Analysis of Penetration Depth of Pipeline on Cohesive Soil Seabed

This paper conducts laboratory tests to investigate detailedly the soil deformation law around the pipeline and its penetration depth under self-gravity. The seabed model is prepared by consolidating saturated soil using vacuum pressure technology, and the pipeline models are specifically designed to possess different radii. Based on the experimental results and digital images, the soil deformation process is analyzed and summarized, a kinematic admissible velocity field is given and an upper bound solution of pipeline penetration depth and soil reaction force is derived and proposed in this paper. In order to verify the accuracy of the upper bound solution deduced in this paper, a comparison is made among some published results and the solution suggested in this paper, the comparison results confirm that the upper bound solution and the soil failure mode are reasonable. Finally two empirical formulas are given in this paper to estimate the soil reaction force of seabed and the penetration depth of pipeline. The empirical formulas are in agreement with the upper bound solution derived in this paper, and the conclusion of this paper could provide some theoretical reference for the further study of the interaction between the pipeline and the soil.


Introduction
When a submarine pipeline is not required to be artificially buried on the seabed, it is necessary to analyze its stability, so the penetration depth of the pipeline under selfgravity is a very important parameter during the design process. Besides, while the pipeline is located on the seabed foundation, White and Randolph (2007) pointed out that its exsitence directly changes the hydrodynamic conditions of the surrounding water current which affects the hydrodynamic calculation on the pipeline. Actually, the pipeline penetration depth is determined by the constraint reaction force of the soil seabed, and the soil reaction force of the seabed foundation is dependent on the deformation law of seabed foundation around the pipeline. These hydrodynamic loading and soil constraints are two important factors to assess the pipeline stability. Under the self-gravity and some additional load, the soil around the pipeline will be compressed and plastic deformation will emerge until the soil satisfies the limit equilibrium state. Merifield et al. (2009) suggested that the soil deformation around the pipeline must be studied in detail. The penetration depth of the pipeline is closely related to the soil reaction force of seabed foundation which depended on the soil failure mode or the soil de-formation law. In order to study the pipeline penetration depth, the ultimate bearing capacity or the soil reaction force of saturated seabed should be determined first.
At present, some research scholars and in site engineers often use two dimensional plane formulas to calculate the soil reaction force and the penetration depth of the pipeline. Murff et al. (1989) studied a shallow-embedded pipeline on undrained soft clay and presented a plasticity solution for vertical load-embedment response. Aubeny et al. (2005) proposed a relationship between penetration depth and soil reaction force in cohesive soil. Martin et al. (2012) used limit analysis method to study the soil reaction force of offshore pipelines. Merifield et al. (2008aMerifield et al. ( , 2008bMerifield et al. ( , 2009) studied the soil deformation law and reaction force of seabed foundation under the circular pipeline by using the finite element method. Dingle et al. (2008) used large deformation method to analyze the pipeline penetration depth. Randolph and White (White and Randolph, 2007;Randolph et al., 2012;Chatterjee et al., 2013) has analyzed the soil reaction force of shallow-buried pipeline in detail, and during the process of theoretical analysis, an assumption has been made that the soil failure mode beneath the pipeline is similar to Prandtl mode, which ignored the effect of the circular interface on the soil reaction force. Zhou et al. (2008) analyzed the penetration depth into soft clay using physical test. Silva et al. (2006) and Kim et al. (2008) separately adopted experimental tests to study the effect of penetration velocity rate on the soil reaction force. Chatterjee et al. (2013) used large deformation finite element methodology combined with modified Cam-clay plasticity soil model to explore the coupled consolidation behaviour among soil and pore pressure beneath the pipeline. Martin and White presented exact horizontal-vertical failure envelopes for "Wished-in-Place" (WIP) pipelines under undrained loading conditions via finite element limit analysis method. By using numerical analysis method, Krost et al. (2011) analyzed the pore pressure change law in the surrounding soil around the pipeline, and simulated the soil deformation law. The above results are often compared with the two dimensional formula derived on the plane strip footing. In fact, the geometric shape of circular pipeline is obviously different from that of the strip footing which is the plane interface between soil and the footing, so the soil reaction force of the seabed beneath the pipeline must be different from the strip footing, then the classical calculation formula of the bearing capacity should be modified or revised when used to estimate the soil reaction force beneath the circular pipeline. Some codes specify that the soil reaction force of underwater pipeline per unit length is equal to , which is similar to the two dimensional Prandtl results (DNV, 2007).
In order to calculate the soil reaction force of seabed foundation and the penetration depth of the pipeline accurately and reasonably, it is necessary to study the soil deformation law and failure mode around the pipeline theoretically. In fact, whether or not to solve the soil reaction force of shallow-buried circular pipeline on the seabed foundation accurately depends on the soil failure mode surrounding the pipeline. The research work in this paper is based on the Hill plastic yielding criterion, and an admissible velocity field of soil around the pipeline is obtained through experimental test. Then the soil reaction force of seabed foundation and the penetration depth of pipeline are analyzed by using virtual power principle, and the corresponding upper bound solution are deduced in this paper. Simultaneously, a detailed comparison is made between the upper bound solution deduced in this paper and some published results, two more practical modified formulas are given finally.

