1 Introduction

Piezocone penetration test (CPTu) is the most commonly used in-situ testing method in marine site investigation. In practice, the standard penetration rate of CPTu is (20 ± 5) mm s⁻¹. At this penetration rate, sandy soils exhibit fully drained behavior and cohesive soils exhibit fully undrained behavior. The effective internal friction angle of sandy soils and the undrained shear strength of clayey soils can be accurately measured by CPTu, but for soils of medium grain size with properties between clay and sand, the soil strength is often measured by regional empirical methods.

The field penetration tests conducted in multiple regions have shown that the penetration rate significantly affects the results of CPTu on intermediate grain size soils (Chung et al., 2006; Salgado et al., 2010; Poulsen et al., 2013; Krage et al., 2014). The effect of penetration rates on CPTu results is manifested in two aspects: 1) the penetration resistance decreases with the increase of penetration rate due to drainage behavior; 2) the penetration resistance increases with the increase of penetration rate due to the viscous effect. Although negligible in both sand and clay, the effect of penetration rate is significant in intermediate grain size soils. The confidence level of geotechnical parameter estimation can be very low under the partial drainage conditions (DeJong and Randolph, 2012; DeJong et al., 2012). Disregarding the effect of penetration rates may distort the inversion results (Suzuki and Lehane, 2015). These problems all call for the study on the effect of penetration rates on CPTu under partial drainage conditions.

Quantitative studies have been carried out by centrifuge tests and calibration tank tests to reveal the influencing factors and the mechanism of the effect of penetration rates on CPTu in intermediate soils. Centrifuge tests have shown that penetration rates are affected by partial drainage behaviors and cohesive effects to different degrees in different soils (Finnie and Randolph, 1994; Lunne and Andersen, 2007; Lehane et al., 2009). As for the dilatancy silt, it is the cohesive effect that causes more significant impact (Silva et al., 2006).

To investigate the effect of penetration rates through numerical simulations, Silva (2005) has modeled the cylindrical cavity expansion with variable penetration rates by CAMFE FE code and found that the test results did not coincide with those of the centrifuge test performed by Randolph and Hope (2004). Jaeger (2012) has analyzed the cylindrical cavity expansion by FLAC3D software with the finite difference method and the results are in agreement with those obtained by Sliva (2005). The one-dimensional axisymmetric geometry adopted for modeling the cylindrical cavity expansion may not be suitable for simulating the actual vertical and horizontal drainage around the cone tip. Yi et al. (2012) and Mahmoodzadeh et al. (2014) have used the large deformation finite element analysis method and effective stress analysis to study partial drainage effect on normally consolidated clay. The results were in good agreement with those obtained by the centrifuge test conducted by Mahmoodzadeh and Randolph (2014). Two-phase material point method, characterized by a contact algorithm, was used to simulate CPTu test under partial drainage conditions, in which soil behavior is described with the MCC model (Ceccato et al., 2016). This method also takes into account the large deformation of soil and the dissipation of pore pressure during penetration, and the results agree well with experimental data. Monforte et al. (2018) applied the particle finite element method, which is capable of tackling large deformation problems, to the parametric analysis of the CPTU test in the material described in a Cam-clay hyperelastic model. Spherical cavity expansion has also been modeled in numerous studies with the finite element analysis software PLAIXS (Xu and Lehane, 2008; Lehane et al., 2009; Tolooiyan and Gavin, 2011). It has been found that after reconsolidation, the dissipation of excess pore pressure and the increase in effective radial stresses start earlier in the spherical scenario than in the cylindrical scenario (Mo et al., 2020). The comparison between the calculated ultimate cavity expansion pressure and the measured value confirmed that spherical cavity expansion analysis was more suitable for understanding the effect of local consolidation after drainage during CPTu (Yu, 2000; Cao et al., 2001). Coupled with the consolidation analysis, spherical cavity expansion analysis could be performed to investigate the effect of variable penetration rates on CPTu (Suzuki, 2010).

