Study on Penetration Grouting Mechanism Based on Newton Fluid of Time-Dependent Behavior of Rheological Parameters

Faculty of Public Safety and Emergency Management, Kunming University of Science and Technology, Kunming, Yunnan 650093, China Chengdu Institute of Mountain Hazards and Environment, Chinese Academy of Sciences and Ministry of Water Resources, Chengdu, Sichuan 610000, China Kunming Survey Design and Research Institute Co. Ltd. of CREEC, Kunming, Yunnan 650200, China Yunnan Institute of Geological Environment Monitoring, Kunming, Yunnan 650216, China 1e 1ird Engineering Co., Ltd. of China Railway Seventh Bureau Group Corporation, Xi’an, Shaanxi 710032, China


Introduction
Grouting, as a widely used geotechnical reinforcement and water shutoff technology, has been used in practical engineering fields involving buildings, highways, subways, mines, tunnels, hydropower services, military applications, railways, and stratum treatments [1]. According to different rheological constitutive formulas, the grouting fluids are divided into three types: Newton fluid, Bingham fluid, and power-law fluid [1][2][3].
e grouting mechanism delivers a great and important influence on grouting effects; for example, the diffusion model of grouting is one of the significant ways to provide theoretical support and guidance for actual grouting construction. Presently, abundant research achievements have been made with respect to the penetration grouting mechanism based on Newton fluid, which are also widely used in practical grouting engineering, such as Maag spherical theory, sleeve valve, and cylinder diffusion models [1]. Additionally, Jiacai [4] studied the formula for maximum diffusion radius of Newton fluid in the fracture surface; Baker [5] deduced the calculation formula for maximum diffusion radius of Newton fluid in the rock mass fracture; Jinfeng et al. [6] explored the diffusion law of Newton fluid in the plane, radial, and circular fracture; Lie Weile [7] deduced the diffusion formula of Newton fluid in the two-dimensional rough fracture; and Weicheng [8] explored the cylindrical diffusion mechanism based on Newton fluid under horizontal and vertical drilling conditions. As for the progress of the time-dependent behavior of grouts. Wenjun [9] built the rheological models of different types of grouts with the effect of time; Quansheng et al. [10] studied the variation of the rheological parameters of cement grouts with time; Roussel [11] investigated the thixotropy and flow behavior of fresh cement grouts; Zhuguo and Cao [12] studied the rheological behaviors of concrete in the vibrated state.
However, the time-dependent behavior of rheological parameters of most of the achievements for the penetration grouting mechanism for Newton fluid is not considered currently, and the rheological parameters are considered to be constants in the whole grouting process while calculating theoretical diffusion size. en, the calculated theoretical diffusion size is far greater than the measured value in actual grouting. erefore, the diffusion of grouts in engineering mainly depends on the workers' experience, which brings uncertainty to engineering grouting and makes it difficult to meet the requirements for engineering practices. Consequently, the penetration grouting mechanism based on Newton fluid of time-dependent behavior of rheological parameters is studied in this paper to provide theoretical support for actual grouting construction. (1)