Experimental program
Before presenting the experimental methodology, the conditions investigated in the experiments are first introduced. The model pipeline is placed on the mudline of the rectangular testing box, using a transparent perspex as viewing window. During the test, the pipeline gradually penet-rates into the soil and the soil incremental movement around the pipeline can be continuously acquired through the viewing window. A digital camera with high resolution (4000×3000 pixels) was placed in front of the perspex window to capture images at a rate of 2 frame per second (20 frame per millimeter of pipeline penetration depth, for pipeline penetration rate of 0.1 mm/s). Apart from monitoring the soil deformation law, the soil reaction force acting on the pipeline are measured by some sensors.

1/10th
10 3 The 1g model test apparatus is shown in Fig. 1, and tests are undertaken using a scaled model pipeline at scale. The corresponding radius of pipeline and shear stress on the interface would reduce ten times, so the soil reaction force acting on the model pipeline has to be multiplied by to recreate full scale behaviour. The experimental program comprises circular pipeline and undrained saturated soft clay. The soil samples are confined within a rectangular strongbox container, whose internal dimension is 1800 mm length, 550 mm in width and 600 mm in depth, containing a uniform soil sample. The circular pipeline has eight kinds of radii namely 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75 and 2.0 dm. Under the self-gravity loads, the penetration depth of pipeline depends on the undrained cohesive soil strength. This paper studies the penetration depth of pipeline in the seabed and the soil deformation law around the pipeline. During the tests, the self-gravity load of pipeline is simulated by usage of servo motor to generate vertical driving force, which is recorded by the digital pressure sensor, and the penetration depth of the pipeline is recorded by the displacement sensor LVDT connected with the pipeline model. The model pipeline is instrumented by five sets of soil pressure gauges, one set of axial pressure gauge and one set of axial displacement gauge to record the vertical load, the penetration depth and the soil reaction force of seabed during testing.

Model pipeline and soil sample preparation
Model tests are performed using circular pipeline with a certain kind of radius as shown in Fig. 2. The model pipeline are made from PVC circular tube with a length of 600 mm, the model pipeline are placed on the level of seabed mudline, and vaseline is applied on the surface of pipeline to simulate the smooth interface between the pipeline and the cohesive soil. In this paper, nine pipeline penetration tests are undertaken in detail, which reflect the change law in the pipeline diameter on the penetration depth.
During the nine series of tests on the saturated soil specimens in the rectangular tank, the seabed soil strength is simulated by the conditions of soil in the tank. Firstly, some homogeneously saturated soft clay with 100% water content is prepared and filled in the vacuum tank. Secondly 1g with vacumm pump to produce negative pressure in the soil sample, the soil sample is fully saturated and then consolidated at state in the tank. The preparation process of soil sample is shown in Fig. 3 and the soil geotechnical properties tested in laboratory are given in Table 1.

Experimental results
In order to analyze the surrounding soil deformation law and the penetration process of the pipeline, the saturated soil seabed is filled by some homogeneous layers with 2 cm interval which are dyed with white kaolin clay as shown in Fig. 4a.
The deformation law of soil around the circluar pipeline is shown in Fig. 4b, the response of a pipeline penetration into a homogeneous soil seabed from the mudline exhibits two stages. To capture the change law of soil deformation during the two stages, a digital camera with high resolution is adopted in front of the perspex window . The two stages include: (1) Stage 1 response is an soil plastic compressive and shear deformation near the pipeline, which is constrained by the pipeline and the outer soil. (2) Stage 2 response is a soil general shear deformation outside the pipeline, which is a kind of classical passive shear failure.
On the basis of the experimental results, a soil plastic failure mode obeying deformation compatibility is pro-   abc acd posed in this paper. The failure mode is comprised with two parts shown in Fig. 5, one is a plastic compressive shear zone as beneath the pipeline, and the other is a general passive failure zone as outside the pipeline. In order to detailedly determine the relationship between the penetration depth of pipeline and the soil reaction force of the soil seabed, Fig. 6 presents the relationship curves recorded from the pressure sensor and the displacement sensor mounted on the experimental set-up. The soil reaction force acted on the pipeline gradually increases with pipeline penetration depth, which shows a nonlinear growth tendency. These experimental data will be used to verify the theoretical upper bound solution deduced in this paper.

Upper bound solution
Virtual power principle shows that for any set of static allowable stress fields and any set of kinematic admissible velocity fields, the virtual power of the external force equals the internal energy dissipation in continuous deformable body. Moreover, the upper bound theorem points out that the genuine solution is the smallest of all the possible results corresponding to the admissible deformation field.