Given that the permeability and mechanical properties of the widely distributed silt layers in the Yellow River Delta are significantly different from that of typical clay and sand layers, there is still a lack of theoretical support for the penetration rate effect of the Yellow River Delta silt. The understanding of the effect of penetration rates on CPTu of silt soils has not been theoretically supported. At present, the CPTu on sediments in the Yellow River Delta region is mainly based on regional empirical coefficients (Guo and Prakash, 1999; Chu, 2017; Dong et al., 2018). A large amount of sediment accumulates at the Yellow River Delta region with over 90% of silt (Feng et al., 2002). Because of spatially non-uniform soil strength (Zhou et al., 2006), submarine geohazards such as landslides (Zhao et al., 2020) and seabed liquefaction (Yang et al., 2019; Song et al., 2020) frequently occur, causing numerous accidents including the suspension or fracture of submarine pipelines and the tilt of offshore platforms.

To study the effect of penetration rates on CPTu in the typical Yellow River Delta silt under partial drainage conditions and provide a reliable evaluation method for field CPTu tests, spherical cavity expansion was modeled with PLAXIS to simulate the penetration process of CPTu according to the spherical cavity expansion theory. Indoor 1 g model tests with different penetration rates were performed for comparison with numerical simulation results. The evaluation method of CPTu test results was proposed and the penetration and consolidation properties of silt under partial drainage conditions were obtained in this paper, which provided some valuable support for marine engineering construction in the Yellow River Delta.

2 Numerical Analysis of the Piezocone Penetration Resistance Under Variable Penetration Rates 2.1 Design and Numerical Calculation

Based on the spherical cavity expansion theory, the piezocone penetration modeling is simplified to a spherical cavity expansion process. The generation and dissipation characteristics of excess water pressure due to spherical cavity expansion can be predicted by using coupled consolidation calculations in PLAXIS software, which is designed to discuss the factors affecting the cone tip resistance under different drainage conditions. The numerical simulations in this paper are based on a nonlinear elastic-plastic hardening intrinsic model with spherical cavity expansion modeling. The method is similar to those proposed by Xu (2007), Xu and Lehane (2008), and Suzuki (2015). The CPTu cone tip resistance under different drainage conditions is measured.

An axisymmetric soil domain (about 1200 elements) is created in the X-Y plane as shown in Fig.1 (effect of mesh coarseness has been shown to be not sensitive to the results at this sufficiently fine mesh). A spherical cavity of 0.1 m in radius (a₀) without lining or interface is set at a depth of 12 m. Spherical cavities of 0.7 m and 2.1 m in radius are also set to partition the axisymmetric soil domain into four class groups. The boundary conditions are completely fixed at the bottom, completely free at the top, and partially free at the left and right boundaries in the vertical direction only. Material properties are imposed on the cavity and the surrounding soil body, with a linear elastic model for the cavity and a hardening soil model (HS model) for the surrounding soil body. A triangular cell mesh is generated and further refined near the cavity where large strains are expected to occur.

The water level line is set at the upper boundary of the soil model and the initial effective stress is generated according to the K₀ = 1− sinφ' consolidation procedure. According to the axisymmetric unit, seven stress points (A – G) and seven nodes (H – N) are selected on the cavity wall and the nodes close to the cavity wall, respectively. Stress expansion curves and pore pressure curves are generated based on the average values of radial displacements at the seven nodes and the average values of radial stresses and excess pore water pressure at the seven stress points.

At the first stage, the initial stresses are calculated with the guarantee that the ground stresses are balanced. At the subsequent stages, the volume strain in the 'Calculation' program (i.e., 10% increase in each stage) is applied to the spherical cavity, resulting in gradual expansion of the spherical cavity. This large deformation problem is solved in PLAXIS by using the update grid option. The cavity must be expanded to at least twice in radius than the initial cavity. Each calculation stage (always starting from the initial stress condition) defines a different consolidation time, representing a different rate of cavity expansion.