Study on Penetration Grouting
According to the literature [9], the time-dependent behavior of rheological parameters of Newton fluid (dynamic viscosity) formula is e rheological formula for Newton fluid of time-dependent behavior can be obtained by combining formulas (1) and (2): where τ is the shear stress; c is the shear rate (c � − dv/dr); η is dynamic viscosity; η 0 is the initial value of dynamic viscosity of Newton fluid, and it approximately equals the fixed dynamic viscosity n in formula (1); η(t) is the dynamic viscosity of Newton fluid at the moment of t; t is the grouting duration; and k is the time-dependent coefficient and it can be measured via the experiment. In a circular tube with the Newton fluid of time-dependent behavior of rheological parameters, the radius is hypothesized to be r 0 . Taking an infinitesimal element of the fluid column and using the tube axis as the symmetry axis, the radius satisfying r < r 0 and the length is dl [13]. It is hypothesized that the pressure distribution of the infinitesimal element of the fluid column dl at the left and right ends is p and p + dp, respectively. erefore, the pressure difference in the infinitesimal element is dp. e shear stress acting on the surface is τ, its direction is on the left, and it is opposite to the direction of flow velocity as shown in Figure 1.
Under the condition that the gravity of Newton fluid is ignored, the force on the infinitesimal element column is shown in Figure 1 which satisfies the following equilibrium equations: Shear stress on the surface of the infinitesimal element column is and combining formula (5) and formula (3), we can obtain e velocity formula of Newton fluid of the time-dependent behavior in the cross section of the circular tube can be solved and obtained at v � 0 by applying the method of separation of variables and considering the boundary conditions r � r 0 of the pipe wall for formula (6): v � − 1 4η 0 e kt dp dl e flow Q P of Newton fluid of the time-dependent behavior for the laminar flow in the circular tube can be obtained: By substituting formula (7) into formula (8), the following results can be obtained: 8η 0 e kt dp dl . (9) erefore, the average velocity of Newton fluid of timedependent behavior on the cross section of the pipe is shown as follows: 8η 0 e kt dp dl . (10) en, the average seepage velocity of the fluid in the grouted medium holds formula V � ϕv, and, according to literature [13], 2 Shock and Vibration e seepage formula based on Newton fluid of timedependent behavior of rheological parameters can be obtained by simultaneous formulas (10), (11), and (12): where ϕ is the porosity of the grouted medium; μ is the viscosity of water; V is the average seepage velocity of Newton fluid of time-dependent behavior in the grouted medium; and other symbols are the same as above.

Penetration Grouting Mechanism of Newton Fluid Based on Time-Dependent Behavior of Rheological Parameters.
According to [13][14][15][16], the following assumptions are adopted in this paper while studying the penetration grouting mechanism based on Newton fluid of time-dependent behavior of rheological parameters: (1) e grouted medium or material is isotropic and homogeneous (2) e fluid is incompressible and its flow pattern remains unchanged during grouting (3) e dynamic viscosity of Newton fluid is time-dependent (increased with time), and the dynamic viscosity is exponentially related to time (4) e flow velocity was small, and the grout flow is laminar except that the local area around the grouting hole is turbulent (5) e tampering method is used for grouting, and cement grout is grouted into the grouted medium from the bottom of the grouting pipe orifice and diffuses in a spherical shape (6) e gravity effect of fluid is ignored in the grouting process e spherical diffusion model for penetration grouting mechanism based on Newton fluid of time-dependent behavior of rheological parameters studied in this paper is shown in Figure 2.
In Figure 2, p 1 is the grouting pressure; p 0 is the groundwater pressure at the grouting point; l 1 is the diffusion radius of Newton fluid of time-dependent behavior at the time of t; and l 0 is the radius of grouting pipe. e total grouting volume Q of Newton fluid of timedependent behavior in the grouting process meets the following formula: where A is the total diffusion surface area of Newton fluid of time-dependent behavior in the grouting area. e total diffusion surface area of the diffusion model is shown in Figure 2 and expressed as follows: By substituting formulas (14) and (15) into formula (13), the following formula is derived: For formula (16), the boundary conditions of grouting are taken into account while the integral of separation of variables method is used, namely, l � l 1 holds when p � p 1 , and l � l 0 when p � p 0 : and combining grouting volume Q � (4/3)πl 3 1 ϕ into formula (16), In practical engineering, l 1 ≫ l 0 is established, that is, Formula (19) is the formula for the penetration mechanism of spherical diffusion based on Newton fluid of timedependent behavior of rheological parameters.
Maag formula for penetration grouting based on Newton fluid may be deduced from (19), without considering the e spherically theoretical diffusion radius of Newton fluid of time-dependent behavior of rheological parameters in the grouted medium can be obtained by formula (19):

Application Scope and Parameter Determination
3.1. Application Scope. Formula (19) or (21) was derived on the basis of laminar flow, so it was concluded to be nonapplicable to the turbulent flow. According to [17,18], the laminar or turbulent flow state of Newton fluid of time-dependent behavior was determined by Reynolds number (Re). Newton fluid was in a turbulent state when Reynolds number >4000; Newton fluid was in a mixed state of turbulent flow and laminar flow when 2000 < Re < 4000 held; and Newton fluid was in a laminar flow state when Re < 2000 held. e Reynolds number could be obtained by calculating thee following formula: where v is the average velocity of Newton fluid of timedependent behavior in the grouted medium or material; d is the spatial range size of the flow of Newton fluid (e.g., pipe radius or diameter, and it means the pore size of the flow of Newton fluid in the grouted medium in this paper); and the meaning denoted by η is shown in the abovementioned text.

