grout flow velocityV is(28)V=−Kμdpdl+ρgsinα1−43ρgsinα−λdp/dl+ρgsinα+13ρgsinα−λdp/dl+ρgsinα4.Together with equations (11) and (28), it yields(29)V=Q4πl2t=−Kμdpdl+ρgsinα1−43ρgsinα−λdp/dl+ρgsinα+13ρgsinα−λdp/dl+ρgsinα4.Considering the pressure gradient−dp/dl≫λ, equation (29) is simplified as(30)V=Q4πl2t=−Kμdpdl+ρgsinα1−43ρgsinα−λdp/dl+ρgsinα.Separating variables and integrating equation (30) yield(31)p=Qμ4πtKlt+13ρgsinα−43λl+C,with the boundary conditions p=p0 and l = l0, and equation (31) leads to(32)p0=Qμ4πtKl0t+13ρgsinα−43λl0+C,(33)C=p0−Qμ4πtKl0t+13ρgsinα−43λl0.Together with equations (17), (31), and (33), the injection pressure p at any position is given as(34)p=p0−ϕμ3tKl0l3+ϕμ3tKl2+43λ−13ρgsinαl0−l.


### 2.2.3. Newtonian Grout

Considering the constitutive equation, the Newtonian grout can be regarded as a special form of Bingham grout whenτ0 is zero, so the starting pressure gradient λ is given as(35)λ=ρgsinα.Then, the injection pressure of Newtonian grout is(36)p=p0−ϕμ3tKl0l3+ϕμ3tKl2+ρgsinαl0−l.


## 2.2.1. Power-Law Grout

The mechanical equilibrium of the grout microelement for different angles ofα is written mathematically as(1)pπr2−p+dpπr2−2πrτdl−ρgsinαπr2dl=0,where p is the injection pressure; τ is the shear stress on surface of the grout column; ρ is the grout density; and g is the gravitational acceleration.Equation (1) is simplified into the following equation:(2)τ=−r2dpdl+ρgsinα.The rheological equation for power-law fluid is defined as follows [43]:(3)τ=Cγn=C−dvdrn,where C is the viscosity coefficient; γ is the shear rate; and n is the rheological index.Together with equations (2) and (3), it yields(4)dvdr=−τC1/n=−−12Cdpdl+ρgsinα1/nr1/n.Then, the flow velocityv at any position of the grout fluid yields(5)v=−12Cdpdl+ρgsinα1/nnn+1r0n+1/n−rn+1/n.Thus, the grout quantityq per unit time is given as(6)q=∫0r02πrvdr=nπ3n+1−12Cdpdl+ρgsinα1/nr03n+1/n.The average flow velocityv¯ can be given as(7)v¯=qπr02=n3n+1−12Cdpdl+ρgsinα1/nr0n+1/n.Solving the Dupuit–Forchheimer equation that defining flow velocityV=ϕv¯ [43], it yields(8)V=nϕ3n+1−12Cdpdl+ρgsinα1/nr0n+1/n,where ϕ is the porosity of porous media.Here, introducing the effective permeabilityKe and effective viscosity μe, which are defined as [43](9)Ke=ϕr022n3n+1,μe=C3n+1ϕr0nn−1,and equation (8) is rearranged into(10)V=Keμe1/n−dpdl+ρgsinα1/n.For a representative sphere case, its total surface area isA=4πl2. Thus, the total grout quantity Q is given as(11)Q=VAt,where t is the grouting time.Equation (11) is rearranged into(12)V=Q4πtl2=Keμe1/n−dpdl+ρgsinα1/n.Separating variables from equation (12) yields(13)p=−Q4πtnμeKe1−2nl1−2n−ρgsinαl+B,with the boundary conditions p=p0 and l = l0 and yields(14)p0=−Q4πtnμeKe1−2nl1−2n−ρgsinαl0+B,where p0 is the injection pressure and l0 is the radius of the grouting pipe.Solving equation (14) yields(15)B=p0+Q4πtnμeKe1−2nl1−2n+ρgsinαl0.Together with equations (13) and (15), it yields(16)p=p0+Q4πtnμeKe1−2nl01−2n−l1−2n+ρgsinαl0−l.Thus, the total grout quantity can be expressed as(17)Q=43πl3ϕ.Together with equations (16) and (17), it leads to(18)p=p0+ϕ3tnμeKe1−2nl01−2nl3n−l1+n+ρgsinαl0−l.