The main model adopted in this paper is the HS model. The relevant input parameters are shown in Table 1. In contrast to the Mohr-Coulomb model, the HS model has advanced features to account for strain-stress dependent Young's modulus. Full description of the HS model is provided by PLAXIS reference manual (Brinkgreve et al., 2012).

In PLAXIS coupled consolidation analysis, the generation and dissipation of excess static pore pressure during cavity expansion can be simulated for a given consolidation time. Thus, the equation for calculating the radial expansion velocity of the cavity is:

$ {v_{{\text{r}}\_{\text{CE}}}} = \frac{{\Delta {r_{{\text{CE}}}}}}{{\Delta {t_{{\text{CE}}}}}}, $

(1)

where ∆r_CE is the radial displacement of the cavity wall; ∆t_CE is the expansion time corresponding to the displacement. In this study, the cavity expanding pressure (σ₁) at a/a₀ = 2 is used as an approximation of the limit pressure. The relationship between the limit pressure (p_limit) and the steady state cone end resistance (q_c) is expressed as follows (Silva et al., 2006; Suzuki, 2015):

$ {q_{\text{c}}} = {p_{{\text{limit}}}} + \sqrt 3 \left({{p_{{\text{limit}}}} - u} \right)\tan \delta, $

(2)

where the interface friction angle δ was assumed as same as the constant volume friction angle φ_cv of the soils.

The cavity expansion at the velocity of 0.0152, 0.152, 1.52, 15.2, 152 and 1520 mm s⁻¹ (over six orders of magnitude) were analyzed by changing the given consolidation time in each phase. A total of 27 cases were analyzed in this paper, which were divided into three groups. The basic parameters used in all analytical cases are listed in Table 2, and the variables of all designed cases are summarized in Table 3.

2.2 Numerical Analysis Results

The total radial stress, excess water pressure and radial displacement of the cavity wall were taken as the average values of all the 7 stress points and 7 nodes. The cavity expansion pore pressure curve and stress curve were obtained to demonstrate the cavity expansion process. The typical results of these two curves are shown in Fig.2.

As a result of varying the cavity expansion velocity from 0.0152 to 1520 mm s⁻¹, the normalized cavity expanding pressures and the normalized excess pore pressures can clearly indicate whether the soil is drained or not. The ∆u/σ'_v0 values decrease rapidly with the decrease of v_{r_CE} from 1520 mm s⁻¹ to 152 mm s⁻¹, then decrease more dramatically when v_{r_CE} furtherly reduces to 152 mm s⁻¹ and 1.52 mm s⁻¹. Significant increases in the (σ₁ − σ_v0)/σ'_v0 values also occurred at v_{r_CE} of 152 – 1.52 mm s⁻¹, indicating that the cavity expansions were likely to be under the partially drained condition in these cases. Fully drained conditions were also confirmed at v_{r_CE} below 0.0152 mm s⁻¹.

2.3 Effect of Key Parameters 2.3.1 Effect of E₅₀ and φ'

Studies investigating the effect of penetration rates under the partially drained condition often use the normalized velocity (V) to unify the data (Finnie and Randolph, 1994), which is defined as

$ {V_{\text{v}}} = \frac{{vd}}{{{c_{\text{v}}}}}, $

(3)

where v is the penetration rate, d is the piezocone diameter, and c_v is the vertical consolidation factor.

Silva et al. (2006) and Lehane et al. (2009) suggested to use a horizontal coefficient of consolidation (c_h), instead of c_v, for 1-D consolidation analysis on a laboratory sample. For the HS model, c_h may be calculated as:

$ {c_{\text{h}}} = \frac{{{k_{{\text{h}}0}}{E_{50}}}}{{{\gamma _{\text{w}}}}}, $

(4)

where E₅₀ was secant stiffness at the initial stress condition and γ_w is the unit weight of the water, then the normalized velocity (V_{h_CE}) can be defined as (Suzuki, 2015):

$ {V_{{\text{h}}\_{\text{CE}}}} = \frac{{{v_{{\text{r}}\_{\text{CE}}}}{d_{{\text{CE}}}}{\gamma _{\text{w}}}}}{{{k_{{\text{h}}0}}{E_{50}}}}, $

(5)

where d_CE are taken as the final cavity diameter (two times of the final cavity radius); k_h0 is assumed to be constant during expansion.