Parameters
Determined. e methods for obtaining each parameter in equation (19) or (21) are determined as follows: e porosity ϕ denotes the ratio of pore volume to total volume in the grouted medium or material, and it can be obtained with the following formula: where ω is the water content of the grouted medium or material; c is the density of the grouted medium or material; and c s is the volumetric weight of soil. Such parameters may be determined according to [19]. e permeability coefficient K of the grouted medium or material reflects its penetration behavior, and it may be determined by an indoor or on-site measurement method. However, the on-site water injecting experiment technique is normally used to obtain such a coefficient in order to truly reflect the permeability of the grouted medium or material. e initial dynamic viscosity η 0 and time-dependent behavior coefficient k can be obtained by one of the following two methods: (1) It can be obtained on the research results of rheological parameters of Newton fluid of time-dependent behavior, such as [2] and [17]; (2) e rheological parameters of Newton fluid can be measured by a rotating viscometer.
e grouting time t can be designed and selected according to the actual engineering situation, and the radius of the grouting pipe can be determined by directly measuring the grouting pipe.
On the basis of obtaining the porosity and permeability coefficient of the grouted medium or material and inquiring the viscosity value of water at different temperatures, then the β value can be calculated by formulas (11) and (12). Now that all the parameters required to be obtained in formula (19) or formula (21) are completely determined and the theoretical diffusion radius l 1 of Newton fluid of timedependent behavior in the grouted medium or material can be obtained under the condition of known grouting pressure and groundwater pressure difference at the grouting point, on the contrary, the theoretical grouting pressure differential Δp can be obtained when the grouting pressure l 1 is known.

Grouting Experiment
In order to verify the applicability of formula for penetration grouting mechanism based on Newton fluid of time-dependent behavior of rheological parameters derived above (i.e., formula (19) or formula (21)), an indoor grouting experiment was designed to verify the formula).

Grouting Experimental Device.
e grouting experimental device is composed of the experiment chamber, pressure supply device, and grouting storage container as shown in Figure 3. e experiment chamber, composed of the steel support and plexiglass plate, has an appearance size of 600 × 600 × 600 mm, and it is used to place the grouted medium, as shown in Figure 3(a). e nitrogen installed in the pressure supply device offers pressure for grouting. e grouting pressure can be monitored by a pressure monitor installed at the upper end of the grouting pipe. rough the nitrogen pressure reducer and grouting control switch, the precise grouting pressure and grouting time can be controlled, as shown in Figure 3(b). e grout storage container is mainly composed of the closed steel cylinder and circularshaped iron frame, in which the closed steel cylinder is sized with a bottom diameter of 15 cm and a height of 40 cm; it is designed to bear the pressure at 2.5 MPa.
e container opened at the top and bottom, respectively. A metal joint is welded to hold Newton fluid of time-dependent behavior. When the experiment is conducted, the metal joint is put into a circular iron frame, the pressurized nitrogen enters the grout storage container from the top through the pipe, and then the grouted fluid is applied with the pressure and enters into the grouted medium. e electronic scale at the lower part is capable of accurately measuring the volume of Newton fluid in the whole grouting experiment process as shown in Figure 3(c).

Grouting Materials and Grouted Medium.
In this paper, gravel (sand) stone bodies with particle size respectively distributed within the range of 1-3 mm, 3-5 mm, and 5-10 mm were selected as the grouted medium for the verifying experiment. To make the grouted gravel (sand) body met the hypothesis for isotropy and homogeneity, the gravel (sand) stone bodies had been washed out in clear water for 3 times before start the grouting experiment. e properties of three types of grouted gravel (sand) stone bodies are shown in Table 1.
By referring to GB/T50123-1999 Standard for Soil Experiment Method issued by the Ministry of Water Resources of China, the proportion c s and water content ω of gravel (sand) stone bodies in Table 1 can be obtained by specific indoor experiment. Density is designed by the actual grouting experiment; then, according to formula (23), the porosity of gravel (sand) body can be calculated. e permeability coefficient was obtained by using type 70 permeameter (the inner diameter of the type 70 permeameter is 100 mm and the maximum particle diameter of the injected gravel (sand) body is 10 mm in this experiment; therefore, φ/d � 10 > 5, and it meets the requirements of grouting permeability test), and the experiment is shown in Figure 4.
Ordinary Portland cement with a grade of #32.5 produced by Kunming Cement Plant was selected as the grouting material. In the verifying experiment of this paper, the grout with a water-cement ratio of 2.0 was used for the grouting verifying experiment. According to [20,21], such a grout was a typical Newton fluid, and its rheological formula of time-dependent behavior of rheological parameter can be found in the research results of [20].

