Study of the Bearing Capacity at the Variable Cross-Section of A Riser-Surface Casing Composite Pile

Reducing the cost of offshore platform construction is an urgent issue for marginal oilfield development. The offshore oil well structure includes a riser and a surface casing. The riser, surface casing and oil well cement can be considered special variable cross-section piles. Replacing or partially replacing the steel pipe pile foundation with a variable cross-section pile to provide the required bearing capacity for an offshore oil platform can reduce the cost of foundation construction and improve the economic efficiency of production. In this paper, the finite element analysis method is used to investigate the variable cross-section bearing mode of composite piles composed of a riser and a surface casing in saturated clay under a vertical load. The calculation formula of the bearing capacity at the variable section is derived based on the theory of spherical cavity expansion, the influencing factors of the bearing capacity coefficient Nc are revealed, and the calculation method of Nc is proposed. By comparing the calculation results with the results of the centrifuge test, the accuracy and applicability of the calculation method are verified. The results show that the riser composite pile has a rigid core in the soil under the variable cross-section, which increases the bearing capacity at the variable cross-section.


Introduction
With the development of the Bohai Bay oilfield, a growing number of marginal, low-productivity fields must be developed, including some small marginal fields. These fields have a minimum of 1−2 wells and a maximum of 5−6 wells, with a considerable total oil storage volume. If these oilfields can be economically and effectively developed in the future, oil production in the Bohai Sea region can be substantial. However, due to the high construction costs, the traditional jacket platform cannot meet the goal of the efficient development of marginal oilfields. In recent years, to reduce development costs, some minimum offshore platform structures have been developed, such as MOSS (Zuo, 1992), SEAHORSE (Zhang, 1994), GUARDIAN, and MANTIS (Feng et al., 2009).
These structures use a riser in the wellhead of the oil production together with the inclined pile and jacket structure to provide support for the upper platform structure, but these minimum offshore platforms still need a pile foundation.
The wellhead structure generally includes a riser, a sur-face casing, an intermediate casing and a production liner. The riser is a steel pipe that goes from the offshore drilling platform to the shallow seabed during the drilling process; its main function is to isolate seawater and form a drilling fluid circulation channel. The surface casing is usually used to isolate the shallow surface water layer and the shallow complex bottom layer to protect the freshwater layer from drilling fluid contamination. The riser and surface casing in the platform wellhead are shown in Fig. 1. The riser and the surface casing are steel pipes, which are very similar to the size of the steel pipe pile. If the riser and surface casing can be effectively used to replace or partially replace the traditional steel pipe pile foundation to provide bearing capacity for the oil production platform, it can reduce the cost of platform construction and make the development of marginal oilfields more economical.
The riser is often installed by the driving method. The depth is generally 30−70 m in the Bohai Sea, the surface casing is installed by drilling, and the length can reach hundreds of meters. A high-strength cementing connection is used between the riser and the surface casing, so it is con-sidered to be a whole structure. Because the diameter of the riser is significantly larger than that of the surface casing, a special structure with an obviously variable cross-section is formed. The study of variable cross-section piles has a precedent in subgrade engineering. Fang et al. (2012) used model tests to reveal the stress distribution of the "nailshaped pile" under a vertical load. Huang et al. (2018) studied the influence of a large-diameter hollow variable-section resistance on the lateral resistance in bridge engineering through theoretical analysis. Manandhar and Yasufuku (2013) studied the end resistance bearing mode and calculation method of wedge-shaped piles in sand. However, the objects of these studies are different from the structure of the composite pile of the riser, and there is no clear bearing mode and calculation method on the variable cross-section.
In this paper, by considering the common size of risers and surface casings in the Bohai Sea region, finite element analysis and centrifuge tests are used to explore the bearing mechanism and bearing capacity calculation method of riser-surface casing composite piles (hereinafter referred to as composite piles) with the variable cross-sections in clay.