Assuming that the cavity expansion limit pressure is equal to the total radial stress of the cavity, the theoretical cone tip resistance value is calculated according to Eq. (2) and normalized as follows:

$ Q = \frac{{{q_{\text{c}}} - {\sigma _{{\text{v}}0}}}}{{{{\sigma '}_{{\text{v}}0}}}}. $

(6)

Whether the drainage occurred or not can be clearly observed by changing the normalized velocity. According to Fig.3, the normalized penetration resistance values decrease significantly when V_{h_CE} is between 0.1 mm s⁻¹ and 100 mm s⁻¹, a rate interval in which partially drained conditions are usually considered to be achieved. The normalized velocity needs to change by at least three orders of magnitude from fully drained to fully undrained conditions.

Also in Fig.3, the significant effect of the strength and internal friction angle of the soil on the cone tip resistance is shown. Taking Fig.3b as an example, there is a good positive correlation between the stiffness and the normalized cone tip resistance in the soil body with a given friction angle. And for the soil body with a given stiffness ratio, the larger the friction angle is, the larger the resistance of the CPTu is. The results are in agreement with those of coupled-consolidation finite-element analyses of Yi et al. (2012).

2.3.2 Effect of K₀

In Fig.4a, it can be found that K₀ severely affects the cone tip resistance at any normalized velocity. Soils in isotropic conditions from HS19 to HS21 exhibit larger Q values at any normalized velocity compared to soils from HS10 to HS12 when K₀ = 1− sinφ'. Especially in the case of equal vertical effective stresses, the latter would produce greater horizontal effective stresses around the cavities. According to the equations defining the parameters of the HS model, the higher initial values of E₅₀ and E_ur need to be produced by the larger horizontal effective stresses. Therefore, for the expansion of cavity, to reach a predetermined radial displacement requires a greater pressure. According to Fig.4b, for a given stiffness and friction angle, the penetration resistance ratio is essentially the same under the drained and undrained conditions, and the ratio is independent of the stiffness modulus.

2.3.3 Effect of permeability

The effect of anisotropic permeability is shown in Fig.5a. The curve of case HS26 shifts to the right when the horizontal permeability (k_h) is doubled (k_v is maintained at 3× 10⁻⁶ m s⁻¹) compared to the curve of case HS08. This implies that the drainage of soil is faster than that of the original (HS08) case. When k_h increases by an order of magnitude (k_v remains unchanged), the curve shifted further to the right, but does not overlap with the HS22 case (k_h and k_v both increase by one order of magnitude). This result confirms that both horizontal and vertical permeability have an effect on the drainage characteristics during cavity expansion, but the effect of k_h is greater than k_v. This is due to the fact that the water flow is mainly horizontal, and it also indicates that k_h may be a better normalized parameter than k_v in defining general drainage conditions for different soil anisotropic permeability. In Fig.5b, except HS26 and HS27, we add five curves obtained at five orders of isotropic permeability to verify the reasonableness of using k_h for velocity normalization. The Q-V_{h_CE} curves approximately coincide in a single curve, which indicates that the effect of permeability is being well taken into account.

2.4 Resistances Under Drained and Undrained Conditions

The normalized cone tip resistance values (Q_D, Q_UD) with and without drainage are extracted from Group I. It is evident that as the soil stiffness and friction angle change, the calculated (Q_D, Q_UD) values also change, showing good linear relationships between logQ_D, logQ_UD, and logE₅₀ /p'₀, as shown in Figs.6a and 6b.

