Centrifuge testing to verify scaling of offshore pipeline ploughs

The behaviour of offshore pipeline and cable ploughing lends itself to physical modelling investigation at various scales as existing empirical models to predict plough tow forces and advance rates rely heavily on empirical factors that are normally back figured from small-scale beach tests (Grinsted, 1985; Reece and Grinsted, 1986) or real ploughing campaigns (Cathie and Wintgens, 2001). For instance, interface friction behaviour for a full-scale plough is often determined by pulling the plough on a beach or in shallow water at low speeds. Even this apparently simple determination is made more difficult as a beach environment may be subject to significant suctions. The plough also typically has a fresh coat of paint that has low interface frictional properties, which is quickly removed on ploughing in granular materials. The plough may also embed and start to generate passive resistance. Therefore, even this simple element (interface friction) of field plough behaviour is not simple to deconvolute from the passive and rate effects that make up the remaining components of plough resistance (Cathie and Wintgens, 2001). This has led to some reluctance to use empirical parameters determined for loose sand that may result in over optimistic predictions of advance rates. The selection of appropriate soil parameters is further complicated where they may only be determined based on site investigation at 1–2 km intervals (OSIG, 2004). One common form of empirical approach used to predict offshore pipeline plough tow forces (F) is that proposed by Cathie and Wintgens (2001) 

where C_w is a dimensionless friction coefficient that may be taken as tan δ, where δ is the soil–steel interface friction angle (Lauder et al., 2013), W′ the submerged buoyant plough weight, C_s a dimensionless passive pressure coefficient, γ′ the submerged unit weight of the soil, D the depth from the original sand surface to the heel of the plough main share, v the velocity of plough advance and C_d is a dynamic or rate effect coefficient with units of t/(m³/h).

Due to the potential difficulties in deconvolution and significant costs of full-scale plough deployment, the study of offshore pipeline and cable ploughing (Robinson et al., 2016a) lends itself to scaled physical modelling at 1g where soil beds of known properties can be prepared in a repeatable fashion. Interface behaviour of plough materials can be deconvoluted directly through element testing of steel–sand interfaces in shear and dry sand testing can be used to study the passive resistance behaviour of a plough as ploughing depth increases. Once these elements are understood, the sand can be saturated and tests undertaken at elevated plough rates and various depths in sands of different permeabilities to understand the rate effect component of behaviour (Palmer, 1999). Working at 1g with appropriate sized models has the advantage of simple testing that uses non-specialist equipment that can be undertaken quickly allowing repetition and thus broad parameter consideration.

Limited controlled laboratory testing data for ploughs is available in the literature. Bransby et al. (2005) examined the behaviour of pipeline ploughing in layered soils using 1:50 scale model ploughs at 1g. Brown et al. (2006) proposed that the results of the modelling could be converted to their full-scale equivalent by considering the appropriate scaling laws (Table 1) although this was only verified by calculating scaled-up tow forces that were of similar magnitude to those encountered at full scale. Subsequent to this, 1g model testing facilities were used to study a variety of different ploughing situations and geohazards, for example ploughing in sandwaves (Bransby et al., 2010a, 2010b), across slopes (Bransby et al., 2018), in fibre-reinforced soil (Brown et al., 2015) and varying plough geometry (Lauder et al., 2013) and vibro ploughing (Zefirova et al., 2012) to increase plough efficiency. Further investigation of the scaling effects and proposed scaling for 1g testing (Bransby et al., 2005; Brown et al., 2006) was not undertaken until the research study by Lauder (2011), who used a modelling approach based on tests at 1g with model ploughs at scales of 1:50, 1:25 and 1:10. This previous work was carried out at 1g and at lower stress levels than experienced in the soil surrounding full-size ploughs. To provide support for the output of such 1g testing requires either costly or hard-to-repeat full-size testing, or controlled tests carried out on a geotechnical centrifuge.

