1 Introduction

With an increase in frequency of operations, the situation that installation of jack-up platforms on sites which contains old footprints is becoming more common and inevitable. According to van den Berg’s statistics (Van den Berg et al. 2004), within Shell EP Europe alone roughly 1,200 footprint points had been registered in geotechnical and footprint datasets. In addition, there are approximately 80 new single footprint points added to the existing datasets every year. Thus, it can be seen that footprints are not rare and they pose a serious and growing threat to operational safety of jack-up drilling platforms. Figure 1 shows when a leg is close to an existing footprint, the non-uniform bearing load caused by the footprint will make the spudcan slide into the footprint in the jacking process, which was proven by Gaudin et al. (2007), Leung et al. (2007). The sliding trend is affected by the leg stiffness, connection between leg and hull, and in-place condition of other two legs, and the size of the trend is measured by the horizontal sliding force and overturning moment (McClelland et al. 1982; Hossain and Randolph 2007; Bouwmeester et al. 2009). If a slide occurs, the legs will incline in different directions, so that the legs may become stuck in the platform and this would mean the platform cannot be raised. The potential risk of slipping is a serious threat to the operational safety of platforms.

Fig. 1
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Schematic diagram of existing footprint problems

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Re-installing a spudcan very close to or partially overlapping existing footprints is generally not recommended in the guidelines (SNAME OC-7 panel. 2007; Hossain and Randolph 2008). In a situation where this is inevitable, the guidelines recommend the use of an identical jack-up (same footing geometries and leg spacing) and locating it in exactly the same position as the previous unit, where possible. However, it is unlikely that two jack-up units have an identical design because the structures of most units are often custom-made and the deployments of units are subject to availability. It is evident that existing guidelines are not adequate for rig operators to install jack-up units in close proximity to existing footprints safely.

Footprint issues involve soil elastoplasticity, material and geometric nonlinearities, fluid–solid coupling, friction contact during spudcan preloading, and difficult convergence of numerical solutions (Hanna and Meyerhof 1980; Kellezi and Stromann 2003; DeJong et al. 2004; Deng and Kong 2005; Leung et al. 2008). Previous research mainly focuses on the spudcan–footprint interaction through the centrifuge model test. Murff et al. (1991), Hossain et al. (2005), Cassidy et al. (2004, 2009), Teh et al. (2010), Gan (2009), Gan et al. (2012), Kong et al. (2010, 2013), Xie et al. (2012) conducted a series of drum centrifuge model tests to investigate spudcan–footprint interaction and the effect of leg stiffness on spudcan–footprint interaction. With the centrifuge model tests, Stewart and his coworkers (Stewart 2005; Stewart and Finnie 2001) studied the effect of bending rigidity of legs on spudcan–footprint interaction and the influence of the distance between the spudcan and the footprint on sliding. Dean and Serra (2004) discussed the effect of equivalent stiffness of legs on spudcan–footprint interaction. Teh et al. (2006) reported a set of test results investigating the effects of sloping seabed (30° inclined to the horizontal) and footprint on loads developed in jack-up legs. They found that the effect of the footprint is much greater than that of the seabed slope. This indicates that the footprint problem is more serious than a sloping seabed. Other researchers have tried to investigate the footprint problem with numerical simulation (Zhang et al. 2011, 2014). Jardine et al. (2002) simplified a three-dimensional model to a plane strain one to deal with footprint issues. The current understanding of this topic is still insufficient, and only a small number of studies of the footprint problem are available in the public domain. Although it is a great challenge to obtain a converged numerical solution, a good numerical model and solution is very important because it is able to achieve more accurate estimation of carrying capacity of spudcans and better explanations for tests. This paper takes various factors including failure process of foundation, nonlinearity, sliding friction contact, and fluid–solid coupling into account. It discusses the finite element model of spudcan–footprint interaction in spudcan re-installation near an existing footprint as well as handling relative parameters. With the model of the spudcan–footprint interaction, the changes of horizontal sliding force on the spudcan at different offset distances between the spudcan and the footprint were analyzed with ABAQUS software. The finite element model was validated by comparing the simulation result with experimental results.