Working Principle and Steps.
e working principle of the experiment is shown in Figure 5.
e experiment steps are shown as follows: (1) According to the density and porosity in Table 1, the volume of grouted gravel (sand) stone body in the experiment chamber (3) was calculated. Four layers of gravel (sand) stone body were designed and laid from the bottom of the experiment chamber, and the pave height of each layer was designed to be 15 cm; then, the theoretical mass of each layer of gravel (sand) stone body in each group of grouting verifying experiment was calculated as shown in Table 2 according to the predetermined density in Table 1.  Table 2, and compact the gravel (sand) stone body on each layer within the marked volume range. Ensure that the density and porosity meet the specified requirements in Table 1; then lay the grouting pipe simultaneously. (4) Prepare cement grouts (5) with a water-cement ratio of 2.0, weigh the required materials, and mix them for over 5 minutes to ensure the uniformity of the grouts. (5) Pour the mixed grout into a grout storage container (2) and install the grouting pipe (7). (6) Check the above steps and prepare for grouting. (7) Start grouting. Turn on the grouting control switch, and start nitrogen pressure reducer (4) to make the grouting pressure quickly hit the designed value; Shock and Vibration then, make sure the grouting pressure in the entire grouting process is the designed value (as shown in Table 2). Observe the flow and diffusion of the grout. (7) Stop grouting. When the grouting duration hits the designed time in Table 2, stop grouting. (8) Mould removal and measurement. When the grout is completely consolidated with the grouted gravel (sand) stone body, measure the three-dimensional diffusion radius of the grouting body. (9) Analyze the experimental data and draw a conclusion.

Experiment Design.
e grouting verifying experiments designed in this paper are shown in Table 2. PVC pipe with a diameter of 15 mm is to be used for the grouting test. Namely, the radius grouting pipe is l 0 � 0.0075 m. Both the indoor environment temperature and water temperature for the experiment are 20°C (the water viscosity at 20°C is μ � 1.01×10 − 3 Pa·s), and some corresponding basic grouting parameters can be calculated by the formula introduced above, as shown in Table 3.
Physical photos of the grouting verifying experiments are shown in Figure 6. Calculate the Reynolds number Re b of the instantaneous mixed cement grouts. Reynolds number of Newton grout of time-dependent behavior in Experiment 1, Experiment 2, and Experiment 3 satisfied Re G1b � 25.9, Re G2b � 32.51, and Re G3b � 38.02. When their viscosity gradually increased, the pore size and path space became smaller or were even completely blocked; also the average movement speed of the grout decreased gradually. en, the instantaneous Reynolds number of grout decreased in the grouting process. Namely, the instantaneous Reynolds number of Newton grout of time-dependent behavior satisfied the relationship

Determination on Flow State of Newton
is result showed that the Newton grout of time-dependent behavior with a water-cement ratio of 2.0 in the grouting process of gravel (sand) stone body coincided with the theoretical basis of the laminar flow state.

Determination on Diffusion Radius of Grouting and
Analysis on Grouting Stone Body. It is found from Figure 6 that Newton grout of time-dependent behavior in the grouted gravel (sand) stone body showed a semiellipsoid diffusion shape, which was close to the theoretical diffusion shape (the completely hemispherical diffusion shape). In order to analyze and compare the difference between actual diffusion shape and theoretical diffusion shape in a better way, the diffusion radii of axis x, axis y, and axis z of the grouting stone body were measured and analyzed.
When the grouting test is completed, disassemble the experiment chamber, measure the diffusion radius of grouting stone bodies in axis x, axis y, and axis z with the ruler at least three times each, and take the average value. e comparative analysis results of the diffusion radius of grouting stone body in axis x, axis y, and axis z directions are shown in Table 4.   It is found from Table 4 that there is an insignificant difference with respect to the size of diffusion radiuses of axis x, axis y, and axis z of grouting stone body. e size of the diffusion radius in the direction of axis y is maximum, and the size in the direction of axis x is minimum, but the difference between such two values was within 5%; this indicates that the shape of the grouting stone body is close to the theoretical model, as shown in Figure 2.

