The analytical solution of the grouting migration height for the post-grouted drilled shaft based on the Herschel-Bulkley model

: The traditional bored pile technology has some arduous problems, such as the sediment at the pile tip, the mud skin along the pile shaft, and the stress release due to borehole construction. The post-grouted technology at the pile tip of bored pile has emerged because of demand. The grouting migration height (GMH) is of great significance to the strengthen and reinforcement of the pile foundation. This paper derives the calculation formula of the GMH based on the theory of the column hole expansion and Herschel-Bulkley model. The influence of relevant parameters on the GMH is discussed. Aiming at the problem of the grouting migration along the pile shaft in layered soils, the iterative calculation method of the GMH is proposed. The correctness of the GMH is verified by an engineering case, which can guide the engineering practice. The result shows that the GMH increases with the increase of the grouting pressure, the pile diameter and the thickness of the mud skin, and the grouting pressure is positively correlated with the GMH. The GMH decreases with the increase of the buried depth, the consistency coefficient and the rheological index. On this basis, the correctness of the GMH is verified by an engineering case.


Introduction
The bored cast-in-place concrete piles have many advantages, such as the high bearing capacity of the single pile, the flexible selection for the pile length and diameter, and the wide adaptability to various soil conditions [1]. Thus, it has been widely used in the pile foundation important role in the overall reinforcement for the soil of the pile shaft and pile tip of the bored cast-in-place concrete pile. Therefore, the grouting split theory is the crux of determining the grouting migration height (GMH). The research of the grouting split theory mainly includes the split mechanism and the propagation mode. The split mechanism is mainly concerned with the starting split conditions and the threshold of split pressure [28,29]. The empirical solutions are obtained by the model test or field test [30][31][32], and the analytical solutions are deduced based on the tension failure, shear failure and other failure criteria to obtain theoretical models [33,34]. For the propagation paths characteristics of the grouting split, Li et al. [35], Zou et al. [36] and Zhang et al. [37] adopt constitutive models of the Newtonian fluid, the Bingham fluid or the Power-law fluid to obtain the relationship between the diffusion radius of the grouting split and the grouting pressure respectively. Because the propagation paths of split have complex dynamic properties and the research is highly complex, tentative studies are carried out using the numerical simulation and model tests. Yang [38], Murdoch [39] and Bezuijen [40] carried out laboratory simulation experiments of the grouting split path propagation using the grouting material separation and the scanning electron microscopy. Many scholars simulate the grouting split process based on the particle flow method and the discrete element software [41][42][43][44].
At present, the GMH is mostly determined by experience and differs greatly from the actual project frequently, which hinders the development of the post-grouted technology seriously.
Therefore, based on the theory of the cylindrical cavity expansion, the Herschel-Bulkley model and the basic equation of the uniform flow, an equation for calculating the GMH along the mud skin of the pile shaft is derived in this paper. The influences of grouting parameters, the grouting pressure, the pile length, the pile diameter and the mud thickness on the GMH are analyzed emphatically. Furthermore, the proposed method is validated through the comparison with the observation data in an engineering case. This study provides the theoretical foundation for further study and the application of the post-grouted technology.

The basic hypothesis of the PSDS
The basic assumptions of the grouting split in soils are as follows: (1) The grouting material conforms to the Herschel-Bulkley model and the flow pattern of the grouting material remains unchanged during grouting. (2) The grouting material is an incompressible homogeneous isotropic fluid without considering the time-varying effect of the grouting material. (3) The fracture aperture and the velocity of the grouting material are small. The flow pattern of grouting material is the laminar flow. (4) When the grouting material flows in the aperture, the no-slip condition of the wall is established, i.e., the velocity of the grouting flow on the upper and lower surface of the aperture is 0. (5) The treated soil is considered as an isotropic homogeneous material. (6) It is assumed that the consistency coefficient of the grouting material is constant in the course of the movement. (7) The pile shaft is a regular cylindrical surface regardless of the roughness of the pile shaft. (8) The aperture width of the grouting split is + , where is the thickness of the mud skin, and is the compression of the soil and mud skin at the pile shaft in section . (9) The grouting material has good grouting ability and no blockage occurs, i.e., the effect of the pressure filtration is not considered.