2 Analytical methods and computing model

During jacking, the deformation of the surrounding soil is very large, which results in changes in pore pressure and then a reduction in the effective strength of the soil. To analyze spudcan–footprint interaction, the coupling of stress/fluid flow in soil should be considered. Undrained total stress analysis is used in the computing model, i.e., the total stress is the sum of effective stress and hydrostatic pressure. Thus, the equilibrium equation in the vertical direction is as follows (Houlsby and Martin 2003):

$$ \frac{{\text{d} \overline{\sigma }_{z} }}{{\text{d} z}} = \left\{ \begin{gathered} \rho g - \gamma_{\text{w}} \left(S_{\text{r}} (1 - n^{0} ) - \frac{{\text{d}} {S}_{\text{r}} }{{\text{d} z}}(z_{\text{w}}^{ 0} - z)\right); \quad z \le z_{\text{w}}^{ 0} \hfill \\ \rho g;\quad z_{\text{w}}^{ 0} \le z \le z^{0} \hfill \\ \end{gathered} \right., $$
(1)

where \( \overline{\sigma }_{z} \) is the vertical stress, Pa; ρ is the soil dry density, kg/m3; γ w is the water gravity density, N/m3; S r is the soil saturation,  %; \( z_{\text{w}}^{ 0} \) is the free water surface elevation, m; z 0 is the elevation of interface between dry soil and partially saturated soil, m; and n 0 is porosity,  %; when \( z \le z_{\text{w}}^{ 0} \) in completely saturated, \( S_{\text{r}} = 1 \), and when \( z_{\text{w}}^{ 0} \le z \le z^{0} \), in partially saturated, \( S_{\text{r}} < 1 \).

The advantage of ABAQUS in soil engineering is that it provides not only various elastic/plastic constitutive models for soil but also coupled analysis of stress/fluid flow in soil. In numerical computation, the finite element mesh is fixed on the soil skeleton, and fluid may flow through the mesh and satisfy the fluid continuity equation. The Forchheimer equation (Zeng and Grigg 2006) is adopted to describe nonlinear flow in soil (porous medium). Since less relative parameters in calculation are needed, the Mohr–Coulomb constitutive model is used (Li 2004), i.e., the soil is considered as a perfect elastic–plastic material, and obeys the noncorrelation flow rule. The Mohr–Coulomb yield criterion is as follows:

$$ s + \sigma_{\text{m}} \sin \phi - c\cos \phi = 0, $$
(2)

where \( s = (\sigma_{1} - \sigma_{3} )/2 \) is half of the difference of maximum and minimum principal stresses, kPa; \( \sigma_{\text{m}} = (\sigma_{1} + \sigma_{3} )/2 \) is the average value of maximum and minimum principal stresses, kPa; c is cohesion, kPa; and ϕ is the internal friction angle, °. Except for over-consolidated soil, clay always shows little dilatancy, and thus the dilatancy angle ϕ = 0. Assume that the deformation modulus is approximately proportional to the undrained shear strength, then \( E = 500s_{\text{u}} \) (s u is the undrained shear strength, kPa).

A vertical plane containing the line connecting the spudcan and the footprint center is chosen and a finite element model is established, as shown in Fig. 2. The diameter and depth of the footprint are D and d, respectively. In order to reduce the boundary effect on accuracy of the numerical simulation, the width and depth of the surrounding soil are taken as 15D and 7d, respectively. The offset distance between the spudcan and the footprint center is denoted as S. The 8-node plane strain and pore pressure element, CPE8PR, is used to simulate the soil element to avoid self-locking phenomena and to increase the computational accuracy in numerical simulation. The active–passive surface contact algorithm is used to deal with the contact interaction and relative displacement between the spudcan and the surrounding soil, and the spudcan surface is taken as the active surface and the soil surface as the passive surface (Zhuang et al. 2005). The principle for choosing an active or passive surface is that the mesh of the passive surface should be finer, and if both mesh densities are similar to each other, the surface of the softer material should be passive. The tangential contact obeys the Coulomb friction law, and the normal contact follows the hard touching mode, i.e., penetration is not allowed between the spudcan element and the soil element, but they are allowed to separate (Zhuang et al. 2005). In order to obtain the correct horizontal sliding force–displacement curve, the displacement control method is used to load. A simplified spudcan, with its side friction ignored because of its relative smaller area, is adopted to reduce the difficulty of convergence in calculation. The friction coefficients for undrained clay and drained granular soil are 0.2–0.3 and tan δ, respectively, where δ is the friction angle between the spudcan and the soil. It must be pointed out that whether setting a reasonable degree of spudcan–soil contact will lead to the calculation converging or not.