Structure of composite piles
During the construction of the wellhead, the riser is generally driven first, the surface casing is installed after drilling in the riser, and finally, cement is used to connect the riser and the surface casing. Common sizes of the risers, drill bits and surface casings in the Bohai Sea region are shown in Table 1.
The upper part of the composite pile is composed of a riser, a cementing ring and a surface casing. The lower part is composed of a cementing ring and a surface casing. The cross-section of composite pile is shown in Fig. 2.
In Fig. 2, D 1 is the outer diameter of the upper part, D 2 is the outer diameter of the lower part, D 3 is the inner dia-meter of the composite pile, L 1 is the pile length of the upper part, and L 2 is the length of the lower part. Zhai et al. (2008) tested the cement-steel and cementsoil interface strengths through shear tests. The cement-steel interface strength is higher than the cement-soil interface strength, and because cement has a rougher surface than steel, the cement-steel interface strength is also higher than the steel-soil interface strength. Therefore, it is reasonable to consider the composite pile as a whole structure and adopt the equivalent elastic modulus. To verify this conclusion, the steel and concrete are set separately in the finite element model, and frictional contact is used between the steel and cement. The interfacial shear force is extracted with different D 1 and undrained shear strength S u , as shown in Fig. 3 and Fig. 4, respectively.    LIU Run, LIANG Chao China Ocean Eng., 2021, Vol. 35, No. 2, P. 262-271 The bonding strength between the steel and grouted cement is larger than 6 MPa (Huang et al., 2009). It can be seen from Figs. 3 and 4 that the shear stress is far smaller than 6 MPa. Based on the above results, the riser, surface casing and cement are considered to be a whole structure, as shown in Fig. 5. The diameter of the upper part is the diameter of the riser, and the diameter of the lower part is the diameter of the drill bit used for grouting cement. The elastic modulus of the riser and surface casing is 210 GPa, and the elastic modulus of the cement ring is 28 GPa. According to Table 1, the equivalent elastic modulus of the upper part and the lower part can be calculated as shown in Eqs. (1) and (2), respectively:

Equivalent section of composite piles
(1) where E steel is the elastic modulus of the riser and the surface casing; E c is the elastic modulus of the cement; E 1 and E 2 are the equivalent elastic moduli of the upper and lower parts of the composite pile, respectively; A r , A c , and A sc are the cross-sectional areas of the riser, cement and surface casing, respectively; and A 1 , A 2 , E 1 and E 2 are the cross-sectional area and equivalent elastic modulus of the upper and lower parts of the composite pile, respectively. Table 2 shows the equivalent elastic modulus of the composite pile under common specifications in the Bohai Sea.
In the finite element analysis model, the "master-slave surface" contact is selected for the pile−soil interaction, the pile with larger rigidity is used as the master contact surface, and the soil body is used as the slave contact surface. The pile−soil contact is divided into the normal contact and tangential contact, the normal contact type is selected as the hard contact, the tangential contact is selected as the Coulomb friction type, and the contact friction coefficient is selected as 0.4 according to the existing research (Yan et al., 2013). In the boundary conditions of soil, horizontal constraints are imposed on the side of the soil, symmetric constraints are used on the symmetry plane, fixed constraints are imposed on the bottom surface, and no constraints are imposed on the top surface.
To ensure the accuracy of the calculation, the influence of different factors of the model on the calculation results was analyzed (Liang et al., 2018), including the soil model size and mesh size. Through the calculation, it is determined that the diameter of the soil model is 20D 1 and the length is L 1 +L 2 +20D 2 to eliminate the boundary effect. The soil model mesh is divided into three parts according to Fig. 6: the radial grid, axial grid 1 and axial grid 2, among which the radial grid is the one of the soil on the pile side along the pile diameter, axial grid 1 is the one within the length of the pile, and axial grid 2 is the one below the pile end.
To ensure the accuracy of the results, the grid influence of 1/4 model is analyzed, and the results are shown in Fig. 7.
With the decrease in the minimum radial grid, the inflection point of the ultimate vertical bearing capacity gradually converges to a fixed position. After analysis of the calculation results, the size of the radial grid is selected as (0.05−1.5)D 1 , and those for the axial grid 1 and axial grid 2 are selected as 1 m and 1−2.5 m to ensure the convergence and timeliness of the model calculation.

