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1. IntroductionThe conductor pipes of offshore wellhead platforms are of-ten installed closed together. In order to reduce the risk of interaction, the peripheral conductor piles can be deviated away from the group. One method that can be used is to provide an inclined driving shoe, thus creating a tendency for the pile to deviate in a particular direction. By adjusting the length and inclination of the driving shoe, the lateral deviation of a conductor pile can be controlled.

In some cases it is important to know the path of the conduc-tor pile. As deviations may not be small, conventional beam-column theory is not applicable and a more general theory must be used. Poskitt1 has developed a method to compute the path of a bent pile during driving. From this theory it is possible to develop a method applicable to conductor piles with inclined driving shoe. The parameters acting on the de-viation of the pile are the axial tip and side resistance of the pile, the lateral soil resistance, and the length and inclination of the driving shoe. In order to understand the influence of these parameters, a sensitivity analysis was performed. Then, the developed method was for a real case, where the deviation was measured, to check the relevance of the model.

2. Poskitt MethodPoskitt has described an incremental method to determine the path of a non-straight pile during driving. This method is based on finite differences and is applied to a pile with constant initial curvature. The path of a pile is function of the tip resistance, the shaft friction and the normal pres-sure. The forces acting on the pile are shown in Figure 1.

Figure 1a corresponds to the forces acting on a perfectly straight pile if the soil is homogeneous. In practice, the piles are never perfectly straight, so lateral and moment forces appear at the tip of the pile deviating the pile away from the vertical (Figure 1b).

2.1 Fundamental equationsFrom Kirchoff’s equations, Poskitt shows that a small curved element of pile of length, ds, is governed by the following equations (see Section 6 for notation definitions):

(1)These fundamental relationships lead to the following equation

(2)with the boundary conditions as follows

(3)

CONTROLLING CONDUCTOR DEVIATION WITH INCLINED DRIVING SHOE

Engueran Tisseau, Christophe Jaeck and David Cathie Cathie Associates, Belgium

AbstractClosely spaced conductor piles are sometimes deviated away from the group to reduce the risk of in-teraction, using an inclined driving shoe.

The theory developed by Poskitt1 for modelling the deviation of bent conductors or piles during driving has been extended to conductor piles with an inclined driving shoe. Variables include the axial tip and side resistance of the pile, the lateral soil resistance, and the length and inclination of the driving shoe. Parametric results are presented to illustrate the sensitivity of the results to input parameters. A com-parison between the theory and the actual deviation of a conductor pile with a driving shoe confirms the reasonable agreement between theory and practice.

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2.2 Incremental methodThe pressure normal to the pile, p, depends on the path fol-lowed by the pile. So the solution must be computed in an incremental manner starting from when the pile first enters the soil. A pile increment is vertically added (Figure 2a), the forces are then applied (Figure 2b) and the displacements calculated iteratively (Figure 2c). Then, a new straight in-crement is added following the tangent of the curve at the last point of the pile (dashed line in Figure 2d). The loads

are applied again and the displacements calculated (solid line in Figure 2d). The method is repeated until reaching the desired penetration.

2.3 Numerical solutionUsing the equations given in Section 2.1 and finite differences at each grid points, a non-linear sys-tem of equations can be derived, governing the path of the pile. Figure 3 shows a pile after adding r increments. The results at all the internal grid

Figure 1: Pile forces – (a) perfectly straight pile and (b) pile with initial curvature

points satisfy equation 2. Replacing the differential deriva-tives with their finite difference equivalents, equation 28 from Poskitt’s work is obtained.

The top and the toe of the pile (j = 0 and j = r) are governed by the boundary conditions (equation 3). Written in finite differences, it becomes equation 4.

(4)Assuming that θ-1 = θ1, a system of r equations with r un-knowns (θ1 to θr) is obtained.

The values pj (normal pressure at grid point j) depend upon the displacements normal to the pile. Therefore, the vertical and lateral displacements, xj and yj, are calculated using the trapezoidal rule, as in equation 5.

(5)The Newton-Raphson method can be used to solve the problem and determine θ = (θ1, ... , θr). Let A(n) be the matrix formed with the gradient vectors of the system of equations (partial derivation relative to [θ1, ... , θr]). A(n) is the Jacobean matrix of the system (taking into account the second and higher products and the pj gradients). The itera-tion routine of Newton-Raphson is given in equation 6.