Fig. 2
figure 2

Schematic diagram of the finite element model

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Since the ultimate bearing capacity would be underestimated if the initial geo-stress equilibrium were not considered in numerical simulation, this paper deals with the initial geo-stress equilibrium first and imports a stress file with an ‘initial conditions’ method. This is instead of the ‘Geostatic’ way, a commonly used geo-stress equilibrium analysis method in general simulation involving in soil that is difficult to deal with for such a complex problem as spudcan–soil interaction with an existing footprint. In addition, because of serious soil deformation under a large spudcan penetration depth, in order to avoid huge warping and ensure accuracy of calculation, ALE self-adaptive meshes are employed.

3 Spudcan–footprint interaction in clay

3.1 Failure process of clay foundations

Let S = 0.75D (D = 6 m, d = 6 m). The mechanical characteristics of uniform soil such as clay are shown in Table 1.

Table 1 Material parameters of single-layer foundation
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The gradual failure process of clay foundation occurs in three stages: elastic balance, plastic expansion, and complete plastic damage (Fig. 3). Figure 3a shows that plastic damage first appears at the bottom edge of the footprint close to the spudcan. Figure 3b shows the expansion of the soil foundation plastic zone from the bottom edge of the footprint toward the farther edge of the spudcan with load increasing. Figure 3c indicates that when the complete plastic damage of clay foundation appears, the plastic zones have expanded to form a continuous sliding surface.

Fig. 3
figure 3

Plastic zone of clay foundation in loading (part around the footprint)

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3.2 Clay foundation yield at different S

Changing only S while keeping other parameters constant, the situations of clay foundation yield at different S are shown in Fig. 4. This indicates that the plastic zone becomes larger with an increase in S and the failure pattern of soil around the spudcan gradually changes from asymmetric to symmetric.

Fig. 4
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The complete plastic damage zone at different S (part around the footprint)

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3.3 Soil movement patterns at different S

When the spudcan arrives at the designed depth, the soil displacement vectors under different S are shown in Fig. 5, from which we see that there is an obvious uplift trend at the bottom of the footprint and the soil near the footprint clearly migrates toward the footprint. The bulge on the farther side surface of the clay foundation changes little with an increase in S. However, the apophysis on the footprint bottom increases significantly and the soil movement patterns on the closer side to the spudcan and below the spudcan change greatly. When S is small, part of the soil below the spudcan moves to the footprint, while another part migrates downward with the spudcan. With the S increasing, the soil under the spudcan bottom basically migrates downward, while most of the soil on the closer side of the footprint moves into the footprint and only a little moves downward with the spudcan edge. This may provide a coping idea for jack-up re-installation close to footprint (which will be discussed in a separate paper).

Fig. 5
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The displacement vector of clay at different S (part around the spudcan)

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3.4 Influence of S on horizontal slip force

The relation between the horizontal slipping force on the spudcan and the spudcan vertical displacement, i.e., depth at different S is displayed in Fig. 6. This shows that at any S, with the depth increasing, the horizontal force on the spudcan increases initially then decreases after it reaches a peak value. The peak values at different S appear at a depth from 2.5 to 4.5 m, and the maximum peak horizontal force is about 0.7 MN when S = 4 m. This indicates that the most potentially dangerous situation is when the spudcan partially overlaps the existing footprint. In order to investigate the overall relationship between the peak horizontal force on the spudcan and S, the peak horizontal forces are sorted at different S in dimensionless form (Table 2).

Fig. 6
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The horizontal force–depth diagram at different S

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Table 2 Peak horizontal forces at different ‘S’
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For the problem with a ‘footprint,’ the horizontal slip force on the spudcan varies with soil strength, footprint dimension, diameter of the spudcan, and the offset distance between the spudcan and the footprint center. Taking these factors into consideration, the expression of the peak horizontal force on the spudcan in dimensionless form can be summarized as

$$ H_{\hbox{max} } = f^{'} \left( {\frac{{D_{\text{s}} }}{{D_{\text{f}} }},\frac{S}{{D_{\text{f}} }},\frac{d}{{D_{\text{f}} }}} \right) \times s_{\text{u}} D_{\text{s}}^{ 2} , $$
(3)

where H max is the peak horizontal force on the spudcan, MN; S u is the soil undrained shear strength; D f is the diameter of the footprint, m; D s is the diameter of the spudcan in future operations, m; S is the distance between the spudcan and the footprint center, m; and d is the depth of the footprint, m.