The current work has been carried out in order to use centrifuge modelling to verify (or otherwise) plough performance parameters based on 1g testing. It includes a total of 23 centrifuge tests undertaken in loose to very dense sand (Table 2) in both dry and saturated conditions, simulating scaled plough velocities of up to 580 m/h. This paper will describe the development of a large displacement linear actuation system for use in a long centrifuge box mounted on a medium-sized beam centrifuge. This system was developed to undertake both model cable and pipeline ploughing at 50g, initially to verify previous scaling assumptions for 1g model testing as part of a wider research study. As well as scaling verification and description of the apparatus, other developments made to assist the study will also be described. This work has been undertaken as part of a joint programme of research, which aims to develop computational techniques to overcome issues with the modelling of large offshore deformation operations such as cable and pipeline ploughing (Cortis et al., 2017).

The scaling laws for 1g model plough testing as initially proposed by Bransby et al. (2005) and verified as of appropriate magnitude against field experience by Brown et al. (2006) are now discussed. The scaling adopted for 1:50 scale is that lengths are reduced by a factor of 50, areas by a factor of 50² and volumes by a factor of 50³. As the model tests are undertaken at 1g the unit weight of the soil is identical to full scale so effective stresses will be reduced by a factor of 50 due to the shallow reduced scale trenching depth, which at field scale may be 1·5–2 m deep typically (for a single pass of the plough). If it is assumed that a Coulomb failure envelope (with c′  =  0 kPa and zero dilation) describes the soil failure envelope then there will be a 50 times reduction in shear strength in the model compared to the prototype. Additionally, the weight of soil above the plough share in the model will be reduced by a factor of 50³. As the tow forces measured during a test are either proportional to the projected area of the plough multiplied by a shear stress or due to soil self-weight, in either case, the model forces will need to be multiplied by 50³ to recreate full-scale behaviour. This is because the area will be reduced by 50² and the shear stress by a factor of 50, thus the model tow forces will be 1/50³ (  = (1/50²)  ×  (1/50)) times the field tow forces (Brown et al., 2006). The scaling adopted for 1g testing is summarised in column one of Table 1.

These scaling laws were adopted by Lauder et al. (2012), who looked at the origins of the empirical correction factors (C_w, C_s and C_d) in Equation 1 in some detail. When investigating the origins of the passive resistance term C_s through dry ploughing in three different sands at various relative densities and plough depths it was discovered that the value of C_s did not vary significantly with relative density (12·2–15·2). This was in contrast to C_s varying from 5 (loose sand) to 20 (very dense sand) as previously back figured from full-scale ploughing (Cathie and Wintgens, 2001). Similarly, the rate effect or dynamic term C_d was found to increase at a reduced rate with permeability than that determined in the field (Lauder et al., 2013). As the 1g testing was undertaken at very low effective stresses relative to the field (typical trench depths of 32 mm compared to 1·6 m prototype) it might be anticipated that dilation would be increased and peak friction angles would be elevated resulting in significant variation of both C_s and C_d over and above that seen in the field. As this was not the case, and no previous consistent verification of scaling had been undertaken, a ‘modelling of models’ study was undertaken by Lauder (2011), which involved the testing of models ploughs at scales of 1:50, 1:25 and 1:10. Due to the size and difficulties of preparing samples for the 1:10 scale plough along with limited pull distances only dry tests were undertaken at this scale limiting the verification of scaling of the rate effects. Results of this study (in loose and dense sands) are shown in Figure 1 where the plough tow forces scaled up to prototype using the scaling factors in Table 1 are compared with the Cathie and Wintgens (2001) approach using the factors for C_s derived from 1:50 scale 1g model testing. The modelling of models suggests that the values of C_s derived from 1:50 scale tests are appropriate for predicting behaviour at 1:10 scale and potentially at prototype scale. It is also clear that anecdotal concerns over adopting excessively low C_s values in loose sand are justified, as measured data suggests this under predicts tow forces significantly. Further discussion of this difference and the appropriateness of adopting 1g testing for ploughing are given in Lauder et al. (2013) and Lauder and Brown (2014) although it is acknowledged that further verification of the scaling and assumptions made is required.