(6)where (n + 1) and (n) indicate the (n + 1)th and nth iterates, respectively, and e(n) is the residual vector obtained by sub-stituting θ(n) into the equations.

3. Adaptation of Poskitt’s Method to a Pile with a Bent Shoe3.1 MethodologyThe Poskitt method can be adapted to a pile with a shoe by calculating the forces applied by the soil on the shoe. The pile is divided into two parts: the body pile and the shoe (as shown in Figure 4a). These two parts are supposed to have a negligible internal curvature angle.

The forces on the shoe are the normal pressure, which is determined from a simplified model of p-y curves, the axial

Figure 2: Incremental path: (a) addition of a new increment; (b) application of the forces; (c) iterative calculation of the dis-placements and (d) addition of a second increment and calcu-

lation of the new shape of the pile

Figure 3: Finite difference scheme

Figure 4: Modelling of the pile with bent shoe: (a) pile ele-ments; (b) forces acting on the shoe and (c) simplified model

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tip resistance and the shaft resistance, as shown in Figure 4b. The normal pressure induces a bending moment at the elbow point in the direction of the shoe. The tip resistance and the shaft resistance induce an upward axial force and a radial force which causes bending in the opposite direction to the normal force. Once these forces and the soil forces acting on the body of the pile are determined, the path of the pile can be solved using the forces as boundary condi-tions in the original method, as shown in Figure 4c.

The forces acting on the pile can be written as in equation 7 (using the same notations as for equation 1).

(7)The subscript elb applies to the elbow between straight pile section and bent shoe. Lshoe is the length of the bent shoe. The incremental solution is then implemented in a Visual Basic for Application (VBA) spreadsheet.

3.2 Calculation of the forces The p-y, t-z and Q-z curves are needed to solve the problem and are drawn from the soil profile using American Petroleum Institute (API) RP2A2. For all analyses performed, the unit shaft resistance value, t, and the axial tip resistance, Pr, were taken as the ultimate values. To simplify the problem, the p-y curves were modelled assuming a linear elastic-perfectly plastic relationship, as shown in Figure 5.

The pu value is the ultimate pressure. The slope of the linear part is taken as the tangent of the p-y curve at very small dis-placements and is called the subgrade modulus K (= pu/yu).

3.3 Sensitivity analysis in a homogeneous soil composed of clayA sensitivity analysis has been performed on an 80m long, 26in OD x 1.25in WT conductor pile, with a typical bent shoe 1.5m long and 0.75° inclined. The soil is taken as clay with an undrained shear strength, su, varying with depth: su = 0.25*σ’v (with σ’v being the effective vertical overburden pressure).

A factor 0.5 is applied on the interface undrained shear strength in order to derive the unit shaft resistance: fs = 0.5*su; this is considered a high estimate parameter, particularly for continuous driving in stiff normally consolidated clay. Reference can be made to typical methods for assessing the soil resistance to driving of tubular piles, e.g. Stevens et al.3

The unit end bearing is taken as nine times the cohesion and the pile is supposed to be plugged. To remain conservative (large lateral soil resistance induces large pile deviation), a high subgrade modulus is selected with the ultimate value reached for a displacement yu of 5ε50D (ε50 is the strain at one-half the maximum deviator stress on laboratory und-rained compression tests of undisturbed soil samples). For comparison, Matlock4 suggests a larger value yu of 20ε50D.

The sensitivity of the results to the pile increment length are first analysed to determine the optimum length, for which accurate results are calculated but within a reasonable cal-culation time. The calculated pile deviation for the base case (bent shoe 1.5m long and 0.75° inclined) are presented in Table 1 for three pile increment values. The results show that an increment length of 1.6m is small enough to obtain acceptable solutions.

The sensitivity of the pile deviation to the following param-eters has then been analysed:• Lateral subgrade modulus• Axial tip resistance Pr• Length of bent shoe Lshoe• Inclination of bent shoe θelb.

3.3.1 Effect of soil resistance (lateral subgrade modulus and axial tip resistance)The effects of the subgrade modulus and the axial tip resist-ance on the deviation of the pile are presented in Tables 2 and 3, respectively. The effect of both parameters is also illustrated on Figure 6.