Kinematic admissible velocity field
h Based on the failure mode presented in Fig. 5 and using the upper bound analysis theorem, given an assumption that the penetration depth of pipeline is , an upper bound solution of pipeline penetration depth is deduced in this paper.   The soil velocity vector is shown in Fig. 7 and some assumptions are summarized as follows.
(1) In this paper, a circular pipeline with unit length is adopted as investigated subject, and the solution of pipeline penetration depth is simplified as a two-dimensional plane strain problem.
(2) The circular pipeline is a rigid body, and the interface between the pipeline and the soil is smooth.
(3) The seabed soil is assumed as rigid plastic body, which satisfies the associated flow rule and obeys the Mohr-Coulomb yielding criterion, and the soil shear strength does not change with depth.
V P (4) The vertical velocity of the pipeline is , and the soil velocity distribution is shown in Fig. 7b.

Upper bound solution
In the process of theoretical derivation, the internal en-ergy dissipation rate includes two parts: (1) the internal energy dissipation rate along the velocity discontinuity surface, (2) the internal energy dissipation rate in the spiral deformation zone.

Geometric relationship of kinematic velocity field
Based on the pipeline penetration depth h shown in Fig. 7a, some geometic relationship can be obtained as follows: (1) 3.2.2 Internal energy dissipation rate of soil abc (1) Internal energy dissipation rate in the plastic deformation zone is a non-integral function to be solved numerically.

bc
(2) Internal energy dissipation rate along the spiral discontinuity surface

cd
(3) Internal energy dissipation rate along the spiral discontinuity surface 3.2.3 External power of self-gravity P Given that the external self-gravity load acting on the pipeline is , the external power produced by gravity can be written as:

Upper bound solution
On the basis of limit analysis theorem, the external force power equals the total internal energy dissipation rate. So an energy equation which indicates the soil reaction force and the pipeline penetration depth is written as follows: pipeline does not exceed the radius R.
In this paper, forty kinds of penetration conditions are analyzed using the upper bound theoretical solution presented, including (1) the radius of the pipeline (R=0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75 and 2.0 m), (2) the penetration depth of pipeline ( , , , and ). Theoretical results are shown in Table 2. In order to verify the accuracy of this upper bound solution proposed in this paper, the theoretical results are compared with some published conclusions. The comparison results shown in Fig. 8give  between the pipeline cross-section on the mudline and its penetration depth. (2) The soil reaction force increases rapidly with the penetration depth at initial stage of pipeline self-gravity, then the soil reaction force reaches the maximum when the penetration depth equals the radius R. (3) The upper bound solution proposed in this paper is bigger than the Murff solutions and the Martin solution. The main reason is that the soil kinematic velocity field shown in Fig. 7a around the pipeline is different from that suggested by Murff and Martin, which would dissipate more plastic deformation energy. The soil near the circular pipeline surface is no longer the active failure mode or the rigid mode shown in Fig. 8; instead, the plastic shear deformation, squeezed by the curved smooth surface of the pipeline, occurs, which is similar to the small-hole expansion; therefore, more plastic shear distortion energy is dissipated than the classical Prandtl failure mode. (4) The upper bound solutions deduced in this paper are smaller than the results from the 1g scaled experimental tests. When the radius of pipeline is 0.5 m, the upper bound solution is smaller than the test results; when the radius is 1.0 m, the upper bound solution is smaller than the test results; when the radius is  1.5 m, the upper bound solution is smaller than the test results; and when the radius is 2.0 m, the upper bound solution is smaller than the test results. The comparison results show that the error decreases with the increase of the pipeline radius .
Finally, to make the upper bound solution deduced in this paper more convenient in engineering, two reasonable modified formulas based on the upper bound solutions are suggested as follows. The suggested formulas are in agreement with the upper bound solution given in this paper, and the comparison data between this paper's solutions and that of the suggested formula are shown in Fig. 9.

Conclusions
A large number of 1g experiments have been done to investigate the penetration depth of pipeline under its selfgravity, then a kinematic admissible velocity field is proposed based on the experimental results. By using the upper bound analysis theorem, an upper bound solution of the soil reaction force is theoretically deduced. Finally, some practical conclusions can be drawn as follows.
(1) The kinematic admissible velocity field obtained through the 1g scaled model tests can express the soil deformation law around the pipeline when it penetrates into the seabed under self-gravity, so the velocity field obtained by the model test could be used to theoretically deduce upper bound solutions of the pipeline penetration depth and the corresponding soil reaction force.
(2) The upper bound solutions proposed in this paper can calculate the soil reaction force and the penetration depth well, and the suggested Eqs. (11) and (12) can be easily used to estimate the pipeline penetration depth and the soil reaction force.
(3) In this paper, the upper bound solution of pipeline penetration depth under self-gravity could provide some theoretical reference for further study of the interaction between the pipeline and soil.
(4) The conclusions and corresponding test method mentioned in this paper could provide some references for further study of the penetration depth and the soil deformation law of layered soil, where more thorough and detailed study is needed including thickness ratio, soil mechanical parameters and the interface characteristics.