As part of the study by Lauder et al. (2013) and with prompting from MF Randolph (2012, personal correspondence) the form of the Cathie and Wintgens (2001) approach was also modified to make the C_d term dimensionless

where C_dn is a non-dimensional form of rate effect coefficient, s the dilation potential (Palmer, 1999), a method for the determination of which is described in Lauder et al. (2012) and c_v the coefficient of consolidation, which was determined at appropriate stress levels for the model testing (Lauder and Brown, 2014). The results in Figure 2 present normalised tow forces (F/F_v=0, where F_v=0 is based on the equivalent target depth tow force in dry sand, which has effectively zero rate effect) found during 1:50 scale and 1:25 scale ploughing tests. For the velocities associated with the 1:50 scale tests (0–240 m/h) there is good agreement at both scales but there appears to be some non-linearity in the 1:25 scale tests (>250 m/h), which is also shown by the variation in C_dn. Based on these results and the lack of verification at the upper model scale of 1:10 it would appear necessary for further verification of the scaling of rate dependency especially at increased ploughing rates. Further verification of the 1g scaling assumptions for both ‘static’ and rate effect behaviour is therefore necessary and is addressed in this paper through centrifuge testing.

Due to the comparatively large displacements required to reach steady state when testing model ploughs (approximately three ploughs share lengths where the share length at model scale was 110 mm in this study), the existing centrifuge model containers available were unable to accommodate ploughing. To overcome this, a new enlarged 1·6 m long model container was designed and fabricated. The container was fabricated from 50 mm thick steel plate and cast acrylic in order to ensure that it had sufficient rigidity to meet the deflection limits suggested by Garnier et al. (2007) to prevent soil disturbance during spin up. The centrifuge container consisted of two end plates of external dimension 650 × 400 mm machined from 50 mm thick steel. The sides of the container were made from 50 mm thick acrylic sheets 650 × 1600 mm. These were braced along the top edge by a solid steel beam element 50 × 50 mm and at mid-length by a vertically orientated solid steel element 50 × 50 mm to further minimise bending (Figure 3(a)). The edge beams were themselves braced together across the top of the container at least every 500 mm by 50 × 15 mm aluminium bars, which form part of the actuator for ploughing.

The actuator consists of a moving instrumentation platform mounted on four pairs of two Igus Drylin WJ200QM-01-16 linear bearings (two bearings at each corner of the platform) each with a vertical capacity of 2·1 kN running on Igus WSQ-16 linear rails attached to a rigid frame (Figure 3(c)). These bearings are used due to their low friction coefficient and also due to the fact that their lubricant-free operation prevents performance issues related to the ingress of sand particles. The platform actuation is provided by a belt system using two Brecoflex 25AT10 steel-reinforced polymer belts on either side of the container each with a tensile capacity of 3·5 kN, driven by a Parvalux SD12-LWS high torque 220 V DC motor (capacity of 63 N m at 13 r/min). At each end of the container, the drive belts are connected to 47 mm diameter AT10 pulleys mounted on a 20 mm diameter steel drive shaft attached to rotary bearings on the frame. To make the design as compact as possible, the belts and pulleys are housed within the two structural aluminium channels (100 mm × 50 mm × 6 mm), which make up the frame. At the motor end, a further motor drive belt with a 95 mm diameter pulley on the drive shaft and a 47 mm diameter pulley on the motor provides the actuation. The combination of pulleys used provides an overall gear ratio of 2 in order to reduce the torque on the motor to an acceptable level and allow the actuator to provide a towing capacity of up to 5 kN, with horizontal displacements of up to 800 mm.

