Penetration Model for Installation of Skirted Foundations in Layered Soils

The flow ($q$) was evaluated at the stress integration point; based on this flow and the applied pressure difference ($H$), the critical gradient was calculated as shown in Eq. (7)

Fig. 4 shows the results of the Pure Flow analyses where the surface boundary is open, together with the new equation presented in Eq. (4). A comparison with the results from Erbrich and Tjelta (1999) and Andersen et al. (2008) is included in Fig. 4(a) for the case where there is a large distance to a lower impermeable layer. The results of the three analyses are practically identical. The small difference seen between the studies may be related to variation in the exact location where the gradient was evaluated. Nevertheless, the good comparison with previously presented studies confirms the validity of the FE model. The analyses show that the seepage flow and thereby the critical suction gradient was affected only by the impermeable layer when the distance between the skirt penetration depth and the impermeable layer was smaller than approximately $D/2$ (i.e., $z_b/D\le 0.5$).

Similar analyses were carried out with an impermeable surface boundary. This boundary condition represented the Part Flow case where the skirt penetrated through an impermeable layer into a permeable layer. Fig. 5 presents the results of these analyses. The critical suction number is higher here compared to the Pure Flow situation, especially for short penetration depths. It is noted that, even though the critical suction numbers for short penetration depths and short distances to the impermeable layer are not included in all the graphs in Fig. 5; these points were used to calibrate the model. They were omitted from the figures because their values were so relatively large that they could not be shown within the present scale.

To illustrate the accuracy of the calibrated flow models, the predicted results are compared with the results from the finite-element analyses in Fig. 6. It can be seen that the simple model calculates the critical suction number with a high degree of accuracy for all investigated analyses, including the very high values omitted from Fig. 5.

The water flow out of the caisson can be used to estimate the plug lift. The flow rate, $q$, was found using the proposed modification to Hvorslev’s theoretical solution shown in Eq. (3). During the Part Flow analysis, the average flow over the caisson area was also evaluated. These flow rates were calculated using Eq. (3). The calibration was similar to previous calibrations except it was ensured that the factors used would predict correct or larger flow rates because this would provide an estimate of the plug lift on the safe side. From this calibration, the following factors, to be used with Eq. (3), were found. These factors are shown in Eq. (8)

The predicted flow results are compared to the results from the finite-element analysis in Fig. 7 to show the quality of the calibration. The model does not show the same good match as seen with the critical suction number prediction, and it appears that the model consistently overestimates the flow. As a result, this model will overestimate the soil plug lift, which is also regarded as acceptable.

All flow analyses were performed using a linear elastic soil model, which does not capture the stress dependency of the soil stiffness and thereby affects the flow and introduces a small error. The fact that the model uses a combination of theoretical and empirical data masks this potential effect. An elastic soil model was also used in the previously proposed model (Andersen et al. 2008), together with the reduction in resistance shown in Fig. 2. The empirical observations are therefore linked to the calculation procedure, meaning that this effect is consider to be insignificant.

The model used to assess reduced penetration resistance due to seepage (Fig. 2) is based on model tests and field observations of suction caisson penetration in permeable soil profiles (Pure Flow). This model is also used here to predict the reduction in penetration resistance due to seepage in permeable layers underlying an impermeable layer (Part Flow). The reduction model presented in Fig. 2 is therefore used outside its calibration range. In order to evaluate the validity of this assumption, different flow conditions were examined. Fig. 8 presents normalized water head potential lines for eight boundary conditions. These eight analyses represent the extreme boundary conditions used in the calibration. Four analyses for the Pure Flow condition and four for the Part Flow condition are considered.

The following observations can be made from Fig. 8:

It can be seen that the mechanism controlling penetration resistance changes from case to case, and it is not possible to unambiguously conclude that the model will always over- or underestimate predictions. Even though the flow cases are different, a suction corresponding to a critical gradient is applied to both and it may therefore be assumed that the reduction model can be used as a reasonably good approximation. Further studies will be needed to confirm this assumption.

Fully coupled analyses were performed assuming a steady-state flow condition to calibrate a model that can predict the critical suction number. In reality, some time is required before a steady-state flow condition is achieved. Eight transient flow analyses were performed for the Pure Flow and Part Flow conditions in order to evaluate the time required to reach a steady state. Table 1 shows the approximate time until steady-state flow is reached. As may be expected, the time to reach steady state is much shorter for the Pure Flow case than for the Part Flow case because of the much shorter drainage path. It should be noted that these analyses are done by applying a pressure difference to a foundation under wished-in-place conditions at a given penetration depth. When a suction caisson is penetrating into the ground, a differential pressure has already been established in the previous installation depth. The actual time required to reach steady-state conditions at a new depth is approximated as the difference between the time it takes to reach steady state at the previous depth and the time to reach it at the new depth.

In order to analyze the FE results further, the time is normalized as proposed in Eq. (9) using the permeability inside the caisson, the oedometer stiffness of the soil ($M$), the effective unit weight (${\gamma }^{\prime }$), and a presumed representative drainage path of the water flow ($z+2D$). Because it was impossible to find a unique drainage length given the different drainage conditions, the presumed drainage length was chosen based on trial and error

Fig. 9 shows the results of the analyses shown in Fig. 8, now as normalized pressure head ($\mathrm{\Delta }H/H$) versus normalized time. It is seen that the chosen normalization does not yield a unique relation for Pure Flow conditions, but appears to work reasonably well for Part Flow conditions. This is again related to the change in drainage path. The proposed normalization is used later to estimate the soil plug lift for the Part Flow condition.