## 2.2.2. Bingham Grout

In this case, substituting equation (2) into the rheological equation for Bingham grout that is defined as τ=τ0+μ−dv/dr [43] yields(19)dvdr=τ0−τμ=1μτ0+r2dpdl+ρgsinα,where τ0 is the yield stress.When−r/2dp/dl+ρgsinα≤τ0, the grout is not subject to shear action and there exists a flow core with a radius of rp in the tube; when 0≤r≤rp, no relative motion exists in the grout fluid, and when rp<r≤r0, the relative motion in the fluid produce shear action. Thus, the radius of flow core can be given as(20)rp=−2τ0dp/dl+ρgsinα.Integrating equation (19) yields(21)v=−1μ14dpdl+ρgsinαr02−r2+τ0r0−r,rp≤r≤r0,v=−1μ14dpdl+ρgsinαr02−rp2+τ0r0−rp,0≤r≤rp.The grout quantity per unit time in the circular tube is given by(22)q=∫rpr02πrvdr+πrp2vp.Equations (21) and (22) lead to the following equation:(23)q=−πr048μdpdl+ρgsinα1−43−2τ0/r0dp/dl+ρgsinα+13−2τ0/r0dp/dl+ρgsinα4.The average velocity of the grout in the tube can be expressed as follows:(24)v¯=qπr02=dpdl+ρgsinα1−43−2τ0/r0dp/dl+ρgsinα+13−2τ0/r0dp/dl+ρgsinα4.When there is no grout flow in the tube, it leads to(25)−2τ0/r0dp/dl+ρgsinα=1.Thus, the starting pressure gradientλ is(26)λ=−dpdl=2τ0r0+ρgsinα.Here, introducing the variableK is defined as(27)K=ϕr028.Thus, the grout flow velocityV is(28)V=−Kμdpdl+ρgsinα1−43ρgsinα−λdp/dl+ρgsinα+13ρgsinα−λdp/dl+ρgsinα4.Together with equations (11) and (28), it yields(29)V=Q4πl2t=−Kμdpdl+ρgsinα1−43ρgsinα−λdp/dl+ρgsinα+13ρgsinα−λdp/dl+ρgsinα4.Considering the pressure gradient−dp/dl≫λ, equation (29) is simplified as(30)V=Q4πl2t=−Kμdpdl+ρgsinα1−43ρgsinα−λdp/dl+ρgsinα.Separating variables and integrating equation (30) yield(31)p=Qμ4πtKlt+13ρgsinα−43λl+C,with the boundary conditions p=p0 and l = l0, and equation (31) leads to(32)p0=Qμ4πtKl0t+13ρgsinα−43λl0+C,(33)C=p0−Qμ4πtKl0t+13ρgsinα−43λl0.Together with equations (17), (31), and (33), the injection pressure p at any position is given as(34)p=p0−ϕμ3tKl0l3+ϕμ3tKl2+43λ−13ρgsinαl0−l.


## 2.2.3. Newtonian Grout

Considering the constitutive equation, the Newtonian grout can be regarded as a special form of Bingham grout whenτ0 is zero, so the starting pressure gradient λ is given as(35)λ=ρgsinα.Then, the injection pressure of Newtonian grout is(36)p=p0−ϕμ3tKl0l3+ϕμ3tKl2+ρgsinαl0−l.