Therefore, based on the HS model cavity expansion analysis, Q_D, Q_UD can be reasonably estimated by using the fitting method proposed by Suzuki (2015).

$ {Q_{\text{D}}} = \frac{{{q_{{\text{cD}}}} - {\sigma _{{\text{v}}0}}}}{{{{\sigma '}_{{\text{v}}0}}}} = 5.974{\left({\sin \varphi '} \right)^{1.428}}{\left({\frac{{{E_{50}}}}{{{{p'}_0}}}} \right)^{0.398}}, {\text{ }}{R^2} = 0.974, $

(7)

$ {Q_{{\text{UD}}}} = \frac{{{q_{{\text{cUD}}}} - {\sigma _{{\text{v}}0}}}}{{{{\sigma '}_{{\text{v}}0}}}} = 4.248{\left({\sin \varphi '} \right)^{0.926}}{\left({\frac{{{E_{50}}}}{{{{p'}_0}}}} \right)^{0.144}}, {\text{ }}{R^2} = 0.99, $

(8)

where q_cD and q_cUD are the cone tip resistances under drained/undrained conditions estimated from Eq. (2). The equation is based on Group I calculations, where the friction angle ranges from 20˚ to 40˚, the stiffness ratio ranges from 20 to 480, the over consolidation ratio OCR is 1, the static earth pressure coefficient K₀ = 1− sinφ', and the shear-induced volume expansion or compression is not considered.

2.5 Prediction of Partially Drained Resistance

A hyperbolic function relating normalized cone resistance (Q) and the normalized velocity (V_{h_CE}) can be expressed as (House et al., 2001):

$ Q = \frac{{q{\text{c}} - \sigma {\text{v}}0}}{{\sigma '{\text{v}}0}} = {Q_{{\text{UD}}}} + \frac{{{Q_{\text{D}}} - {Q_{{\text{UD}}}}}}{{1 + a{{({V_{{\text{h_CE}}}})}^{0.80}}}}, $

(9)

where Q_D and Q_UD can be determined from Eqs. (7) and (8), a is a constant.

The effects of E₅₀ /p'₀ on Q/Q_UD ratios in the partially drainage range are well explained by Eq. (9) (for the soils with φ' = 24˚ and 30˚). As seen in Fig.7, the values of fitting parameter a also tend to increase with E₅₀ /p'₀. It can be seen that the hyperbolic function can effectively predict the cone resistances. In addition, it can be seen that the soil stiffness has a significant effect on the parameters in the Fig.7.

3 The1 g Model Simulations of CPTu with Variable Penetration Rates 3.1 Model Test Equipment

To obtain the variable penetration rate, a penetration rack unit was designed with indoor segmented penetration rate control equipment, as shown in Fig.8. With a diameter of 10 mm, a cone top angle of 60˚, a cone tip cross-sectional area of 78.54 mm² and a friction sleeve area of 1162.39 mm², the miniature piezocone is mounted on a servo motor driver, which can penetrate through the predrilled hole in the top plate into the soil samples at the controlled rate of 0.005 – 20 mm s⁻¹. The mini-CPTu can measure the cone tip resistance q_c, the sleeve friction resistance f_s and the cone shoulder pore water pressure u₂.

3.2 Preparation of Soil Samples

Two set of samples from Yellow River Delta silt and Malaysia kaolin were prepared. Basic geotechnical tests, one-dimensional consolidation tests and triaxial shear tests were performed on the two soil samples. The physical and mechanical properties, consolidation properties and strength of the remodeled samples were measured.

Table 4 summarizes the physical properties of Malaysia kaolin and Yellow River Delta silt, and Table 5 summarizes the consolidation coefficient c_v at effective stress levels in the range of 25 to 150 kPa.

The results of the undrained test at a consolidation stress of 100 kPa are shown in Fig.9. The shear properties are strongly influenced by the clay fraction. Based on the undrained triaxial shear test, the range of apparent peak friction angle of the Yellow River Delta silt φ'_p was estimated to be 28˚ – 32˚, with a mean value of 30˚.