Increment Length (m)

Segment Number

Pile Tip Deviation (m)

Angle of Deviation (°)

2.5 32 1.07 0.77

1.6 50 1.15 0.83

1.0 80 1.17 0.84

Subgrade Modulus, K

Pile Tip Deviation (m)

Angle of Deviation(°)

Kref/2 0.56 0.40

Kref 1.15 0.83

2*Kref 2.22 1.59Figure 5: Modelling of the p-y curves

Axial Tip Resistance, Pr

Pile Tip Deviation (m)

Angle of Deviation(°)

Pr,ref/2 1.18 0.84

Pr,ref 1.15 0.83

2* Pr,ref 1.11 0.79

Table 3: Effect of axial tip resistance, Pr, 26in OD x 1.25in WT pile, 80m penetration (bent shoe 1.5m long and 0.75° inclined)

Table 2: Effect of lateral subgrade modulus, K, 26in OD x 1.25in WT pile, 80m penetration

(bent shoe 1.5m long and 0.75° inclined)

Table 1: Effect of increment length: 26in OD x 1.25in WT pile, 80m penetration (bent shoe 1.5m long and 0.75° inclined)

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The lateral subgrade modulus has a large influence on the pile deviation. For the case studied, the calculated angle of deviation is 0.4° for half the reference subgrade modulus, but is increased to 1.6° for twice the subgrade modulus. The pile deviation is increased by a factor of 4.

On the contrary, the value of the axial tip resistance has a limited impact on the results, as shown in Table 3 and Figure 6. Therefore, for clay conditions, considering either coring or plugged behaviour during driving should have a limited effect on the pile deviation.

Figure 6: Effect of the subgrade modulus and the axial tip re-sistance on pile deviation

Figure 7: Effect of shoe geometry on the deviation of the pile

Shoe Inclination, θelb (°)

Pile Tip Deviation (m)

Angle of Deviation(°)

0.25 0.38 0.28

0.50 0.77 0.55

0.75 1.15 0.83

1.00 1.54 1.10

1.50 2.21 1.58

2.00 2.47 1.77

3.00 2.62 1.88

4.00 2.62 1.88

5.00 2.59 1.86

10.00 2.31 1.66

Length of Bent Shoe (m)

Pile Tip Deviation (m)

Angle of Deviation(°)

0.5 –0.01 –0.01

1.5 1.15 0.83

3 9.17 6.61

Table 5: Effect of length of bent shoe, Lshoe: 26in OD x 1.25in WT pile, 80m penetration (bent shoe 0.75°

inclined, reference soil resistances, Kref and Pr,ref)

Table 4: Effect of inclination of shoe, θelb: 26in OD x 1.25in WT pile, 80m penetration (bent shoe 1.5m

long, reference soil resistances, Kref and Pr,ref)

3.3.2 Effect of shoe geometry (inclination and length)The effect of the inclination and length of the bent shoe on the deviation of the pile is presented in Tables 4 and 5, respectively. The effect of both parameters is also illustrated in Figure 7.

The deviation of the pile is significantly influenced by the length of the bent shoe. For a short shoe (0.5m long), al-most no deviation is calculated for the considered inclina-tion at the elbow (i.e. 0.75°). But increasing the length of the shoe from 1.5 to 3m leads to a deviation at pile tip multiplied by a factor 8 (from less than 1° to nearly 7°).

As shown on Figure 7, the pile deviation versus bent shoe inclination curves can be subdivided into three parts: first the deviation increases with the value of the inclination at the elbow; then the deviation reaches a plateau; and finally the pile tip deviation slowly decreases with increasing incli-nation of the bent shoe.

Figure 8: Initial and computed deviated shapes of a 30in OD x 1in WT conductor pipe (offshore Nigeria)

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Depth (m)

Description

0.00–7.10 Very soft clay (su = 1 to 8kPa)

7.10–15.60 Medium dense sand

15.60–30.50 Firm to stiff clay (su = 50 to 130kPa)

30.50–40.00 Loose sand

40.00–60.50 Dense to very dense sand

60.50–69.70 Medium dense sand

69.70–78.00 Very stiff clay (su = 200kPa)

Penetration (m)

Pile Tip Deviation (m)

Angle of Deviation (°)