The design allows actual towing velocities of 14 mm/s (50 m/h), although due to the accelerated (50×) pore-pressure dissipation at 50g this corresponds to a scaled plough velocity of 1 m/h if water were to be used as the pore fluid. Using hydroxypropyl methylcellulose (HPMC – Methocel) as a viscous pore fluid (300 cP) allows this scaled velocity to be increased to 300 m/h. This velocity range matches that typically used in full-scale ploughing of up to 300 m/h (Lauder et al., 2013); however, in reality, velocities of up to 560 m/h may occur (Cathie and Wintgens, 2001). Increasing the Methocel viscosity further to allow this range to be investigated was not possible due to difficulties in saturating the model with the extremely viscous fluid. To overcome this, a series of additional tests with the gear ratio of the pulley system reduced to 1 were carried out, allowing scaled velocities of up to 600 m/h. This was done by changing the pulleys on the motor drive belt to be of equal diameter (78 mm).

The actuator is controlled by a DC motor speed controller, allowing the towing velocity to be set for each test, which is in turn activated by a National Instruments (NI) 9482 computer-controlled relay capable of providing 240 V mains AC power to the speed controller. The relay is mounted within an NI CompactRIO 9024 chassis, which networks the relay by way of an ethernet connection through the centrifuge's fibre optic data slip ring to a PC in the control room with a LabView interface. Hardware-based limit switches at each end of the container are used to protect the system from excessive travel in the event of a software failure.

Data acquisition was provided by an eight-channel Fylde μAnalog 2 system connected to an onboard PC by USB. Instrumentation was connected to the data logger by way of an Igus E08 series e-chain to control the cabling over the full travel distance of the actuator.

The horizontal displacement of the instrumentation platform was measured by a miniature Multicomp SP1-50 draw-wire transducer (DWT) mounted within the frame. Measurements of the vertical displacement of the plough were obtained from a Honeywell MLT004 linear position transducer with a 101 mm range secured to the instrumentation platform, allowing the plough depth to be measured throughout the test. The stainless-steel shaft of the position transducer rested directly above the back face of the plough on a 3 mm thick Perspex plate, which was parallel to the plough base (Figure 3(b)). This particular sensor was used due to its compact size, with the housing being only 9·5 mm in diameter and 132 mm long. The shaft was not spring loaded, and was free-falling so as not to apply additional force to the plough. The mass of the shaft was negligible at only 6·8 g. The force required to tow the plough was measured by a 5 kN Tedea Huntleigh type 616 ‘S’ beam load cell on the towing arm, to which the plough was attached by 2·5 mm diameter stainless-steel wire rope.

Measurements of the pitch and roll of the plough provide valuable data to assess the plough performance. Lauder et al. (2013) used a pair of Accustar clinometers to measure rotations in the two axes, however, these had an impact on the self-weight of the plough and were not suitable for centrifuge use due to their measuring system. A replacement system for measuring plough pitch and roll was used here, based around an Analog Devices ADXL377Z three-axis accelerometer mounted on the top surface of the plough. The accelerometer measures accelerations in the x, y and z axes in the same micro-electro-mechanical systems (MEMS) sensing unit, with a measurement range of ±200g. Coupled with the fact that it weighed only 50 g including mountings and measured 25 mm × 25 mm, the unit was ideal for centrifuge use. Technical specifications for the MEMS sensor are shown in Table 3.

Equations 3 and 4 were used to derive the pitch and roll from the measured accelerations (ADI, 2010). As the maximum horizontal acceleration of the plough at the start of the test was only 0·05 m/s² (four orders of magnitude lower than the enhanced gravity in the centrifuge) the impact of this dynamic effect on the pitch and roll measurements was deemed negligible. Using the Fylde Data Acquisition system, the MEMS tilt sensor provided a resolution of better than 0·05°. The accuracy of the sensor calibration could also be determined in flight by calculation of the total measured acceleration using Equation 5. Manual measurements of the orientation of the plough were taken using a hand-held inclinometer to verify the performance of the MEMS accelerometers at the start and end of the centrifuge test, and the results provided good agreement.