## 2.3. Prediction Model of Grout Concretion Dimension

Considering the Bingham and Newtonian grouts are the most commonly used types for grouting practices, thereby the Bingham grout was chosen to establish the prediction model for grout concretion dimension in this section.Together with equation (34) and boundary conditions p=0 and l = lmax, which yield(37)p0−ϕμ3tKl0lmax3+ϕμ3tKlmax3+43λ−13ρgsinαl0−lmax=0,where lmax is the maximum diffusion radius.Whenα is 0, π/2, and 3π/2, respectively, the diffusion radius is lmax,0, lmax,π/2, and lmax,3π/2. Thus, the following equation can be obtained:(38a)p0−ϕμ3tKl0lmax,03+ϕμ3tKlmax,03+43λl0−lmax,0=0,(38b)p0−ϕμ3tKl0lmax,π/23+ϕμ3tKlmax,π/23+43λ−13ρgl0−lmax,π/2=0,(38c)p0−ϕμ3tKl0lmax,3π/23+ϕμ3tKlmax,3π/23+43λ+13ρgl0−lmax,3π/2=0,where lmax,0, lmax,π/2, and lmax,3π/2 is the maximum diffusion radius at horizontal direction, vertical upward direction, and vertical downward direction, respectively.l
max
,
0, lmax,π/2, and lmax,3π/2 can be obtained by solving equations (38a), (38b), and (38c), and then the grout concretion dimension at different directions are(39a)Dx=Dy=2lmax,0,(39b)Dz=lmax,π/2+lmax,3π/2,where Dx, Dy, and Dz indicated in Figure 1 are the dimensions of grout concretion at transverse direction, longitudinal direction, and vertical direction, respectively.


## 2.4. Analysis of the Injection Pressure Distribution Law

Some findings of the previous study by Ruan [44] have been utilized to explore the spatial distribution law of injection pressure with diffusion radius variations. In this study, the cement grout was identified as three flow patterns, i.e., power law (w/c ratio = 0.5–0.7), Bingham (w/c ratio = 0.8–1.0), and Newtonian grouts (w/c = 2.0–10.0) based on the variations of the w/c ratios. The cement grout with the w/c ratios of 0.5, 1.0, and 2.0 together with the aforementioned established models have been used to discuss the relationship between diffusion radius at different directions under different injection pressures. The grout parameters are listed in Table 1, and Figure 3 presents the experimental results.Table 1
Parameters of three grouts used for the study.
Grout type
μ
e (Pa·s)
τ
0 (Pa)
ρ (g/cm3)
K
e (cm/s)
n
l
0 (cm)
t (s)
p (kPa)
Power law
—
—
1.86
0.03
0.4
2.5
600
200
Bingham
0.0096
1.563
1.49
0.03
0.4
2.5
600
200
Newtonian
0.0047
0
1.30
0.03
0.4
2.5
600
200
Note. n is porosity.Figure 3
Injection pressure distribution of grouts at different directions. (a) Power-law grout, (b) Bingham grout, and (c) Newtonian grout.
(a)
(b)
(c)Figure3(a) indicates that injection pressure decreases linearly with increase of diffusion radius. Also, attenuation rates of injection pressure at different directions vary marginally in terms of a smaller rheological coefficient. From Figures 3(b) and 3(c), variations of injection pressure along with diffusion radius of Bingham and Newtonian grouts significantly differ from the power-law grout. Nonlinear decreases of injection pressure are found in both two grouts with the increase of diffusion radius. Particularly, compared with the Bingham grout, marginal attenuation in injection pressure was observed within 10 cm of the radius for Newtonian grout, while the slight difference emerges when radius is beyond 10 cm. The injection pressure distributes diminishingly at directions of 3π/2, 0, and π/2 in terms of the same diffusion radius.