During sample preparation, the water content was adjusted by adding water to twice the liquid limit. The samples were stirred under negative pressure for 4 h with a vacuum stirring device before slowly poured into the sample preparation drum. Then the samples were compressed by a consolidation system which had an inner diameter of 60.0 cm, a wall thickness of 10.0 mm, and a height of 80.0 cm. Geofabrics was used to separate the top and bottom plates from the samples. A drainage conduit with a valve was installed on the bottom of the drum and round holes were cut on the top plate so that water can be drained from both ends. A layer of fine sand approximately 30 mm thick was applied on top of the Malaysia kaolin clay to accelerate the consolidation. The consolidation system was constructed with a standard cylinder, a threaded rod and a counterforce frame. Pressure was applied evenly to the samples by an air pump. After consolidation, the sample container was placed on the test rig under a loading pressure of 50 kPa for over 24 h to maintain normal consolidation condition of the samples. The whole preparation process is shown in Fig.10.

3.3 CPTu Tests and Results at Different Penetration Rates

Due to conducted in a relatively small vessel, the boundary effect of the penetration tests cannot be neglected. The finite element numerical simulation on the spherical cavity expansion was performed and the possible boundary effects of this test was analyzed based on the parameters of Yellow River Delta silt. The detailed numerical modeling procedure has been presented in Section 2.

The radius r of the miniature piezocone is 5 mm, and the minimum distance D from the probe center to the inner wall of the cavity is about 100 mm. To simulate the critical case where D/r = 20, the soil area radius R is set to 4 m in the axisymmetric numerical analysis. The cavity radius was expanded from a₀ = 0.1 m to a_f = 0.2 m (i.e., R/a_f = 20).

As shown in Fig.11, the excess pore pressures (∆u) and the cavity limit pressures (σ₁) close to the cavity wall are little affected by R/a_f at different rates of cavity expansion. Fig.11a shows the radial pore water pressure distribution at a depth of 12 m after undrained cavity expansion, indicating almost no boundary effect on the cavity expansion pressure when R/a_f = 60. The predicted values when R/a_f = 20 is approximately 7% higher than that when R/a_f = 60 as radial water flow was restricted by the lateral boundary. As shown in Fig.11b, at a certain range of cavity expansion velocities, the cavity expansion pressure on the cavity wall is slightly affected by the distance away from the boundary. When R/a_f = 20, the cavity expansion pressure is 5% higher than that when R/a_f = 60, so 20 times of the piezocone radius was used for R in the 1 g model test.

Thirteen penetration tests distributed as shown in Fig.12 were performed on each sample. The interval between two tests was 24 h to minimize the influence of excess pore pressure generated during previous penetration. Considering the boundary conditions, the penetration holes are at least 100 mm (20 times the piezocone radius) away from the cylinder wall or each other. To ensure the accuracy of the results and the homogeneity of the samples, four sets of repeatability tests were carried out, and the results showed that the test was repeatable and the samples were homogeneous.

The penetration tests were conducted at a constant penetration rate at the start and once a predetermined penetration depth was reached, the penetration rate began to change from 0.005 to 20 mm s⁻¹ (the piezocone vibrated at 0.005 m s⁻¹). Adopting a relatively faster penetration rate at the shallower depths was to check the consistency of CPTu measurements between each penetration test, thus ensure that the interaction between the different penetrations was minimal. A corrected total cone resistance (q_t) was calculated by using the following relationship, which has involved the unequal pore pressure effect (Lunne et al., 1997b):

$ {q_{\text{t}}} = {q_{\text{c}}} + {u_2}(1 - \alpha), $

(10)

where q_c is the measured cone resistance, u₂ is the pore pressure measured at the cone shoulder and α is the unequal area ratio (0.8 for the CPTu).