The centrifuge tests were carried out in the Geotechnical Centrifuge Facility at the University of Dundee, using the 3 m radius beam centrifuge (Davies et al., 2001; Milne et al., 2012). The centrifuge is capable of applying accelerations of up to 120g to model packages of 1 t, although an acceleration of 50g was used for this series of tests.

The pipeline plough tested was a 1:50 scale model of a type of plough commonly used offshore (Figure 3(b)). The model plough has a mass of 1·4 kg including instrumentation, corresponding to 175 t at prototype scale. The plough is the same as that used by Lauder et al. (2013) in the non-forecutter case, and has a maximum possible plough depth of 40 mm (2 m at prototype scale). The depth of the plough is controlled by varying the angle of the front skid arms with a shallower angle resulting in a deeper target depth. The skid arms were accurately set to the required angle prior to each test using a Vernier protractor, before being secured using thread-lock adhesive. The front skids themselves are connected to the skid arms by a free-moving pinned connection. While the plough is set at a target depth, the actual depth achieved is dependent on the test conditions and the balance of forces on the plough share, with higher sand densities and faster velocities (in saturated tests) leading to the plough running at a shallower depth (Lauder et al., 2013).

The 150 mm deep sand bed in which the plough is tested was prepared by air pluviation using a linear slot pluviator mounted on rails running above the container (relative densities from 40 to 85% were achieved). The sand bed was pluviated until a height just above the required depth was reached, before the sand surface was levelled. This was done using a Perspex surface scraper which was laser cut to the specific dimensions of the container. Next, the actuator system was placed and bolted into the container and the plough placed at the starting position.

Once the centrifuge had reached the required acceleration of 50g, a rest period of 5 min was allowed to let the sensors stabilise and to minimise any creep of the sand bed. The motor relay was then activated until a typical horizontal displacement of 750 mm (37·5 m prototype) had been applied at a speed of 6 mm/s. Wireless GoPro cameras mounted on the instrumentation platform and above the container allowed the plough progress to be monitored. A further 5 min rest period was allowed before the centrifuge was spun down, and the actuator removed from the container. The final depth of the plough relative to the sand surface was measured by hand, pitch and roll of the plough relative to the container was recorded using a digital inclinometer with a resolution of 0·1°.

A new low-cost contactless three-dimensional (3D) sand surface scanning system developed as part of this research was also used to capture the final surface and trench geometry of the entire model. The system is based around a 3D systems sense 3D scanner, and is capable of generating a 3D representation of the model accurate to within ±0·5 mm. The 3D output from the system can be imported in CAD software, allowing visualisation, extraction of accurate trench cross-sections and calculations of areas and volumes. A full description of the scanning system is given by Robinson et al. (2016b) along with verification of its performance and accuracy. The 3D scans also allowed more accurate measurement of the final plough depths than was possible with hand measurements. This was done by applying matt adhesive tape to the Perspex platform on the top of the plough, rendering it visible to the scanner.

As mentioned, while the apparatus can apply towing speeds of up to 50 m/h or 100 m/h depending on the gear ratio used, the enhanced gravity (N times) in the centrifuge caused the scaled plough velocities to be N times lower than would be achieved in a 1g test. This is due to the elevated hydraulic gradient in the pore fluid caused by the enhanced gravity, meaning that pore fluid flows and pore pressures dissipate N times faster. To account for this, a viscous pore fluid with a viscosity of N times water (1 cP) could be used to instead of water. In this case where N  =  50, methylcellulose with a viscosity of 50 cP could be used to match the 1g scaled velocities. However, to extend the range of scaled plough velocities to that used offshore, methylcellulose viscosities of up to 300 cP were used to achieve scaled velocities of up to 600 m/h. The scaled plough velocity (not to be confused with the normalised velocity, V) is determined by

where v_scaled is the scaled plough velocity, v_tow the applied towing speed, η_methocel the Methocel solution viscosity, η_water the viscosity of water and N the scaling factor, is the ratio of the enhanced gravity used relative to 1g.