Result Analysis.
According to the penetration grouting mechanism based on Newton fluid of time-dependent behavior of rheological parameters, the theoretically calculated value of diffusion radius, the actually measured value of diffusion radius, and the difference analysis between them are shown in Table 5. For comparison, the diffusion radius of Maag formula [1] without considering the time-dependent behavior of the viscosity of Newton fluid was calculated and the diffusion radius and difference between the Maag value and experimental value are also listed in Table 5.
To verify and analyze the applicability of formula (19) or (21) in a more effective manner, the minimum measured diffusion radius in three-dimensional directions (i.e., the diffusion radius size in the direction of axis x) is compared to the theoretically calculated value.
Analyzing Table 5, it is concluded that the theoretical value of the grouting diffusion radius calculated by the formula based on Newton fluid of time-dependent behavior of rheological parameters derived in this paper basically    coincides with the measured value of the grouting verifying experiment, and the difference is within 15% between the two values, which is much less than the change in 80% between the measured value and the value calculated by the Maag theoretical formula. is indicates that the penetration grouting mechanism based on Newton fluid of time-dependent behavior of rheological parameters established in this paper can reflect the penetration grouting mechanism based on Newton fluid of time-dependent behavior of rheological parameters, and the results are closer to the actual value. Such results can provide theoretical support and guidance for actual grouting construction. ere are three reasons for the theoretical diffusion radius by using the formula for penetration diffusion mechanism based on Newton fluid of time-dependent behavior of rheological parameters derived in this paper being larger than the actually measured value in grouting test.
(1) Multiple factors affect the permeability and diffusion radius of Newton fluid of time-dependent behavior of rheological parameters in the grouted medium. For example, the grout is considered to be an unstable liquid due to excessive water separation rate while being prepared, but the Newton fluid is assumed to be a stable grout liquid when the theoretical value is calculated by using the formula for penetration diffusion mechanism based on Newton fluid of time-dependent behavior of rheological parameters; for example, sedimentation and blockage may occur when the grout diffuses in the grouted medium. (2) e gravel (sand) bodies were still different from the isotropic and homogeneous hypothesis after washing three times. (3) Presently, although some research achievements have been made in the time-dependent behavior of Newton fluid, the research on the time-dependent behavior of the grouts is still in the experimental stage. e rheological models are instructive to the research, but their time-dependent behavior law cannot be truly reflected.

Conclusions
(1) On the basis of the basic rheological formula for Newton fluid and time-dependent behavior of its rheological parameters, the rheological formula of Newton fluid of time-dependent behavior is established.
(2) e seepage motion formula for Newton fluid of time-dependent behavior is studied, and the formula for penetration grouting diffusion mechanism based on Newton fluid of time-dependent behavior of rheological parameters is deduced. A specific determination method and application scope of each parameter in the formula are analyzed. (3) e formulas for penetration grouting diffusion mechanism based on Newton fluid of time-dependent behavior of rheological parameters are verified by grouting experiments. e results show that the theoretical value of the grouting diffusion radius calculated by the formula based on Newton fluid of time-dependent behavior of rheological parameters derived in this paper basically coincides with the measured value of the grouting verifying experiment, and the difference is within 15% between the two values, which is much less than the change in 80% between the measured value and the value calculated by Maag theoretical formula. is indicates that the penetration grouting mechanism based on Newton fluid of time-dependent behavior of rheological parameters established in this paper can reflect the penetration grouting mechanism based on Newton fluid of time-dependent behavior of rheological parameters, and the results are closer to the actual value. Such results can provide theoretical support and guidance for the actual grouting construction.

Data Availability
All data generated or analyzed during this study are included in this manuscript.

Conflicts of Interest
e authors declare that they have no known conflicts of financial interests or personal relationships that could influence the work reported in this paper.