Theoretic derivation
(1) Theoretical deduction of the soil at the pile shaft In the process of the grouting split, the lateral displacement of the soil at the pile shaft is produced. According to the theory of the cylindrical cavity expansion, the calculation of the lateral displacement can be simplified as the axisymmetric plane problem, whose calculation model is shown in Fig. 1-(a) and Fig. 1- (b). There is no shear stress on the plane, i.e. = = 0.
Furthermore, the stress change in the direction is not considered. According to the force balance in the direction of radius , Equation (1) can be obtained.
where is the radial stress, is the tangential stress, is the radius of the calculation point and is the shear stress, which denotes the normal direction of the stress action plane and denotes the direction of the stress action.
By simplifying Equation (1) and omitting higher-order infinitesimal quantities, we can obtain    According to the geometric equation, we have where is the radial strain, is the tangential strain, and is the radial displacement.
According to the generalized Hooke's law, we obtain where is the poisson ratio and is the modulus of the elasticity.
The boundary conditions are where is the radius of the inner bore, is the grouting pressure, and 0 is the initial stress of the soil, which can be expressed as follows Combined with Equation (2) to Equation (6), the displacement of the soil under the elastic state can be obtained.
where is the lateral displacement of the soil, and is the shear modulus, which can be given as follows = 2(1 + ) ⁄ .
Assuming = = 0 + , it can be approximated to obtain ≈ 0 because of ≪ 0 in Equation (7). The displacement of the soil at the pile shaft can be simplified as follows where 0 is the radius of the bored cast-in-place concrete pile.
(2) The deduction theory of the GMH At present, the Bingham model and the Power-law fluid model are mostly adopted to describe the rheology of the cement slurry at home and abroad, which cannot describe the grouting material with the yield value and the pseudo-plasticity [45,46]. The Herschel-Bulkley model is a three-parameter rheological model with static shear stress [47], whose constitutive equation is where is the shear stress, 0 is the yield stress, is the consistency coefficient, is the shear rate, and is the rheological index.
The Herschel-Bulkley model can express the characteristics of the Newton model, the Bingham model and the Power-law fluid model, which are shown in Table 1.
where is the flow rate of the grouting material.
The computational model is shown in Fig. 2. When the grouting material penetrates, the driving force and the resistance force will be equal. What's more, the grouting pressure will be zero when the GMH reaches its maximum height. According to the above conditions, Equations (11) and (12) are valid, which is shown as follows  ∆ · · 2 ( 0 + ) = 2 · 2 ( 0 + ) · ℎ, and ∆ · ( + ) · 2 ( 0 + + ) = 2 · 2 ( 0 + + ) · ℎ, where ∆ is the difference of the grouting pressure, and is the shear stress on the surface of the grouting material, whose flow direction is opposite to the velocity direction.
Simplifying Equations (11) and (12), we obtain where ℎ is the GMH, and is the shear stress at any point.
By combining Equations (9) and (10), the variables are separated by , and then it is integrated along the direction. The results are as follows where is a constant.
The boundary conditions are as following Substituting Equation (17) into Equation (15), we obtain To facilitate the derivation and the calculation of the following deduction, Equation (18) is converted into an integral form, Combining with Equations (13) and (14), we have Substituting Equation (20) into Equation (19), the integral variable changes from to , and we obtain We can derive from Equation (21), Substituting Equations (13) and (14) into Equation (22), we have The flow of the unit time is The in Equation (24) can be determined by where is the length of the aperture.

(27)
Substituting Equation (8) If the influence of the grouting material gravity is considered, Equation (29) can be obtained The difference of the grouting pressure ∆ can be calculated by Equation (29).
(3) The Determination and Solution of the GMH According to the grouting split theory, only if the grouting pressure is greater than the threshold of the split pressure, the rock and soil can be split and grouted. When the grouting pressure located at ℎ is greater than the horizontal static earth pressure 0 , the grouting material can overcome the resistance force to split the rock and soil continuously due to the fact that the cohesion of the mud skin on the pile shaft is very small. Therefore, the GMH increases continuously. When the grouting pressure is equal to the horizontal static earth pressure, the grouting material cannot overcome the resistance force of the rock and soil. At the moment, the GMH reaches to its maximum.