Bearing mode at the variable cross-section
The largest difference between a composite pile and an ordinary single pile is that the composite pile has a variable cross-section. The axial force of the pile at different posi-tions is extracted by the free body slicing function provided by ABAQUS. Fig. 8 shows the axial force distribution of the composite pile under different load conditions. It can be seen in Fig. 8 that at the position of 40 m of the composite pile's variable cross-section, the axial force of the pile is obviously abrupt, and part of the load is transferred into the soil from the variable cross-section. As the load on the pile increases, the degree of axial force attenuation gradually increases until the composite pile reaches the ultimate bearing capacity. The axial force difference between 39.9 m and 40.1 m can be used to calculate the resistance at the variable cross-section. Fig. 9 and Fig. 10 respectively show the displacement contour map and the distribution of the skin friction resistance at the variable cross-section under the ultimate load. Fig. 9 shows that due to the existence of the variable cross-section, part of the soil and the variable cross-section have a synergistic effect, and the rigid core of the soil appears below the variable cross-section. The soil around the pile tends to move away from the pile due to the squeezing action of the soil rigid core. Fig. 10 shows that since the soil rigid core has the same displacement as the pile below the variable cross-section, it is difficult to excite the frictional resistance. Fig. 11 shows that the soil failure surface is located on the surface of the soil rigid core. The bearing mode at the variable cross-section of the composite pile can be drawn, as shown in Fig. 12.    Fig. 9. Displacement at the variable cross-section.
LIU Run, LIANG Chao China Ocean Eng., 2021, Vol. 35, No. 2, P. 262-271 265 In Fig. 12, q u is the unit bearing capacity exerted on the variable cross-section, p u is the normal stress acting on the surface of the rigid nucleus, τ is the shear stress on the rigid core failure surface, τ = S u under limit load conditions, and β is the rigid core failure angle, which is the angle between the rigid core failure surface and the horizontal direction.
3.3 Calculation method of the bearing capacity at the variable cross-section According to the API specification (API, 2014), the calculation of the unit bearing capacity of the pile end in clay is where N c is the bearing capacity coefficient, and N c = 9.0 is adopted for an ordinary steel pipe pile with fixed cross-sec-tion.
From the force analysis of Fig. 12, the bearing capacity at the variable cross-section is Then, The bearing capacity coefficient at the variable crosssection can then be expressed as:

Bearing capacity coefficient at the variable cross-section
Eq. (6) shows that N c is related to p u and β. In this section, the influencing factors and calculation methods of p u and β are discussed, and the bearing capacity coefficient at the variable cross-section of the composite pile is determined.

Calculation of p u
It can be seen from the bearing mode of Fig. 12 that p u can be calculated according to the spherical hole expansion theory proposed by Vesic (1972): ν where I r is the stiffness index, K is the shear strength of the material, K = S u in clay, E 0 is the soil deformation modulus, and is Poisson's ratio of the soil. As seen from Eqs. (7) and (8), p u is related to E 0 and S u . In previous studies, E 0 is usually multiples of S u (Feng et al., 2014), E 0 =xS u , and x is usually 200−500; then, p u can be expressed as: where p u is only related to S u , E 0 and x. When soil conditions are determined, p u can be obtained from Eq. (9).

Influencing factors of β
The variable cross-section ratio η is defined as the ratio of the diameters of the upper part and lower part, η = D 1 /D 2 . According to the common riser and surface casing sizes in the Bohai Sea, a calculation model for the vertical bearing capacity of composite piles under different η values is established. According to previous studies (Liu, 2001), the undrained shear strength of the clay at a depth of 30−50 m in the Bohai Sea is mostly between 50 kPa and 100 kPa. Therefore, four different shear strengths are selected for calculation. The calculation conditions are shown in Table 5, where v and ρ′ are selected according to Table 3.    Under the condition that the composite pile reaches the ultimate bearing capacity, Figs. 13−15 show the contour maps of the soil position at the variable cross-section under different η, S u , and variable cross-section embedding depths, respectively.
It can be seen in Figs. 13−15 that the shape of the soil rigid core changes significantly with the change in η, S u and the variable cross-section depth; that is, β is affected by the above three factors. The bearing capacity at the variable cross-section is obtained by the axial force difference of the pile body near the variable cross-section, and the β value under each working condition in Table 4 is calculated according to Eq. (4). The results are shown in Table 6. The results in Table 6 show that when the composite pile reaches the ultimate bearing capacity, β shows a decreasing trend with the increasing η and S u and shows an increasing trend with the increasing variable cross-section depth.

Empirical calculation method of β
By considering the change in the variable cross-section depth versus the change in the effective overburden earth pressure at the calculation point, the calculated results in Table 6 are normalized using the parameter S u / , as shown in Fig. 16.
As seen in Fig. 16, the relationship between β and S u / under different η can be expressed linearly: where A and B are fitting parameters; through regression analysis of the data in Fig. 16, the relationship between parameters A, B and η can be fitted: where m and n are fitting constants; m=−1.69, and n=0.027. Therefore, when calculating N c at the variable cross-sec-  LIU Run, LIANG Chao China Ocean Eng., 2021, Vol. 35, No. 2, P. 262-271 267 tion of the composite pile, β can be expressed as: where 0<β<π/2.

