J. Cent. South Univ. (2017) 24: 1190-1196

DOI: 10.1007/s11771-017-3522-9

Grouting diffusion of chemical fluid flow in soil with fractal characteristics

ZHOU Zi-long(周子龙)¹, DU Xue-ming(杜雪明)¹, CHEN Zhao(陈钊)², ZHAO Yun-long(赵云龙)¹

1. School of Resources and Safety Engineering, Central South University, Changsha 410083, China;

2. Guangxi Double Elephant Construction Limited Liability Company, Nanning 530029, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2017

Abstract: The chemical fluid property and the capillary structure of soil are important factors that affect grouting diffusion. Ignoring either factor will produce large errors in understanding the inherent laws of the diffusion process. Based on fractal geometry and the constitutive equation of Herschel-Bulkley fluid, an analytical model for Herschel-Bulkley fluid flowing in a porous geo-material with fractal characteristics is derived. The proposed model provides a theoretical basis for grouting design and helps to understand the chemical fluid flow in soil in real environments. The results indicate that the predictions from the proposed model show good consistency with the literature data and application results. Grouting pressure decreases with increasing diffusion distance. Under the condition that the chemical fluid flows the same distance, the grouting pressure undergoes almost no change at first and then decreases nonlinearly with increasing tortuosity dimension. With increasing rheological index, the pressure difference first decreases linearly, then presents a trend of nonlinear decrease, and then decreases linearly again. The pressure difference gradually increases with increasing viscosity and yield stress of the chemical fluid. The decreasing trend of the grouting pressure difference is non-linear and rapid for porosity f >0.4, while there is a linear and slow decrease in pressure difference for high porosity.

Key words: grouting; diffusion; Herschel-Bulkley fluid; porous media; fractal; grouting pressure

1 Introduction

Grouting is a proven but complex method for sealing and stabilizing soils in various engineering applications. The effect of grouting diffusion depends not only on the characteristics and types of fluid material but also on the pore structure of soils.

At present, most researches on grouting diffusion concern the behavior of grouting fluid. Fur types of typical fluid models, the Newton model [1, 2], the power law model [3, 4], the Bingham model [5-8] and the Herschel-Bulkley (H-B) model [9], have been used in existing studies. Among them, the H-B model is an improved model that considers the cohesion force and rheological properties of fluid. With a rheological index n=1, the H-B model is reduced to the Bingham model. With yield stress τ₀=0, the H-B model can be reduced to the power-law model, and it is reduced to the Newton model when n=1 and τ₀=0. Thus, the H-B model is most appropriate and most frequently used for describing the fluid behavior of grouting fluid.

However, few studies have considered the effect of the micro-structure of soil in grouting. In fact, geomaterials, such as soil and rock have large amounts of void space and fissures, which form a pore structure that influences fluid diffusion. Studies of rock and soil mechanics indicate that the pore structure of soil and rock has fractal characteristics [10-12]. Fractal geometry can be used to describe and interpret these types of pore structure and their permeability [13, 14]. For example, TANG et al [15] studied the structure of porous metals based on fractal theory and deduced the relationships between the fractal dimension and the porosity. ALAIMO and ZINGALES [16] deduced the fractional- order transport equation for the laminar flow through fractal porous materials and studied the transport law of a viscous fluid. SEDEH and KHODADADI [17] investigated the effects of fluid flow in porous media with two fractal dimensions and deduced the analytical equation of water pressure in a fractal reservoir. However, little is known regarding the grouting diffusion law of fluid flow in porous media based on the fractal geometry model.

Based on fractal geometry and the constitutive equation for H-B fluid, a fractal model for H-B fluid flowing in a porous geomaterial is derived that accounts for the capillary parameters of the soil medium, the rheological characteristics of chemical fluid and the grouting parameters. The proposed model provides a

theoretical basis for grouting design and helps to understand the chemical fluid flow in soil in engineering applications.

2 Fractal characteristics of porous media

A porous medium consists of a large number of capillaries, which are a set of voids or gaps separated by a solid skeleton, whose cumulative size and tortuosity distributions have been proven to follow the fractal scaling law [18-22]. The scaling relationship between the number of cumulative capillaries and diameter is given by the following equation:

                           (1)

where λ is the diameter of a random capillary and λ_max is the maximum diameter. D_f is the fractal dimension of the capillary size, or the size dimension, with 0f<2 in two dimensions and 0f<3 in three dimensions. The size dimension of the capillary is characterized by the uniformity of pore structure. The smaller the fractal dimension, the more uniform the pore structure. N is the cumulative number of capillaries whose diameter is not less than λ.