## 2.5. Comparison of Diffusion Radius with and without considering Self-Gravity Effect

The proposed models were further used to discuss diffusion radius in aspects of considering and not considering self-gravity effects of the three grouts, and results are shown in Figure4.Figure 4
Diffusion radius of different cement grouts. (a) Power-law grout, (b) Bingham grout, and (c) Newtonian grout.
(a)
(b)
(c)It can be seen from Figure4 that three grouts diffuse approximately in an “ellipsoidal” shape. Though values of the diffusion radius considering gravity effects in horizontal direction exhibit little or even no differences compared with not considering gravity effects, the remarkable differences of the diffusion radius in vertical direction have proved the necessity to take into account the grout gravity effects. The emergence of gravity leads to the decrease of the diffusion radius in the upper part while that increase in the lower part. The Newtonian grout produces the maximum diffusion radius compared with the other two grouts whether considering the gravity effects or not.


## 3. Experimental Campaign

### 3.1. Materials

The cement grout used in experiments was made from the ordinary Portland cement P.I 42.5 (commercial cement produced in Shandong Lucheng Cement Company) and water. The manufactured coarse sand with dimension varying from 0.25 mm to 2.0 mm was used to produce the injected medium, which is presented in Table2. The displayed shear strength indexes of the used sand were determined by a series of laboratory triaxial compression tests.Table 2
Physical and mechanical parameters of the used sand.
G
s (g/cm3)
n
ω (%)
K
e (cm/s)
φ (°)
c (kPa)
C
u
C
c
2.65
0.4
4
0.0344
34.9
4
2.74
1.29
Note. Gs is the specific gravity, n is porosity, ω is water content, φ is internal friction angle, c is cohesion, Cu is nonuniformity coefficient, and Cc is curvature coefficient.


### 3.2. Experimental Program

The grouting test with three different injection pressures andw/c ratios was designed in the experiments. Total 9 groups of grouting tests were conducted, in which the injection pressure is 80 kPa, 100 kPa, and 120 kPa, respectively, while w/c ratio is 1.0, 1.5, and 2.0, respectively. The detailed information of the designed tests is presented in Table 3. The whole grouting model test system consists of four parts: the first part is the model box system; the second part is the grouting system (including grout tank, grouting pipe, and grouting pump); the third part is the pressure system (including air compressor and intake pipe); and the fourth part is the measurement system (including electronic scale and pressure gauge), which are presented in Figures 5 and 6. The model box was made of plexiglass with a steel girder as the support member, and the geometric length, width, and height of the box are 70 cm, 45 cm, and 65 cm, respectively. During the grouting test, the pressure with a certain value that measured by the gauge is generated by the air compressor. The high-pressure gas is connected with the grouting pump, and then the grout is forced to flow through the grouting pipe and finally injected into the sand.Table 3
Material property of the used cement grouts.
No.
w
/
c ratio
p
0 (kPa)
μ
e (Pa·s)
τ
0 (Pa)
ρ (g/cm3)
K
e (cm/s)
n
l
0 (cm)
t (s)
1
1.0
80
0.0096
1.563
1.49
0.0344
0.4
0.5
120
2
1.0
100
0.0096
1.563
1.49
0.0344
0.4
0.5
120
3
1.0
120
0.0096
1.563
1.49
0.0344
0.4
0.5
120
4
1.5
80
0.0073
0.3568
1.37
0.0344
0.4
0.5
120
5
1.5
100
0.0073
0.3568
1.37
0.0344
0.4
0.5
120
6
1.5
120
0.0073
0.3568
1.37
0.0344
0.4
0.5
120
7
2.0
80
0.0067
0.0971
1.30
0.0344
0.4
0.5
150
8
2.0
100
0.0067
0.0971
1.30
0.0344
0.4
0.5
150
9
2.0
120
0.0067
0.0971
1.30
0.0344
0.4
0.5
180Figure 5
Schematic diagram of the grout injection system.Figure 6
Real map of the test devices.