The iterative calculation of the GMH in layered soils
In Section 2, the formula is derived on the basis that the soil at the pile shaft is homogeneous.
In practical engineering, however, the rock and soil of the PSDS are often complex and can be simplified to multi-layered soil. The difference of mechanical properties of soils will result in the difference of the thickness of the mud skin, which will eventually affect the GMH.
The theoretical model is established to study this effect, as shown in Fig. 3. The soil along the pile shaft can be divided into layers, and the grouting material along the pile shaft is divided into layers according to the soil layer. The force analysis of each calculation element is also carried out. It is assumed that the grouting pressure is equal to the junctions of different calculation elements. The bottom of layer is equal to the top of layer + 1, i.e., where is the bottom of layer and +1 is the top of layer + 1. The grouting pressure at the bottom of the layer soil is equal to that at the end of the pile, i.e., = .
n-1 According to the grouting split theory, when the grouting pressure equals the horizontal static earth pressure, the threshold of the split pressure is where 0 is the coefficient of the static earth pressure, is the threshold of the split pressure at the interface of the pile-soil, and is the weighted effective weight of the soil layer above the According to Equation (29), the grouting pressure at the top of section can be calculated by Equation (32).
In layered soils, the calculation method for the GMH is as follows, which is shown in Fig. 4.
(1) The soil along the pile shaft is divided into layers, and the mud skin is divided into layers according to corresponding soil layers, as shown in Fig. 3. The thickness of soil layers satisfies following conditions, [ℎ ] ≤ ⁄ . The interface of soil layers and the groundwater surface should be regarded as a stratified surface, which is shown in Fig. 3. The thickness of the calculation element in each soil layer is less than or equal to ℎ .
(2) According to the condition of the grouting continuity, the grouting pressure at the bottom of section is = . Assuming that the initial grouting hydraulic pressure on the top surface of the lowest subsection is 0 , the 1 can be calculated by Equation (32).

The new is calculated by
Step (2)  (4) According to the condition of the grouting continuity, the grouting pressure at the bottom of calculation element − 1 is −1 = .
(5) When the grouting pressure at the top of calculation element is greater than the threshold of split pressure at the pile-soil interface of calculation element , the next calculation element continues to calculate. If the grouting pressure +1 at the top of calculation element + 1 is greater than or equal to the grouting pressure at the bottom of calculation element , i.e., ≤ , the calculation terminates. The is assigned to , and the GMH of calculation element , denoted as ℎ , can be calculated by Equation (32).

Parameter analysis
To study the influences of different parameters on the GMH, Equation (30)

Yes
No Yes Assign +1 to The grouting pressure of ith calculation +1 is calculated by Equation The soil layer of is calculated The grouting pressure at the bottom of the nth layer satisfies = End Start modulus of soils are shown in Table 2.

The relationship between the rheological index and the GMH
The grouting parameters are supposed that the grouting pressure is 4 MPa, the depth of the pile shaft is 40 m, the radius of the pile 0 is 0.5 m, the thickness of the mud skin along the pile shaft is 0.01 m, the volume of the grouting material is 0.1 L/s, the consistency coefficient of the grouting material is 0.5 kPa • s, and the yield stress of the grouting material 0 is 2 Pa. The relationships between the GMH and rheological indices are shown in Fig. 5. Ruan [48] studied the basic properties of the grouting material by experiments, whose results showed that the rheological index corresponding to the water cement ratio of 0.5 to 0.7 was less than 0.1 in practical engineering. It shows that the GMH decreases with the rheological index gradually in Fig. 5. The category of the soil along the pile shaft has a great influence on the GMH. The GMH of the clay is the largest, the GMH of the silt is secondly, and the GMH of the gravel is the smallest, which is possible to show that the GMH has a great relationship with the seepage coefficient of the rock and soil.

The relationship between the consistency coefficient and the GMH
The grouting parameters are assumed that the grouting pressure is 4 MPa, the depth of the pile shaft is 40 m, the radius of the pile 0 is 0.5 m, the thickness of the mud skin along is 0.01 m, the volume of the grouting material is 0.1 L/s, the rheological index is 0.1, and the yield stress of the grouting material 0 is 2 Pa. The relationship curves between the GMH and the consistency coefficient of the grouting material are shown in Fig. 6. The GMH decreases with the consistency coefficient gradually, which indicates that the GMH is closely related to the consistency coefficient of the grouting material. When the consistency coefficient of the grouting material is less than 0.6 kPa • s, the GMH varies significantly.
When the consistency coefficient of the grouting material is greater than 0.6 kPa • s, the change of the GMH is small.

The relationship between the grouting pressure and the GMH
The grouting parameters are supposed that the depth of the pile shaft is 40 m, the radius of the pile 0 is 0.5 m, the thickness of the mud skin along the pile shaft is 0.01 m, the volume of the grouting material is 0.1 L/s, the consistency coefficient of the grouting material is 0.5 kPa • s, the rheological index is 0.1, and the yield stress of the grouting material 0 is 2 Pa.
The relationship between the grouting pressure and the GMH is shown in Fig. 7. The GMH increases with the grouting pressure, which shows the positive linear relationship between them.
Under certain conditions, the GMH is closely related to the permeability coefficient of the rock and soil.