Differentiating on both sides of Eq. (1) with respect to λ results in the population of capillaries,

                   (2)

A porous medium consists of a large number of tortuous capillaries. The lengths of capillaries have been proven to follow the fractal scaling law [23-25],

                               (3)

where L_t and L₀ are the actual lengths and characteristic lengths of the capillary, respectively, and D_T is the fractal dimension of capillary tortuosity, or tortuosity dimension, with 1T<2 in two dimensions. The tortuosity dimension of the capillary is characterized by the tortuosity degree of capillaries. The smaller the tortuosity fractal dimension, the closer the capillary to a straightline.

Differentiating on both sides of Eq. (3) with respect to L₀ results in the actual lengths of capillaries, then

                        (4)

3 Grouting diffusion of Herschel-Bulkley fluid in fractal media

Herschel-Bulkley fluid diffuses into the porous medium under the action of the grouting pressure (Fig. 1), taking the horizontal direction as the flow direction of the chemical fluid. Here, the chemical fluid is assumed to be incompressible and homogeneous. The density of the chemical fluid is constant, and the gravity of the chemical fluid is negligible.

A Herschel-Bulkley fluid is a non-Newtonian fluid with yield or shear stresses, and its constitutive equation has the form [26],

                               (5)

where τ_w is the shear stress; τ₀ is the yield stress; μ is the viscosity coefficient; γ is the shear rate of fluids; n is the rheological index. As described before, when the initial yield stress is zero, the Herschel-Bulkley fluid can be reduced to the power-law fluid. When the rheological index of the fluids is one, the Herschel-Bulkley fluid acts like a Bingham fluid. The Herschel-Bulkley fluid can be reduced to the Newton model when n=0 and τ₀=0. These relationships are shown in Fig. 2.

For a Herschel–Bulkley fluid, the volumetric flowrate through a single tortuous capillary can be predicted by[27,28]

                  (6)

Combining Eqs. (5) with (6), we have:

Fig. 1 Schematic of fluid diffusion in grouting:

Fig. 2 Rheology of different kinds of fluids

   (7)

From the stress aspect, the yield stress τ_w is given by [29]

                                  (8)

where p is the grouting pressure.

The total flow rate Q(λ) through a unit volume consists of a set of individual capillaries of flow rate q(λ) that cover the whole range of capillaries whose diameters are within the range between λ_min and λ_max. From Eqs. (2), (4), (6), (7) and (8), we have

             (9)

The total volume of capillary V_P per unit volume includes the total flow volume V_u of individual capillaries with diameters ranging between λ_min and λ_max. Thus, we have

                         (10)

                                (11)

                                   (12)

where V_u and S are the capacity of the capillary and the cross-sectional area of the capillary, respectively.

By interpolating Eqs. (11), (12) into Eq. (10) and integrating, we obtain

             (13)

According to Refs. [30, 31], the porosity f is expressed as

                            (14)

The total flow rate Q(λ) through the pores is equal to the total volume V_P of chemical fluid filling the pores, namely,

                                (15)

Combining Eqs. (9), (13) and (15), we have

   (16)

The maximum capillary diameter λ_max can be predicted by a function of particle diameter d and f [30].

                             (17)

Substituting Eq. (17) into Eq. (16), we have

             (18)

Under the condition of L₀=0 and p₀=1 MPa, and according to Eqs. (16) and (18), we can obtain two different functions for grouting pressure:

                        (19)

            (20)

The total capillary area A_P can be given by

      (21)

The starting pressure (p_q) is a reflection of the fluid viscoplasticity. Only when the driving pressure reaches the starting pressure can the fluid flow in the capillary of the porous medium. When the chemical fluid flows in a porous medium, its flow speed will gradually slow as a result of energy consumption. When the flow rate reaches zero, the pressure is defined as the starting pressure.

According to Eqs. (9) and (21), the velocity of the fluid is

                                  (22)

Hence, the starting pressure of a fluid depends on the yield stress and also on the structure parameters of the capillary, d, f, L₀ and D_T. In this work, p_q is assigned a value of 0.32 MPa for theoretical analysis.