## 3.1. Materials

The cement grout used in experiments was made from the ordinary Portland cement P.I 42.5 (commercial cement produced in Shandong Lucheng Cement Company) and water. The manufactured coarse sand with dimension varying from 0.25 mm to 2.0 mm was used to produce the injected medium, which is presented in Table2. The displayed shear strength indexes of the used sand were determined by a series of laboratory triaxial compression tests.Table 2
Physical and mechanical parameters of the used sand.
G
s (g/cm3)
n
ω (%)
K
e (cm/s)
φ (°)
c (kPa)
C
u
C
c
2.65
0.4
4
0.0344
34.9
4
2.74
1.29
Note. Gs is the specific gravity, n is porosity, ω is water content, φ is internal friction angle, c is cohesion, Cu is nonuniformity coefficient, and Cc is curvature coefficient.


## 3.2. Experimental Program

The grouting test with three different injection pressures andw/c ratios was designed in the experiments. Total 9 groups of grouting tests were conducted, in which the injection pressure is 80 kPa, 100 kPa, and 120 kPa, respectively, while w/c ratio is 1.0, 1.5, and 2.0, respectively. The detailed information of the designed tests is presented in Table 3. The whole grouting model test system consists of four parts: the first part is the model box system; the second part is the grouting system (including grout tank, grouting pipe, and grouting pump); the third part is the pressure system (including air compressor and intake pipe); and the fourth part is the measurement system (including electronic scale and pressure gauge), which are presented in Figures 5 and 6. The model box was made of plexiglass with a steel girder as the support member, and the geometric length, width, and height of the box are 70 cm, 45 cm, and 65 cm, respectively. During the grouting test, the pressure with a certain value that measured by the gauge is generated by the air compressor. The high-pressure gas is connected with the grouting pump, and then the grout is forced to flow through the grouting pipe and finally injected into the sand.Table 3
Material property of the used cement grouts.
No.
w
/
c ratio
p
0 (kPa)
μ
e (Pa·s)
τ
0 (Pa)
ρ (g/cm3)
K
e (cm/s)
n
l
0 (cm)
t (s)
1
1.0
80
0.0096
1.563
1.49
0.0344
0.4
0.5
120
2
1.0
100
0.0096
1.563
1.49
0.0344
0.4
0.5
120
3
1.0
120
0.0096
1.563
1.49
0.0344
0.4
0.5
120
4
1.5
80
0.0073
0.3568
1.37
0.0344
0.4
0.5
120
5
1.5
100
0.0073
0.3568
1.37
0.0344
0.4
0.5
120
6
1.5
120
0.0073
0.3568
1.37
0.0344
0.4
0.5
120
7
2.0
80
0.0067
0.0971
1.30
0.0344
0.4
0.5
150
8
2.0
100
0.0067
0.0971
1.30
0.0344
0.4
0.5
150
9
2.0
120
0.0067
0.0971
1.30
0.0344
0.4
0.5
180Figure 5
Schematic diagram of the grout injection system.Figure 6
Real map of the test devices.


## 4. Results and Discussion

### 4.1. Analysis of the Grout Quantity

In this section, the grout quantity changing with time under different injection pressures andw/c ratios is analyzed by recorded results, which are shown in Figure 7.Figure 7
Grout quantity vs. time under different injection pressures andw/c ratios. w/c ratios of (a) 1.0, (b) 1.5, and (c) 2.0.
(a)
(b)
(c)From Figure7, it is indicated that the grout quantity sharply increases at the initial phase, then slowly increases, and finally tends to be stable. More time is required to reach a steady state under a larger injection pressure. Differing from the two other ratios, the w/c ratio of 1.0 brings about an obvious two-stage change in grout quantity over time. Specifically, grout quantity increases sharply within 0.5 minutes at the first stage, and then remains stable in a certain time. At the second stage, the quantity increases slightly in a short time and eventually tends to be stable. As for the two larger ratios, the grout quantity mainly reaches its maximum in 4 minutes and remains stable. When the w/c ratio is set as 1.5, the grout quantity under an injection pressure of 100 kPa is less than that the quantity under an injection pressure of 80 kPa. This may be because compaction process leads to difference between the number and arrangement of pore passages in the soil.