Fig.13a shows the results of a typical penetration test on a Malaysia kaolin soil sample. The penetration depth of the miniature piezocone is about 300 mm, and the penetration rate v starts to change at the depth of 150 mm. The measurements reached stable after a period of time following the change of penetration rate. At the penetration rate v of 1 mm s⁻¹, the corrected cone tip resistance q_t of the Malaysia kaolin sample was relatively constant at 75 ± 5 kPa. The q_t when v = 0.005 mm s⁻¹ exceeds that when v = 1 mm s⁻¹ (approx. 60 kPa) and reaches a constant value of approximately 110 kPa. When v = 0.005 mm s⁻¹, pore water pressure u₂ of the Malaysia kaolin sample at the target depth is approximately 10 kPa.

Fig.13b shows the typical penetration test results for the Yellow River Delta silt samples. When v = 10 mm s⁻¹, q_t was stable at 150 kPa. Pore water pressure u₂ at a depth of 50 mm was 60 kPa. The q_t and u₂ measured when v = 1 mm s⁻¹ were similar to those measured when v = 10 mm s⁻¹. When v = 0.01 and 0.005 mm s⁻¹, q_t and u₂ are significantly different from those at higher penetration rates.

Fig.14 shows the comparison of penetration rate effect on CPTu results between the two soil samples in terms of normalized cone tip resistance (Q = q_net /σ'_v0, the net cone resistance q_net was calculated as: q_net = q_t − σ_v0, where σ_v0 is the total vertical stress) and normalized pore pressure ratios (∆u/σ'_v0, where u = u₂− u₀, u₀ is the hydrostatic pore pressure about 1 to 2 kPa and σ'_v0 is vertical effective stress). Based on Fig.14a, Q values are rather different between the two soil samples when normalized penetration rate V_v < 10.

According to Fig.14, since the value Q at V_v of 1 is similar to those when V_v > 10 for the Malaysia kaolin sample, the transition from undrained to partially drained conditions based on the change in normalized cone tip resistance is not fully consistent with that based on the change in pore pressure u₂ at the cone shoulder, which is referred to as the 'offset effect' by Kim (2005).

4 Discussion 4.1 Validation of 1 g Model Tests

Results from 1 g model tests at the different penetration rates designed in this paper are compared with the experimental results obtained by centrifuges and calibration tanks.

Fig.15 shows the result comparison in Malaysia kaolin between 1 g model test and two centrifuge tests performed by Randolph and Hope (2004) at 100 g and Schneider et al. (2007) at 160 g. The drained resistance in the 1 g test is a slightly lower than those in the centrifuge test, mainly because the maximum stress generated in the centrifuge test was approximately twice than in the 1 g test. As shown in Fig.15b, the Malaysia kaolin undergoes a transition from undrained to partially drained at the normalized penetration rate V_v of 100 and from partially drained to fully drained at the normalized penetration rate V_v of 0.1.

Fig.16 shows the comparison between the 1 g model test in the Yellow River Delta silt and the experimental studies of Jaeger et al. (2010) and Kim et al. (2008) on the mixture of silt with 25% kaolin. As the normalized penetration rate reduces by 2 to 3 orders of magnitude, all normalized penetration resistances Q increase significantly and the corresponding pore pressure ratios decrease significantly. The normalized penetration rate V_v values to make the soil under undrained and partially drained conditions were in the range of 10 – 30, and the normalized penetration rate V_v values to make the soil change from partially drained to fully drained conditions were in the range of 0.03 – 0.06. Similar transitions of drainage conditions can be observed in two different test methods, 1 g test and centrifuge test, based on the differences in measured drained resistance and undrained resistance, which result from the differences in density and preparation techniques of soil samples.