The relationship between the depth of the pile shaft and the GMH
The grouting parameters are supposed that the grouting pressure is 4 MPa, the radius of the pile 0 is 0.5 m, the thickness of the mud skin along the pile shaft is 0.01 m, the volume of the grouting material is 0.1 L/s, the consistency coefficient of the grouting material is 0.5 kPa • s, the rheological index is 0.1, and the yield stress of the grouting material 0 is 2 Pa.
The relationship between the GMH and the depth of the pile shaft is shown in Fig. 8. The GMH decreases with the depth of the pile shaft , which is the negative correlation between them.
When the pile shaft is in the gravel layer, the influence of the depth of the pile shaft on the GMH is more obvious. When the pile body is in the clay, silt and sand layer, the depth of the pile shaft has little effect on the GMH.

The relationship between the pile diameter and the GMH
The grouting parameters are supposed that the grouting pressure is 4 MPa, the depth of the pile shaft is 40 m, the thickness of the mud skin along the pile shaft is 0.01 m, the volume of the grouting material is 0.1 L/s, the consistency coefficient of the grouting material is Pa. The relationship between the GMH and the diameter of the pile 0 is shown in Fig. 9. The GMH decreases with the diameter of the pile, which indicates that the diameter of the pile has a greater influence on the GMH from Fig. 9.   Fig. 9. The relationship curve between the GMH and the diameter of the pile.

The relationship between the thickness of the mud skin and the GMH
The grouting parameters are supposed that the grouting pressure is 4 MPa, the depth of the pile shaft is 40 m, the radius of the pile 0 is 0.5 m, the volume of grouting material is 0.1 L/s, the consistency coefficient of the grouting material is 0.5 kPa • s, the rheological index is 0.1, and the yield stress of the grouting material 0 is 2 Pa. The relationship between the GMH and the thickness of the mud skin is shown in Fig. 10. The GMH increases with the thickness of the mud skin rapidly. For the bored cast-in-place concrete pile with the thick mud skin, the mud skin and soil can be strengthened by the post-grouted technology. The thicker the mud skin is, the better the grouting effect is. It can be seen that the GMH of the gravel is always less than that of the sand, silt and clay  Fig. 5 to Fig. 10. The reason may be that the permeability coefficient of the gravel is bigger, and the grouting pressure is easy to diffuse, which results in the smaller GMH. Nevertheless, the GMH of the clay is always higher than that of the sand, silt and clay. From this point of view, it shows that the clay is more conducive to the post-grouted technology.

The analysis of an engineering case
Referring to an engineering example in practical [49], the distribution of the soil layer along the pile shaft is shown in Table 3. The diameter and length of the test pile are 1.5 m and 41 m, respectively. The bearing stratum at the pile tip is the gravel, and the length of the pile is 8 m, which enters gravel stratum. The water-cement ratio of the grouting material is / = 0.5 − 0.6, and the average grouting pressure is about 4 MPa. When the foundation is excavated to 11m, the hardened cement ring with the thickness of 10-50 mm is found along the pile shaft, which indicates that the GMH along the pile shaft is about 30 m.
The calculation method deduced in this paper is used to calculate and analyze the GMH in this practical engineering. The grouting pressure is 4 MPa, the volume of the grouting material is 0.1 L/s, the thickness of the mud skin is 0.01 m, the consistency coefficient is 0.2 kPa • s and the rheological index is 0.05. According to Equation (32), the GMH of the actual project is 27.6 m, which is close to the observative value of 30 m. The research results show that the conclusions derived in this paper are valid. 6. Discussion

Hypothetical conditions
The Bingham model, the Carson model, the Power-law model and other two-parameter rheological models have better accuracy at high and medium shear rates. Nevertheless, the models Because the mud skin layer is located between the pile and soil, its strength is lower than the rock and soil around the pile. Therefore, the grouting material squeezes the soil along the pile shaft under the action of the pressure and causes the interspace around the pile diameter. The grouting material will split and move upward along the weak surface, which will compact the mud skin and the soil around the pile shaft, or even destroy the mud structure. The grouting material fills voids after compaction and forms cement stones with the high strength after the consolidation, which enlarges the diameter of the pile. The movement of the grouting material in the weak layer can be summarized as follows: splitting the weak surface → the grouting material rising → compaction → re-splitting → re-rising → re-compaction. The above process is not categorized by the chronological order, and these processes may occur simultaneously in fact.