### 4.2. Analysis of the Grout Concretion Dimension

The diagram for the dimension measurement of grout concretion from grouting test at three directions (i.e., transverse, vertical, and longitudinal) is shown in Figure8. Also, the theoretical values obtained based on the dimension prediction model are also used for the comparative analysis and verification.Figure 8
Diagram of the dimension of grout concretion.
(a)
(b)
(c)


#### 4.2.1. Transverse Dimension

As detailed in Figure9, the experimental and theoretical values of the transverse dimension of grout concretion are approximately linearly correlated with injection pressure, and the growth rate is roughly the same. The theoretical values at the w/c ratios of 1.0, 1.5, and 2.0 are ca. 3.0, 2.0, and 1.5 times, respectively, of the experimental values. It is predicted that the increase of w/c ratio can reduce difference between the experimental and theoretical values and finally obtain more reliable results.Figure 9
The transverse dimension of grout concretion under differentw/c ratios. w/c ratios of (a) 1.0, (b) 1.5, and (c) 2.0.
(a)
(b)
(c)


#### 4.2.2. Vertical Dimension

The vertical dimension of grout concretion changing with the injection pressure is shown in Figure10. The results suggest that vertical dimension presents a linear correlation with the injection pressure for the theoretical values, while the correlation under the experimental conditions shows nonlinear characteristics. Specifically, experimental transverse dimension at a ratio of 2.0 decreased slightly when the injection pressure exceeds 100 kPa, which verifies the randomness of the grout diffusion at vertical direction. The theoretical values at the w/c ratios of 1.0, 1.5, and 2.0 are ca. 3.0, 2.0, and 2 times, respectively, of the experimental values. It appears that the difference between experimental and theoretical values tends to decrease with the increase of w/c ratio, but this trend will gradually weaken.Figure 10
The vertical dimension of grout concretion under differentw/c ratios. w/c ratios of (a) 1.0, (b) 1.5, and (c) 2.0.
(a)
(b)
(c)


#### 4.2.3. Longitudinal Dimension

The longitudinal dimensions of grout concretion under experimental and theoretical conditions are presented in Figure11. The data confirm a linear correlation between the longitudinal dimension and injection pressure in both experimental value and theoretical value. It seems that the experimental value of grout concretion at longitudinal direction is gradually close to the theoretical value with the increase of w/c ratio, which indicates that the theoretical model can be effectively used to predict the grout concretion dimension.Figure 11
The longitudinal dimension of grout concretion under differentw/c ratios.
(a)
(b)
(c)It is noted that there is a considerable difference between theoretical and experimental results at the same injection pressure, and this is because of pressure loss, increases of grout viscosity, and filtration effect. Specifically, friction effect caused by grout flowing in the grouting pipe results in pressure loss. The insufficient tightness of the pipe joint also leads to a loss of injection pressure. The viscosity of cement-based grout is featured by the time-varying property. During cement hydration, the viscosity and viscous resistance of grout increase over time, causing an adverse impact on grout diffusion [41]. When grout is injected into the soil, part of cement particles is blocked by skeleton of soil particles, resulting in stagnation and filtration of these cement particles as well as decrease of grout concentration and blockage of the soil pores. The grouting effect is significantly influenced by limited diffusion channels of the grout [28, 45–47].