Measured Onsite

~74.1 ~11.7 9.0

Computed Solution

74.5 14.5 11.8

Table 7: Measured and calculated CP tip deviation: 30in OD x 1in WT pile, 75m depth (bent shoe 1.5m long and

0.75° inclined)

Table 6: Location of well-head platform offshore Nigeria: soil stratigraphy

fs unit shaft friction

h length of a pile increment

p normal force per unit length of pile

s distance along pile axis

t tangential force per unit length of pile

x, y coordinates (x = vertical; y = lateral)

E elastic modulus of the pile

F shear force

I moment of inertia of the pile

K lateral subgrade modulus

L embedded length of pile

Lshoe length of the pile shoe

M bending moment

P axial force in pile

R radius of curvature of pile

R0 initial radius of curvature of pile

su shear strengthθ angle between tangent to the pile and the verticalθelb angle of the shoe with the body part of the pile

σ’v effective overburden stress

For small inclination values, the force generated by the nor-mal soil pressure on the shoe increases more quickly than the radial force applied at the elbow, Felb, which is influenced by the axial tip resistance and the shaft friction on the shoe.

When the angle becomes more important, the normal soil pressures on the shoe reach their maximum value, pu. Therefore, the two opposite forces (normal pressures on the shoe and radial force applied at the elbow) tend to progres-sively counterbalance each other. This corresponds to the second part of the curves.

For large shoe inclination values, the radial force applied at the elbow continues to increase while the normal pressures applied on the shoe remain constant (having reached the ultimate value).

4. Comparison with Real Installation CaseAt a well-head platform located offshore Nigeria, 30in OD x 1in WT conductor pipes with a bent shoe 1.5m long and 0.75° inclination were driven to a depth of 75m. The soil conditions at this location are summarised in Table 6. For one of the driven conductor pipes, the inclination of the pile tip at final penetration was measured and found to be 9° at a deviation of approximately 11.7m.

The computed solution is presented in Table 7 together with the measured deviation. The subgrade modulus, which is the most critical soil parameter, is assessed using the simplified model of p-y curves described in Section 3.2. The ultimate values are calculated in agreement with API RP2A2. In clays, the ultimate lateral resistance is reached for a displacement yu of 5ε50D (as indicated in Section 3.3). In sands, the stiffness at small displacement is considered for the linear elastic part of the p-y curves.

The two values are in reasonable agreement with a differ-ence of less than 3m of deviation and of 2.8° in terms of angle of deviation. The computed shape of the deviated pile is plotted in Figure 8.

5. Conclusions and RecommendationsThis paper has presented a simplified methodology to assess the pile tip deviation during driving of conductor piles with an inclined driving shoe. Such conductor piles are some-times considered to reduce the risk of interaction between closely spaced conductor piles.

The incremental solution developed by Poskitt[1] for mod-elling the deviation of bent conductors or piles during driv-ing, as briefly described in Section 2, is extended to conduc-tor piles with an inclined driving shoe (see Section 3.2). The required variables include the axial tip and side resistance of the pile, the lateral soil resistance, and the length and incli-nation of the driving shoe. The incremental solution is then implemented in a VBA spreadsheet.

A parametric study is conducted to illustrate the sensitivity of the results to the input parameters. The results of this study are presented in Section 3.3 and show that the most sensitive soil parameter is the lateral soil resistance, while the axial tip resistance has a small effect on the pile deviation in clay.

A comparison between the theory and the actual deviation of a 30in OD x 1in WT conductor pile with a driving shoe (wellhead platform installed offshore Nigeria) confirms the reasonable agreement between theory and practice (see Section 4).

6. Notation

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References1. Poskitt TJ. (1996). The deflection of piles during driving.

Géotechnique 46(2), 235–243.2. American Petroleum Institute (API). (2000). Recommended prac-

tice for planning, designing and constructing fixed offshore platforms, 21st edition (RP2A). Washington, DC: API.

3. Stevens RS, Wiltsie EA and Turton TH. (1982). Evaluating pile driveability for hard clay, very dense sand, and rock. OTC 4205, Proc. Offshore Technology Conference, Houston, USA, 465–481.

4. Matlock H. (1970). Correlation for design of laterally loaded piles in soft clay. OTC 1204, . Offshore Tech. Conf., Houston, USA. Preprints, Vol. 1, 557–588.

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