Viscous drag forces due to the use of methylcellulose as a pore fluid have been estimated, and found to be insignificant compared with the measured tow forces. The use of methylcellulose to modify the pore fluid viscosity in centrifuge modelling has become commonplace following studies such as Stewart et al. (1998). It has previously been shown that the use of highly viscous pore fluids has no significant effect on either the shear stiffness degradation (Ellis et al., 2000) or the strength and stress–strain response (Zeng et al., 1998) of sands.

The methylcellulose used was Alfa Aesar HPMC with a specified viscosity of 10 750 cP in 2% aqueous solution at 20°C. The relationship between solution viscosity and HPMC concentration is described by (DCC, 2002)

where η is the methylcellulose solution viscosity and C the HPMC powder concentration by mass in which the viscosity constant, α is specific to each HPMC manufacturing batch. Testing using a Brookfield LV viscometer identified the viscosity constant for the HPMC used as 1·131.

The particular grade of methylcellulose used has a higher viscosity constant than that normally used in centrifuge modelling, meaning that a lower concentration of HPMC powder is required to achieve the same solution viscosity. This had the advantage of making the mixing of the high-viscosity pore fluids easier due to the lower solids content. Methylcellulose solutions maintain a constant viscosity up to a given strain rate, after which the solution viscosity reduces as the strain rate is increased. The lower concentration of the HPMC used in this study allows this constant viscosity to be maintained at higher strain rates than possible using methylcellulose of the same viscosity prepared using a higher concentration of HPMC powder with a lower viscosity constant (DCC, 2002). This makes it well suited to use in modelling of ploughing, where high strain rates occur.

To allow full hydration of the HPMC it must be dispersed in solution at temperatures above 90°C, in which case it is insoluble. Then the solution requires to be cooled to below 20°C to fully hydrate (DCC, 2002). The solution was created in 6 litre batches by adding the required amount of HPMC with one-third of the total amount of water as boiling water and mixing for 2 min to ensure full dispersion. At this point, the remaining water was added (one third as ice and the final third at room temperature) in order to reduce the temperature to 10–15°C (below the 20°C required for hydration to start), and the solution was mixed again. This was then stored in a sealed container until required. The viscosity of the final HPMC was again checked using the viscometer once the temperature had reached room temperature and was found to vary by no more than 2% from the target viscosity.

The sand bed was prepared in the same manner as the dry tests, except with the addition of a 35 mm thick layer of gravel below the sand bed, which was covered with a layer of geotextile membrane. The overall height was kept constant meaning the sand layer was 115 mm thick. Once the model was loaded onto the centrifuge gondola and the plough and actuator were in place, the model was saturated with Methocel solution by a valve on one end of the container located at the mid-height of the gravel layer. This allows the solution to saturate the model evenly across its entire length from the base up. The saturation was carried out slowly over a period of approximately 12 h by using the valve to regulate the flow of the solution, so as not to cause uplift of the sand bed. The Methocel solution level was brought to 50 mm above the sand surface to ensure that the spoil heaps generated during ploughing remain fully covered, and the test was run in the same manner as the dry tests at the required towing speed (Table 4).

To allow surface scanning of the saturated test beds, the pore fluid was allowed to drain off at 1g until the pore fluid was below the sand surface prior to undertaking scanning as described above. This was done before the container was unloaded from the gondola in order to prevent disturbance.