Applicability of the bearing capacity at the variable cross-section
After p u and β being obtained by Eqs. (7) and (13), respectively, the bearing capacity coefficient N c at the variable cross-section can be obtained by Eq. (6), and the conditions in Table 5 are calculated. Fig. 17 shows that the results calculated using Eq. (6) are in good agreement with the numerical simulation results, indicating that Eq. (6) is suitable for calculating the bearing capacity at the variable cross-sections.
The above results consider the soil deformation modulus to be E 0 =500S u . To verify the applicability of Eq. (6) under different soil deformation moduli, two calculation conditions of E 0 =200S u and E 0 =350S u are selected. The calculation is performed under different η and S u / conditions. Fig. 18 shows the comparison between the calculation result of Eq. (6) and the numerical calculation result.
It can be seen in Figs. 17 and 18 that when E 0 changes, the calculation results of Eq. (6) are in good agreement with the numerical calculation results, indicating that E 0 has little effect on N c .

Applicability of the bearing capacity of fixed crosssection piles
As seen in Figs. 17 and 18, N c shows a decreasing trend as η and S u / increase, and as η or S u / increases, the N c value is close to 9.0, which is recommended by the API specification. From Meyerhof deep foundation bearing capacity theory (Meyerhof, 1951), it can be seen that β of the soil rigid core without a variable cross-section is 45°+0.5φ, and φ is the internal friction angle of the soil. In saturated soft clay, φ=0, and β=45°. In saturated clay, the general fixed cross-section pile is calculated using Eq. (6), referring to the Meyerhof pile end bearing mode, assuming that β is 45°. With Eq. (6), the bearing capacity coefficients of different soil deformation moduli at β=45° were calculated and compared with previous research results (Skempton, 1951), as shown in Fig. 19.
It can be seen from the results in Fig. 19 that Eq. (6) can be used to adequately calculate the end bearing capacity coefficient of fixed cross-section piles, which proves the rationality and applicability of the calculation method in this paper. Based on the above conclusions, it can be considered that when the values of η and S u / are large, the pile end bearing mode at the variable cross-section is similar to the Fig. 16. Relationship between β and S u /p 0 ' under different η. Fig. 17. Comparison of the numerically and theoretically calculated results. pile end bearing mode of the fixed section pile. Then, β in Eq. (13) can be assumed to be 45°≤β<90°.

Centrifuge test verification
To verify the accuracy of the above calculation method, a centrifuge model test of a composite pile was carried out. The test was conducted at the Geotechnical Center of Tianjin Research Institute for Water Transport Engineering. The test centrifuge used a single-walled gondola with a maximum radius of 5 m and a maximum capacity of 500 g·t.

Pile model
The prototype dimensions of the composite pile simulated by the test were D 1 =0.914 m, D 2 =0.444 m, L 1 =10 m, and L 2 =14 m. With the small diameter of the composite pile, in the centrifuge test, the model pile was made according to the scale of 1:40, the model pile material was 6061 aluminum alloy, the elastic modulus was 72 GPa, the ultimate tensile strength was 124 MPa, and Poisson's ratio was 0.31. To test the axial force distribution of the pile under the axial load, BFH120-3AA-X30 strain gauges were placed on the model pile, for a total of 14 sections from pile top #1 to pile end #14. Each section consisted of four strain gauges from a full-bridge circuit for testing. The outside of the strain gauges was evenly coated with 1 mm thick epoxy resin for waterproof protection. The model size and strain gauge layout are shown in Fig. 20. Before the test, a pressure test machine was used to test and calibrate the pile strain gauges, as shown in Fig. 21.