4 Results and discussions

In previous studies, without considering the fractal characteristics of porous media, scholars often assumed that the porous medium is uniform and the capillary is linear. In this case, D_T=1.00 and D_f=2 if the above established model is used. To verify the established model with the existing data, this case is investigated first. As D_T=1.00 and D_f=2 cannot be satisfied simultaneously according to Eq. (19) and Eq. (20), D_f is set to 2 and D_T is set to 1.001, which is slightly larger than 1.0. Figure 3 gives the grouting pressure of this condition. The uppermost lines of Fig. 3 indicate good consistency of the new model predication with data from the literature when D_T=1.001 and D_f=2 [32]. This result indicates that the previous studies that did not consider the fractal characteristics of porous media were actually a special case of the new model established in this study.

Fig. 3 Curve of grouting pressure versus diffusion distance at n=0.3, μ=0.3 Pa·sⁿ, γ=0.07, f=0.3 and τ₀=0.06 Pa

With the model, more details can be deduced. From Fig. 3, it can be seen that the grouting pressure decreases with increasing diffusion distance, and the capillary diameter has a great influence on the grouting pressure. The smaller the capillary diameter is, the deeper the decrease gradient of the grouting pressure with diffusion distance is. The pressure decreases slowly and linearly along the direction of diffusion when the capillary diameter is large. However, as the capillary diameter grows smaller, the pressure decreases and diminishes non-linearly and rapidly along the diffusion direction. When λ=5×10^{-4 m, the grouting pressure reaches the starting pressure at 0.6 m, after which the chemical fluid will not spread. When λ=1×10-3 m, the grouting pressure does not reach the starting pressure until 1.2 m. The shear force of fluid, the surface tension of the fluid and soil medium should be overcome during the diffusion of fluid; therefore, the grouting pressure presents the law of diminishing.}

Figure 4 gives the relationship of the grouting pressure difference versus the rheological index at different D_T and D_f.. It can be seen that the pressure difference decreases linearly with increasing rheological index when the rheological index is less than 0.4. Above this value, the pressure difference presents a trend of nonlinear decrease with increasing rheological index. Then, at still higher values of the rheological index, the pressure difference decreases linearly again. Figure 4 also reveals that when the capillaries of the poro medium have the same size dimension but different tortuosity dimensions, the grouting pressure difference has a clearly different changing law with the rheological index. At the same time, when the capillaries of the porous medium have the same tortuosity dimension D_T, the grouting pressure difference has a different changing law with the rheological index. However, the former is larger than the latter degree of difference.

Fig. 4 Curve of grouting pressure difference versus rheological at μ=0.3 Pa·sⁿ, γ=0.07, τ₀=0.06 Pa, f=0.3 and L₀=1.5 m

Figure 5 gives the relationship of the grouting pressure and the tortuosity dimension of the capillary (D_T). It can be seen that overall, the grouting pressure decreases with increasing tortuosity dimension of capillary D_T. When the tortuosity dimension is less than 1.2, the grouting pressure undergoes almost no change. With further increase of the tortuosity dimension, the grouting pressure decreases nonlinearly. Because the values of the tortuosity dimension reflect the tortuosity of the flow path of the grouting fluid. The larger the tortuosity dimension is, the more tortuous and complex the capillary is. The more tortuous the capillary, the more the resistance existing on the chemical fluid flow, and the greater the energy consumption. Figure 5 also showsthat when the tortuosity dimension is greater than 1.32, the grouting pressures of some capillaries with diameter less than 1.0×10^{-5 m become even smaller than the starting pressure, which indicates that the chemical fluid flow would stop in these capillaries.}

Fig. 5 Curve of grouting pressure versus tortuosity fractal dimension of capillary at n=0.3, μ=0.3 Pa·sⁿ, γ=0.07, τ₀=0.15 Pa, f=0.3 and L₀=1.5 m