## 4.1. Analysis of the Grout Quantity

In this section, the grout quantity changing with time under different injection pressures andw/c ratios is analyzed by recorded results, which are shown in Figure 7.Figure 7
Grout quantity vs. time under different injection pressures andw/c ratios. w/c ratios of (a) 1.0, (b) 1.5, and (c) 2.0.
(a)
(b)
(c)From Figure7, it is indicated that the grout quantity sharply increases at the initial phase, then slowly increases, and finally tends to be stable. More time is required to reach a steady state under a larger injection pressure. Differing from the two other ratios, the w/c ratio of 1.0 brings about an obvious two-stage change in grout quantity over time. Specifically, grout quantity increases sharply within 0.5 minutes at the first stage, and then remains stable in a certain time. At the second stage, the quantity increases slightly in a short time and eventually tends to be stable. As for the two larger ratios, the grout quantity mainly reaches its maximum in 4 minutes and remains stable. When the w/c ratio is set as 1.5, the grout quantity under an injection pressure of 100 kPa is less than that the quantity under an injection pressure of 80 kPa. This may be because compaction process leads to difference between the number and arrangement of pore passages in the soil.


## 4.2. Analysis of the Grout Concretion Dimension

The diagram for the dimension measurement of grout concretion from grouting test at three directions (i.e., transverse, vertical, and longitudinal) is shown in Figure8. Also, the theoretical values obtained based on the dimension prediction model are also used for the comparative analysis and verification.Figure 8
Diagram of the dimension of grout concretion.
(a)
(b)
(c)


### 4.2.1. Transverse Dimension

As detailed in Figure9, the experimental and theoretical values of the transverse dimension of grout concretion are approximately linearly correlated with injection pressure, and the growth rate is roughly the same. The theoretical values at the w/c ratios of 1.0, 1.5, and 2.0 are ca. 3.0, 2.0, and 1.5 times, respectively, of the experimental values. It is predicted that the increase of w/c ratio can reduce difference between the experimental and theoretical values and finally obtain more reliable results.Figure 9
The transverse dimension of grout concretion under differentw/c ratios. w/c ratios of (a) 1.0, (b) 1.5, and (c) 2.0.
(a)
(b)
(c)


### 4.2.2. Vertical Dimension

The vertical dimension of grout concretion changing with the injection pressure is shown in Figure10. The results suggest that vertical dimension presents a linear correlation with the injection pressure for the theoretical values, while the correlation under the experimental conditions shows nonlinear characteristics. Specifically, experimental transverse dimension at a ratio of 2.0 decreased slightly when the injection pressure exceeds 100 kPa, which verifies the randomness of the grout diffusion at vertical direction. The theoretical values at the w/c ratios of 1.0, 1.5, and 2.0 are ca. 3.0, 2.0, and 2 times, respectively, of the experimental values. It appears that the difference between experimental and theoretical values tends to decrease with the increase of w/c ratio, but this trend will gradually weaken.Figure 10
The vertical dimension of grout concretion under differentw/c ratios. w/c ratios of (a) 1.0, (b) 1.5, and (c) 2.0.
(a)
(b)
(c)


### 4.2.3. Longitudinal Dimension

The longitudinal dimensions of grout concretion under experimental and theoretical conditions are presented in Figure11. The data confirm a linear correlation between the longitudinal dimension and injection pressure in both experimental value and theoretical value. It seems that the experimental value of grout concretion at longitudinal direction is gradually close to the theoretical value with the increase of w/c ratio, which indicates that the theoretical model can be effectively used to predict the grout concretion dimension.Figure 11
The longitudinal dimension of grout concretion under differentw/c ratios.
(a)
(b)
(c)It is noted that there is a considerable difference between theoretical and experimental results at the same injection pressure, and this is because of pressure loss, increases of grout viscosity, and filtration effect. Specifically, friction effect caused by grout flowing in the grouting pipe results in pressure loss. The insufficient tightness of the pipe joint also leads to a loss of injection pressure. The viscosity of cement-based grout is featured by the time-varying property. During cement hydration, the viscosity and viscous resistance of grout increase over time, causing an adverse impact on grout diffusion [41]. When grout is injected into the soil, part of cement particles is blocked by skeleton of soil particles, resulting in stagnation and filtration of these cement particles as well as decrease of grout concentration and blockage of the soil pores. The grouting effect is significantly influenced by limited diffusion channels of the grout [28, 45–47].