Further correction of the GMH
The GMH is related to the thickness and strength of the weak layer, the grouting ability, the grouting pressure and the volume of the grouting material. The grouting material is easy to diffuse in the soil layer with good grouting ability, and the grouting pressure is low, which causes the smaller GMH. The grouting is difficult to diffuse in the soil layer with poor grouting ability, and the grouting pressure is high, which causes the bigger GMH. The thicker the thickness of the weak layer is, the smaller the resistance force is, and the higher the GMH is. The pore water pressure increases when the soil is subjected to the grouting pressure. When the pore water pressure dissipates after grouting, the strength of the soil around the pile shaft increases due to compaction.
Considering the actual situation of the grouting pressure at the pile tip, the GMH should be revised to make the calculation more exact. The grouting pressure at the pile tip is generally less than that measured on the ground. Therefore, considering the loss of the grouting pressure during transportation, the grouting pressure measured on the ground should be re-estimated where is the grouting pressure at the pile tip, is grouting pressure measured on the ground, and 1 is the correction coefficient of the grouting pressure considering the loss of the grouting pressure during transportation.
The parameter 1 can be obtained by the ratio of the grouting pressure measured at the pile tip and the grouting pressure measured on the ground, whose value is in the range of 0.3-0.95.
As to the bored cast-in-place concrete pile in the practical engineering, the profile of the pile shaft is not regular cylindrical surfaces. The flow distance of the grouting material along the profile of the pile shaft will increase significantly, which will reduce the GMH. Therefore, the GMH calculated by Equations (30) and (32) should be reduced. The roughness coefficient of the pile shaft is introduced, and the GMH is as follows where ℎ ′ is the GMH considering the concave and convex of the pile shaft, is the roughness coefficient of the pile shaft and it satisfies the condition ≥ 1, is the total length of the profile of the pile shaft, and is the vertical height of the pile.

Suggestions for the further study
(1) Studying the mechanism of the grouting-soil interaction. The action mode of the grouting-soil is an important factor affecting the soil reinforcement and an important prerequisite for analyzing the reinforcement mechanism. Nonetheless, the distinction between the action modes of the grouting-soil interaction is vague or insufficient. The grouting method can be divided into the seepage grouting, the compaction grouting and the splitting grouting based on the mechanism of the grouting material diffusion. The above classification is based on the relationship between the grouting-soil interaction under ideal conditions. However, two or three grouting methods may exist simultaneously in the actual grouting engineering. Therefore, the conditions for the existence of the grouting-soil interaction and its transformation process should be understood and mastered in detail.
(2) Establishing a reasonable model to simulate the mechanism of the grouting material at the pile tip. The grouting process involves the interaction of the pile, the soil and the grouting material.
The stress and deformation between the pile, the soil and the grouting material are non-linear.
What's more, the heterogeneity of the material is very prominent and the boundary conditions are complex. The facts mentioned above make the traditional method to simulate the grouting at pile tip very different from the realities and lose the theoretical guiding significance. Therefore, it is necessary to find a new and reasonable mathematical model to simulate the grouting at the pile tip.
(3) Study on the bearing behavior of the PGDS. After grouting at the end of the pile, the resistance forces of the pile shaft and the pile tip have varying degrees of improvement. How to analyze and establish the bearing characteristics of the bored cast-in-place cement pile after grouting and the distribution law of the side friction and end resistance of the pile through the simple and reliable test and the relevant data of the grouting process. There are many methods to evaluate the grouting effect, and most of them are judged qualitatively, which are lack of the unified criteria for quantification. The quantitative evaluation criteria, which are reliable and identical, and meet different engineering requirements, should be further studied.

Conclusion
Based on the Herschel-Bulkley model and the basic equation of uniform flow, the analytical solutions of the grouting material diffusion along the aperture between the pile and soil are derived. Meanwhile, the parameters, such as the rheological index of the grouting material, the consistency coefficient of the grouting material, the length of the pile, the diameter of the pile, the thickness of the mud skin and the grouting pressure, are analyzed. The analyses shows that these parameters have significant influences on the calculated result of the GMH. In practical engineering, therefore, these factors should be considered comprehensively for calculating the GMH to reduce the hidden risk of engineering. Besides, aiming at the GMH in layered soil, the iterative method for calculating the GMH is proposed through the study of the soil stratification around the pile shaft. An engineering example verifies the applicability of this method.
Author Contribution：In the process of this paper, each author is responsible for the following