Table 4 shows the key aspects of the tests conducted, consisting of 22 centrifuge tests in total. Typical data obtained from the centrifuge tests is shown in Figures 4 and 5. This is similar to that obtained from previous 1g testing (Lauder et al., 2013) and shows a transition zone where the ploughs increase in depth to their target depths, which are dictated by the skid settings. The end of the transition zone coincides with reaching a constant plough depth and near-constant tow force. Here, steady state is reached after 350 mm or 3·2 plough share lengths. Figure 5 shows this transition behaviour from the perspective of the plough kinematics where the plough inclination tends to about 0·2° (or close to zero) at transition. The inclination of the plough determined from the MEMS accelerometer system closely tracks the change in plough depth measured by the LVDT resting on the rear of the plough. The data also show that the plough is slightly deeper at its rear than the final trench depth determined post-test by the surface scanning system with a ratio of plough-to-trench depth of 0·88–0·93. This is caused by localised trench failure and the shape of the base of the plough. The plough fully transitioned to steady state within 350 mm in all tests. Once the steady-state zone is achieved this is then used to extract average tow force and plough depth for comparison of different plough tests and conditions. These key results from each of the tests are shown in Table 5.

The results presented in Figure 6 summarise the tow force–depth relationship for the dry centrifuge tests undertaken in loose, medium dense and very dense tests (D_r  =  25–27, 44–47 and 84–85%, respectively) undertaken using the 1:50 scale plough. A fit to the data using Equation 1 has been carried out to generate values of C_s to allow comparison with the previous 1g testing. The values of C_s varying from 11·8 to 13·4 (loose to very dense) are similar to those presented by Lauder et al. (2013), who suggested C_s varied from 12·2 to 13·6 in loose to dense sand. The values at 1g seem slightly higher and this may be as a result of low effective stress dilation but as previously described by Lauder et al. (2013) and Lauder and Brown (2014) this contribution to resistance (or scaling) effects will not be as large as anticipated due to the large amount of post-peak shearing occurring once steady state has been reached. Again previous concerns over the difference between 1g model testing and field-derived C_s values are also unfounded with a similar range of limited C_s values being found during centrifuge testing. Robinson et al. (2017) suggested that for cable ploughs C_s could be represented by 3K_p or K_p² (similar to lateral pile consideration) where K_p is the passive earth pressure coefficient. For the sand used here at critical state using K_p², C_s would equal 10·6, which is close to the value of C_s between 11·8 and 13·4 found in this study.

Comparison of the centrifuge tests with the 1g tests by Lauder (2011) shows relatively good fit to the data in terms of variation of tow force with depth (Figure 7) but it appears to under predict the overall magnitude of the tow force. This is because the 1g set-up used by Lauder (2011) included heavier ploughs due to the inclusion of the bulky Accustar clinometers and associated mounting details (replaced in the centrifuge testing by the lightweight MEMS accelerometer arrangement). Figure 8 shows the prototype tow force corrected for the individual plough weights (varying due to different onboard instrumentation), which suggests that the scaling used in 1g testing (and proven at various scales) is appropriate. It is noted that the 1:10 scale plough tests show increased shallow depth tow force but become better at increasing depth. This was due to subtle shape differences between the shapes of the base of the plough running from the heel to the toe. These differences in manufacture were not noticed until after testing and highlight the necessity for strict quality control on model manufacture when using models of different scales.

By obtaining very similar values using the two testing methods and various scales, these data show that parameter C_s may be derived from 1g testing without significant errors induced by the low effective stress levels in these tests.