## 4.2.1. Transverse Dimension

As detailed in Figure9, the experimental and theoretical values of the transverse dimension of grout concretion are approximately linearly correlated with injection pressure, and the growth rate is roughly the same. The theoretical values at the w/c ratios of 1.0, 1.5, and 2.0 are ca. 3.0, 2.0, and 1.5 times, respectively, of the experimental values. It is predicted that the increase of w/c ratio can reduce difference between the experimental and theoretical values and finally obtain more reliable results.Figure 9
The transverse dimension of grout concretion under differentw/c ratios. w/c ratios of (a) 1.0, (b) 1.5, and (c) 2.0.
(a)
(b)
(c)


## 4.2.2. Vertical Dimension

The vertical dimension of grout concretion changing with the injection pressure is shown in Figure10. The results suggest that vertical dimension presents a linear correlation with the injection pressure for the theoretical values, while the correlation under the experimental conditions shows nonlinear characteristics. Specifically, experimental transverse dimension at a ratio of 2.0 decreased slightly when the injection pressure exceeds 100 kPa, which verifies the randomness of the grout diffusion at vertical direction. The theoretical values at the w/c ratios of 1.0, 1.5, and 2.0 are ca. 3.0, 2.0, and 2 times, respectively, of the experimental values. It appears that the difference between experimental and theoretical values tends to decrease with the increase of w/c ratio, but this trend will gradually weaken.Figure 10
The vertical dimension of grout concretion under differentw/c ratios. w/c ratios of (a) 1.0, (b) 1.5, and (c) 2.0.
(a)
(b)
(c)


## 4.2.3. Longitudinal Dimension

The longitudinal dimensions of grout concretion under experimental and theoretical conditions are presented in Figure11. The data confirm a linear correlation between the longitudinal dimension and injection pressure in both experimental value and theoretical value. It seems that the experimental value of grout concretion at longitudinal direction is gradually close to the theoretical value with the increase of w/c ratio, which indicates that the theoretical model can be effectively used to predict the grout concretion dimension.Figure 11
The longitudinal dimension of grout concretion under differentw/c ratios.
(a)
(b)
(c)It is noted that there is a considerable difference between theoretical and experimental results at the same injection pressure, and this is because of pressure loss, increases of grout viscosity, and filtration effect. Specifically, friction effect caused by grout flowing in the grouting pipe results in pressure loss. The insufficient tightness of the pipe joint also leads to a loss of injection pressure. The viscosity of cement-based grout is featured by the time-varying property. During cement hydration, the viscosity and viscous resistance of grout increase over time, causing an adverse impact on grout diffusion [41]. When grout is injected into the soil, part of cement particles is blocked by skeleton of soil particles, resulting in stagnation and filtration of these cement particles as well as decrease of grout concentration and blockage of the soil pores. The grouting effect is significantly influenced by limited diffusion channels of the grout [28, 45–47].


## 5. Conclusions

(1)
Mathematical models of permeation grouting for power law, Bingham, and Newtonian grouts are constructed. The self-gravity effect on grout permeation and diffusion has been taken into consideration.
(2)
For the power-law cement grout, the injection pressure decreases linearly with increase of diffusion radius, while nonlinear decreases of injection pressure were confirmed in Bingham and Newtonian grouts with the increase of diffusion radius.
(3)
Three grouts approximately diffuse in an “ellipsoidal” shape. The grout self-gravity is closely related to the variations of diffusion radius in vertical direction. The Newtonian grout produces the maximum diffusion radius compared with the other two grouts whether the gravity effects were considered or not.
(4)
The grout quantity under a smallerw/c ratio exhibits significant differences and undergoes two increasing stages, whereas the quantity simply tends to be stable after it reaches its maximum under the two larger w/c ratios.
(5)
The proposed prediction model for grout concretion dimension shows good agreements with the experimental data, which verify its practicability.



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*Source: 7968240-2019-11-27.xml*