Test soil
The soil used for the test was kaolin soil, with a plastic limit of 22.5% and a liquid limit of 46.2%. Before the test, first the kaolin in the soil tank was mixed well with water at twice the liquid limit. The soil was consolidated, and a load was applied using a weight. After consolidation to a predetermined strength, the kaolin within 10 cm of the surface of the soil groove was removed, and a water head of 10 mm was filled on the surface of the soil to ensure the saturation of the soil used in the test. The soil strength was tested with a vane shear tester, and the test results are shown in Fig. 22. Fig. 22 shows that soil strength increased linearly with the increasing depth, S u = 1.1z + 7.5 kPa; the soil surface strength S um = 7.5 kPa; and the soil undrained shear strength increased by gradient k = 1.1 kPa/m. After the test was completed, a hole was formed at the predetermined point by the soil sampler with the aperture equal to the lower part diameter of the pile model, and then the model pile was pressed into the specified depth.

Test steps
In the test, hydraulic jacks were used for constant displacement loading, with a loading speed of 0.06 mm/min. A YP-H58 pressure sensor was connected under the hydraulic jack. The sensor range was 500 kg, and the vertical dis-    LIU Run, LIANG Chao China Ocean Eng., 2021, Vol. 35, No. 2, P. 262-271 placement of the pile top was measured by using a linear variable differential transformer (LVDT), as shown in Fig. 23. To recover the disturbance caused to soil around the pile during the process of pile driving, the soil was consolidated for 30 min under a 40 g acceleration condition, and the soil began to be loaded after the data of all sensors were stable.

Analysis of the test results
The load-displacement curve measured by the test (converted into prototype) is shown in Fig. 24, where Q is the loading force at the pile top, and S is the settlement at the pile top.
It can be seen in Fig. 24 that when the loading force of the pile top exceeded 648.5 kN, the settlement of the pile top substantially increased, and an inflection point appeared in the Q−S curve; 648.5 kN can be taken as the ultimate load value. Because the length of the composite pile can reach more than 400 m in an actual project, it is difficult to transfer the load of the pile top to the end of the lower part, so the ultimate bearing capacity of the composite pile is considered only as the external friction resistance and the bearing capacity of the variable cross-section. The axial force measured in the section #14 of the composite pile model was approximately equal to the end resistance of the lower part. Additionally, the ultimate load was measured by subtracting the end resistance of the lower pile part as the actual ultimate bearing capacity of the composite pile; the ultimate bearing capacity of the composite pile was 637.3 kN.
Strain gauges #12 and #13 at the bottom of the model pile demonstrated had no response in the tests, and the other data from normally working strain gauges were processed to obtain the pile axial force of the composite pile under different loads, as shown in Fig. 25. Fig. 25 shows that at the position of 10 m of the variable cross-section, the axial force changed significantly, which was consistent with the numerical simulation results. When the ultimate bearing capacity was reached, the axial force difference between the two adjacent strain gauges at the variable cross-section was calculated, and the bearing capacity at the variable cross-section was 72.1 kN. Eq. (6) was used to calculate the bearing capacity at the variable cross-section of the test pile, and the API specification method was used to calculate the external friction resistance. The calculated results were compared with the test results, as shown in Table 7.
As seen from the results in Table 7, the relative error between the bearing capacity of the variable cross-section calculated by Eq. (6) and the test value is 7.9%, and the relative error between the ultimate bearing capacity of the pile and the test value is only 3.08%, which proves the accuracy   of the calculation method proposed in this paper.

Conclusions
Based on finite element analysis, a model of the bearing capacity of a variable cross-section of a composite pile with clay is established, and a calculation method of the bearing capacity coefficient of a variable cross-section is proposed. Combined with centrifugal test results, the reliability of the method is verified. The conclusions are as follows.
(1) Under the vertical load, compared with the ordinary pile without a variable cross-section, the axial force of the composite pile exhibits an obvious mutation phenomenon in the variable cross-section, and a rigid core synergistic with the variable cross-section is formed in the soil to bear the load together.
(2) The bearing capacity coefficient N c at the variable cross-section of the composite pile increases with the decreasing η and S u / and is significantly larger than the bearing capacity coefficient of the fixed cross-section pile, which is 9.0. This result indicates that the special structure of the variable cross-section of the composite pile has a large constraint on the soil, thus increasing its bearing capacity. With the increasing S u / , N c gradually approaches the bearing capacity coefficient of the fixed cross-section pile.
(3) Through the regression analysis of the numerical calculation results, the relationship between η and S u / is established. Based on the theory of spherical cavity expansion, a method for calculating the bearing capacity of the composite pile at the variable cross-section is proposed, which has universal applicability to cohesive soil with different buried depths and strength characteristics. The calculation results are compared with centrifuge test results to verify the accuracy of the calculation method.