4.2 Comparison Between Numerical Analysis and 1 g Model Tests

The results from the 1 g test on Malaysia kaolin in this paper were compared with that of the PLAXIS cavity expansion calculations. The normalized penetration rate V_v was used in the 1g model tests. The horizontal consolidation efficiency was considered equal to the vertical one due to the use of remodeled soils with one-dimensional consolidation, which are assumed to have isotropic permeability (Wang et al., 2015). Therefore, V_v is able to be compared with the normalized penetration rate V_h in the numerical simulation. In the numerical simulation for a cone with a vertex angle of 60˚, the cone penetration velocity (v_cone) in the vertical direction is half of v_{r_CE}. Due to the assumption that the volume of the soil pushed out during the cavity expansion is equivalent to the volume of the cone tip, the diameter of the cavity (d_CE) can be considered as approximately 0.8 times the diameter of the cone penetrometer (d_cone). Then the normalized penetration rate V_h, which is linked to the CPTu based on Eq. (5), can be calculated as:

$ {V_{\text{h}}} = \frac{{{v_{{\text{cone}}}}{d_{{\text{cone}}}}{\gamma _{\text{w}}}}}{{{k_{{\text{h}}0}}{E_{50}}}} . $

(10)

The Malaysia kaolin samples have the friction angle of 24˚ and stiffness modulus (E₅₀/p'₀) of about 42 to 62, according to the measurement in triaxial compression tests by Stewart (1990).

The PLAXIS cavity expansion analysis provides the good simulation to the trend of net cone tip resistance with the penetration rate and reflects the transition in drainage conditions. Moreover, the stiffness modulus has a more significant effect on the trend of CPTu test results with the penetration rates.

The results of PLAXIS cavity expansion calculation are compared with the results of 1 g test for the Yellow River Delta silt. Base on the triaxial tests, the Yellow River Delta silt has a friction angle of 30˚. The compression index and recompression index of Yellow River Delta silt are about one fifth and one sixth of Malaysia kaolin, respectively. These relative magnitudes suggest a modulus ratio (E₅₀/p'₀) in the range of 200 to 240.

The normalized penetration rate for the transition from drained to undrained conditions determined by the PLAXIS cavity expansion analysis (0.12 – 120) is slightly larger than the 1 g model test results (0.06 – 30), which may be due to the overestimated shear strength and cone resistance at fast (undrained) expansions caused by Dilation cut-off option in HS model. Also the model (ψ = 0) does not provide realistic predictions for highly compacted soils.

5 Conclusions

In the engineering survey of the Yellow River Delta region, the partial drainage of silt layer is a key factor in affecting the CPTu test results. The curves of soil properties under the partial drainage conditions can provide some valuable reference for CPTu data interpretation in the Yellow River Delta. In this paper, the range of penetration rate to make the silty soils partial drainage in the Yellow River Delta is determined by the numerical simulation and the 1 g model test. The reference drainage curve is given and the reference resistance under the normalized penetration rate (zero dilation and zero viscous effects) is predicted. The main conclusions are summarized as follows:

1) The parameters measured during CPTu penetration are closely related to the drainage degree caused by the change of penetration rates. When the normalized penetration rate decreases by 2 – 3 orders of magnitude, the normalized penetration resistance Q increases significantly and the pore pressure ratio decreases greatly. The results of 1 g model test show that the normalized penetration rate V_v to make the Yellow River Delta silt under the undrained and partially drained conditions is between 10 and 30, and the values of V_v to make the soil change from partially drained to fully drained conditions is from 0.03 to 0.06.

2) Numerical simulation of spherical cavity expansion coupled with consolidation analysis by the HS soil model showed that the normalized penetration rate V_h in which can trigger the partially drainage of the Yellow River Delta silt changed at least three orders of magnitude (approximately 0.12 – 120). Both stiffness and friction angle have significant effects on the Q_UD and Q_D for normally consolidated soils assumed to expand freely, and the influence of stiffness is relatively greater.

3) Numerical simulation provides the reference resistance corresponding to zero expansion and zero viscous at any normalized velocity, which makes the data obtained on the actual silt soils in Yellow River Delta can be interpreted in terms of penetration rates.

Acknowledgements

The study is supported by the National Natural Science Foundation of China (Nos. U1806230, U2006213), and the Fundamental Research Funds for the Central Universities (No. 201962011).