Figure 6 shows the relationship of the grouting pressure difference versus viscosity for capillaries with different diameter λ. The lowermost lines again indicate good agreement between the proposed model and the data from Ref. [32]. It can also be seen that the pressure difference gradually becomes greater with increasing viscosity of the chemical fluid because the higher the viscosity of the chemical fluid is, the greater the frictional resistance occurring in the chemical fluid flow is. Figure 6 also reveals that the capillary diameter has great influence on the grouting pressure difference. When the capillary diameter is 6×10^{-3 m, the grouting pressure difference hardly changes with the chemical fluid viscosity. In contrast, when the capillary diameter is 1.8×10-5 m, there is a great change in grouting pressure difference with chemical fluid viscosity.}

Fig. 6 Curve of grouting pressure difference versus viscosity at n=0.3, γ=0.07, τ₀=0.06 Pa, f=0.3 and L₀=1.5 m

Figure 7 shows the curve of grouting pressure difference versus the yield stress at different capillary diameters. The lowermost lines also show good consistency between the proposed model and the data from Ref. [33]. It can also be seen that the grouting pressure difference increases with increasing yield stress because when the yield stress is higher, more energy is consumed. Figure 7 also reveals that the grouting pressure difference depends not only on the yield stress but also on the capillary diameter in certain cases. The grouting pressure difference increases nonlinearly and rapidly with increasing yield stress τ₀ at a smaller capillary diameter. However, as the capillary diameter increases, the grouting pressure difference increases linearly and slowly.

Figure 8 presents the relationship of the grouting pressure difference and the porosity of the soil. The lowermost lines again indicate good consistency of the proposed model with the data from Ref. [31]. It can also be seen that the grouting pressure difference decreases with increasing porosity. The pressure difference is infinity when the porosity tends to zero, and it is impossible for chemical fluid to flow in this state. The pressure difference decreases nonlinearly and rapidly with increasing porosity when the porosity is less than 0.4 and then decreases very slowly and linearly with the further increase of the porosity. This behavior occurs because when the porosity increases, the available flow paths for the chemical fluid are increased, and the resistance is decreased. Figure 8 also reveals that the diameter of soil particles has a great influence on the grouting pressure difference. The larger the particle, the smaller the grouting pressure difference, as the capillary diameter increases with increasing particle diameter.

Fig. 7 Curve of grouting pressure difference versus yield stress at n=0.3, μ=0.3 Pa·sⁿ, γ=0.07, f=0.3 and L₀=1.5 m

Fig. 8 Curve of grouting pressure difference versus porosity at n=0.3, γ=0.07, τ₀=0.06 Pa, μ=0.3 Pa·sⁿ and L₀=1.5 m

5 Conclusions

1) A new model was developed to investigate the diffusion law of Herschel-Bulkley fluid in soils with fractal geometry. The parameters of pore structure (size of capillary, tortuosity dimension, porosity, particle diameter of the soil) and fluid (yield stress, viscosity, shear stress and rheological index) were all found to influence the diffusion law of grouting. In particular, the structure parameters of the soil medium, which were previously ignored, are very important factors in determining the grouting diffusion of fluid at the microscopic level.

2) The result indicates that the grouting pressure has no obvious change with the increase of the tortousity fractal dimension of capillary (D_T) when D_T is not more than 1.2, and nonlinearly decreases rapidly with the increase of D_T when D_T is greater than 1.2.

3) The result indicates that the distance of diffusion has little changed with the increase of the rheological index (n) at the low rheological index, and increases nonlinearly with the increase of the rheological index at the high rheological index.

4) It should be noted that the proposed model is based on the assumption that the surface roughness of the flow path can be ignored. However, the flow path in natural soil might not be smooth, which may also influence the diffusion law of fluid. To more deeply understand the fluid flow in soil in real environments, further study in this area is in progress.

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(Edited by DENG Lü-xiang)

Cite this article as: ZHOU Zi-long, DU Xue-ming, CHEN Zhao, ZHAO Yun-long. Grouting diffusion of chemical fluid flow in soil with fractal characteristics [J]. Journal of Central South University, 2017, 24(5): 1190-1196. DOI: 10.1007/s11771-017-3522-9.

Foundation item: Project(2015CB060200) supported by the National Basic Research Program of China; Project supported by the R-D program of Gangxi Province of China; Project(201622ts093) supported by the Fundamental Research Funds for the Central Universities, China

Received date: 2015-08-27; Accepted date: 2016-03-17

Corresponding author: DU Xue-ming, PhD; Tel: +86-18603735672; E-mail: 2007-dxm@163.com