To investigate the velocity dependence on plough performance, tests were carried out at a range of normalised velocities as summarised in Table 4, using the method described above. It was decided to limit the testing for rate effects investigation to medium dense and very dense sand due to limited centrifuge access time and the required number of repeat tests. Figure 9 shows the results of saturated testing in medium dense and very dense sand. It is clear that the rate effect increases significantly with increasing depth in the very dense sand (with initial target depths of 23 and 33 mm, respectively). The non-linear increase in the tow force with plough speed is a result of the plough tending to reduce in depth as tow force increases to maintain moment equilibrium (Brown et al., 2015). Figure 10 compares the normalised force or rate effect with the normalised plough velocity as proposed by Lauder et al. (2012) with data presented from the previous 1g testing (Figure 2). It can be seen that there is considerable scatter in the data with the dynamic rate effect coefficient determined for the centrifuge tests (C_dn  =  12·3) being significantly lower than the values previously associated with 1g tests (C_dn  =  20·4). The scatter in the data although is as a result of the change in depth during ploughing, which reduces with increasing tow force and results in lower values of F_v=0 with increasing rate of testing (Figure 11). Thus, the normalisation of the tow force based on the static force at the target depth is not appropriate and this calculation needs to reflect the steady-state depth during the tests (F^*_v=0). Figure 12 shows similar data but with the normalised force calculated based on the relationship derived in Figure 11. The best-fit lines to the corrected 1g data and centrifuge data result in very consistent average behaviour with C_dn values of 20·4 and 19·8, respectively. One significant outlier point although is still apparent for the medium dense test at 35 mm target depth. This test was disturbed during handling and after saturation, resulting in (unknown) positive pore pressures and subsequent consolidation to a lower voids ratio resulting in an unknown dilation potential (s in Equation 2). It is anticipated that the sand would have densified and would thus have a higher value of s and would therefore be consistent with the other data shown in Figure 12. After that all other centrifuge tests were saturated on the centrifuge arm to minimise handling effects.

To verify scaling assumptions prior to further small 1g ploughing investigations, the equipment was developed to allow this large displacement event to be simulated on a medium-sized beam centrifuge using a specially developed long centrifuge box and actuation system. This system was capable of plough tests of 800 mm displacement, which was adequate to allow plough transition and determined steady-state behaviour. With the use of the purpose-built actuation it was possible to achieve scale velocities of 1 m/h using water and through the use of methycellulose solution it was possible to increase this much further to 600 m/h to study the scaling of rate effects over the range of speeds normally encountered during real ploughing projects. This testing was further enhanced through the use of MEMS accelerometers to measure plough pitch and roll and a newly developed low-cost 3D sand surface scanning technique.

The results of this centrifuge study have verified that in the case of offshore ploughing it is possible to use small-scale ploughs at 1g even in very dense sands without encountering significant difficulties with dilation effects. It also shows that previously proposed methods of scaling are valid at 1g although care needs to be taken during rate effects studies to consider the appropriate depth of the plough during testing as this may vary significantly from target depths due to increased rate-dependent tow forces. The study shows that there are significant benefits in physical modelling of complex full-scale events where it is difficult to deconvolute behaviour. This has led to verification of improved parameters for the prediction of pipeline plough performance.

The model seabed ploughing described in this paper has been undertaken for the EPSRC funded project Seabed ploughing: modelling for infrastructure installation (EP/M000362/1 and EP/M000397/1). The 3D surface scanning technique and centrifuge ploughing actuation systems have been developed as part of the Scottish Marine and Renewables Test (SMART) Centre at the University of Dundee, funded by the European Regional Development Fund (ERDF).

A

acceleration change measured from micro-electro-mechanical systems sensor excluding enhanced gravity

C

methylcellulose powder concentration by weight

C_d

dynamic or rate effect coefficient

C_dn

dimensionless dynamic or rate effect coefficient

C_s

dimensionless passive pressure coefficient

C_w

plough interface friction ratio

c′

apparent cohesion

c_v

coefficient of consolidation

D

plough depth

D₁₀

representative particle size

D₅₀

mean sand particle size

D_r

sand relative density

D_v=0

equivalent plough depth at zero velocity

E′₀

one-dimensional Young's modulus

F

total plough tow force

F_v=0

plough tow force at the equivalent of zero velocity or determined during dry testing

F^*_v=0

plough tow force at the equivalent of zero velocity corrected for plough depth changes

F_w

interface friction dependent component of tow force

g

standard gravitational acceleration

K_p

coefficient of passive earth pressure

N

scaling factor

s

dilation potential

V

normalised plough velocity

v

plough velocity

W

plough weight

α

hydroxypropyl methylcellulose (HPMC) viscosity constant

γ

unit weight of the soil

η

methylcellulose dynamic viscosity