In this paper, only the influence of the offset distance on the peak horizontal slip force on the spudcan is considered, as given in Table 2. The horizontal force on the spudcan will be zero when S = 0 as the spudcan is located exactly in the footprint. Using Matlab to fit the numerical simulation results, the peak horizontal force on the spudcan is obtained as follows:

$$ H_{\hbox{max} } = 4.1248 \cdot \left( {\frac{S}{{D_{\text{f}} }}} \right)^{1.3439} \times \exp \left( { - 1.9555\frac{S}{{D_{\text{f}} }}} \right), $$
(4)

The fitting curve of Eq. (4) and the numerical simulation results are shown in Fig. 7. This demonstrates that the curvature tolerance of Eq. (4) is very small and it could reliably represent the relationship between the peak horizontal sliding force on the spudcan and the offset distance S. The peak horizontal force reaches a maximum value when S/D = 0.6. The horizontal force increases quickly before it reaches the maximum value and then gradually decreases. The rate of decrease is far less than the rate of increase. In order to observe the successive change of the peak horizontal force, the horizontal force is calculated at larger ‘S according to Eq. (4), and the whole relation between the peak horizontal sliding force and the offset distance is given in Fig. 8. When S/D ≥ 5, the peak horizontal force becomes almost zero, which means in this case that the influence of the existing footprint could be ignored.

Fig. 7
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The fitted curve between the peak horizontal force and S

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Fig. 8
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The whole relation between the peak horizontal force and S

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4 Verification of numerical simulation results

Based on the University of Western Australia centrifuge model test (Table 3; Gan 2009), we built 2-dimensional and 3-dimensional simulation models (Fig. 9) to conduct finite element simulation. Results at different S (0.25D, 0.50D, 0.75D, 1.0D) are shown in Figs. 10 and 11. Comparisons of results from the 2-dimensional or 3-dimensional simulation models and from the experiments indicate that the simulation results are in good agreement with experimental results, and the results from the 3-dimensional model are a little closer to the test results than those from the 2-dimensional model. However, with the 3-dimensional model, not only the computing time needed is much longer, but also the calculation is much more difficult to converge. Using the 2-dimensional model built in this paper would significantly reduce the necessary computing time, and the simulation results are in good agreement with experimental results, which shows that the 2-dimensional model built in this paper is feasible and reliable.

Table 3 List of major experimental parameters (after Gan 2009)
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Fig. 9
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3-dimensional finite element model

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Fig. 10
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Simulation and experimental results

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Fig. 11
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2-dimensional and 3-dimensional numerical simulation results

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5 Conclusions

  1. 1.

    In the initial loading stage, plastic damage first appears at the bottom edge of the footprint close to the spudcan. Then the plastic zone expands with increasing load and finally it forms a continuous sliding surface.

  2. 2.

    With an increase in the distance between the spudcan and the footprint, the soil failure pattern gradually changes from asymmetric to symmetric.

  3. 3.

    The soil migration patterns on the closer side of the footprint and below the spudcan change greatly at different offset distances. With the distance increasing, the soil on the spudcan bottom basically migrates downward, while most of the soil on the closer side of the footprint moves into the footprint, and only a little moves downward with the spudcan edge. This means “stomping” (repeated raising and lowering of the jack-up leg) may be a successful solution for the jack-up installation close to a footprint.

  4. 4.

    The peak horizontal sliding forces on spudcan at different offset distances modeled with Matlab to fit the numerical simulation results and the possible dangerous ranges during re-installation have been obtained. The peak horizontal force reaches its maximum value when S/D = 0.6. When S/D ≥ 5, the horizontal sliding force becomes almost zero, which means in this case that the influence of the footprint could be ignored.

  5. 5.

    The numerical simulation results show good agreement with experimental results, indicating clearly that the finite element model built in this paper can be used to solve the problems of spudcan–footprint interaction with sliding friction contact, fluid–solid coupling, nonlinear elastic–plastic deformation, and convergence problems.