From Fig. 9, it can be seen that, for Part Flow conditions, the pressure decreases continuously. At a normalized time, $T=0.1$, it is about 30%; at $T=1$, it is about 15%; and at $T=10$, it has reached a steady state. Noticeably, the presence and distance of an underlying and gradually approaching impermeable layer seems to have no effect on the result. Although the analyses were performed with an elastic soil model that did not capture stress dependency, it is believed that the suggested approach provides a reasonable basis for assessing the time to reach-steady state conditions.

One of the main assumptions of the Part Flow model is that seepage flow is developed in the permeable sand layer, which is overlain by an impermeable layer. This can be achieved either by developing cracks in the soil plug above this permeable sand layer or by lifting the complete soil plug within the skirt compartments and thereby triggering the seepage flow required to reduce stresses in the soil and hence penetration resistance. It may not be acceptable, however, that the plug lift/heave is so high that a gap forms below the sealing layer or that an excessive loosening of the sand layer occurs. Thus the plug lift/heave potential needs to be assessed and has to be limited to an acceptable value. To estimate the plug lift, a simple, yet reasonably accurate method that generates sufficient flow is proposed.

The plug lift is evaluated using the average flow required to generate sufficient flow, $q$, from Eq. (3) for the critical suction ($S_{N,\mathrm{cr}}$) value from Eq. (4) and the additional time it takes to reach steady state at the corresponding depth ($\mathrm{\Delta }{t}_{\mathrm{ss}}$). The heave is evaluated using the time to reach steady state multiplied by the corresponding steady-state flow. This is a simplification because the rate prior to steady state will be different from the steady-state flow rate. Both the time and the flow are functions of current penetration depth into the permeable layer ($z$). Based on Fig. 9, it is reasonable to expect that steady state is reached at a normalized time equal to $T=1$. Using this time together with Eq. (9) enables calculation of the time required to generate steady-state conditions at the new depth. This is used together with the flow which is a function of the differential pressure required at the top of the permeable layer ($\mathrm{\Delta }{u}_{\mathrm{eq}}$).

To calculate the plug lift needed to generate sufficient flow, the following methodology is proposed:

Calculation of the plug lift is included in the general penetration algorithm; the required pressure difference together with the estimated plug lift is then an outcome of the analysis. The allowable plug lift and pressure difference needs to be evaluated from case to case. This calculation will overestimate the plug lift because it assumes that the skirt will penetrate only after reaching steady-state conditions. The penetration will most likely start before reaching steady-state conditions.

When a skirt penetrates into a permeable layer with seepage flow, the soil inside the caisson will loosen. The combination of reduction in effective stress, seepage gradient and shearing, will loosen the permeable soil layer trapped within the caisson. This loosening of the soil plug is important to understand not only because it is related to change in the permeability ratio $k_i/k_o$, which is an input parameter to the model, but also because the state of the soil trapped inside the caisson after installation and some distance below the skirt will control the in-place performance of the caisson. For both Pure Flow and Part Flow conditions, the vertical effective stresses at the top of the permeable layer inside the caisson are zero. The geometry of the foundation will also be the same between the two cases, but the flow condition will be different. Even though the flow cases are different, a pressure difference corresponding to a critical gradient is applied in both cases; it is therefore assumed that the mechanism controlling the loosening of the soil plug is identical for the Pure Flow and the Part Flow cases. For this reason, it is recommended to calculate the loosening of the soil based on the geometry of the caisson and the dilatancy of the soil. This method is described in Andersen et al. 2008.

Saue et al. (2017) presented the installation records for a suction anchor installed in a soil profile that is summarized in Table 2. The soil profile consists of two clay units (II and III) overlying a sand unit (IV). In Saue et al. (2017), the given strength profile was used together with a best estimate sensitivity factor of 2.2 in the clay layers (Units II and III) and strength anisotropy ratios of $s_u^{DSS}/s_u^C=0.8$ and $s_u^{AV}/s_u^C=0.6$. In the sand layer, an earth pressure coefficient of $K=0.8$ was used together with a ratio between inside and outside permeabilities estimated to be $k_i/k_o=5$ and with an effective friction angle derived from the measured tip resistance ($q_c$) of the CPT and the database of sands presented in Andersen and Schjetne (2013). An oedometer modulus of $M=\mathrm{40,000}\mathrm{kPa}$ together with an outside permeability of $k_o={10}^{-5}\mathrm{m}/\mathrm{s}$ was used for the plug lift analysis. The following parameters were not presented in Saue et al. (2017) and, as a result, were estimated for this analysis: $k_i/k_o$, $M$, $k_o$, and $t$.

The diameter of the anchor was $D=9\mathrm{m}$, with an assumed skirt wall thickness of $t=0.06\mathrm{m}$ and a skirt length of $s=12.5\mathrm{m}$. During installation, cycling of the differential water pressure was started after 7 m of penetration to reduce the differential pressure between the inside and the outside of the skirt. Thus, only penetration resistance to a depth of 7 m prior to the onset of cycling is considered in the following. For more details see Saue et al. (2017).

Fig. 10 shows a comparison between the recorded measurements from installation, the predictions using the No Flow model, and the predictions using the extended Flow model. As expected, it can be seen in the clay layer that the extended Flow and the No Flow models are identical. However, when the skirt tip starts to penetrate into the sand layer, the two models start to deviate. The No Flow model assumes that the clay above the sand layer makes a seal that prevents any seepage flow around the skirt tip, whereas the extended Flow model assumes that the clay plug is lifted and generates seepage flow in the underlying sand layer. The responses of the two models are fundamentally different. The resistance increases markedly in the No Flow model and decreases in the extended flow model. The reason for the increase in resistance in the No Flow model is that the sand has a very high tip resistance. The reason for a drop in resistance in the extended Flow model is that it assumes that the flow in the soil reduces stresses so much that tip resistance disappears. In this particular case, it can be seen that the equivalent weight of the anchor is $W_{\mathrm{eq}}^{\prime }=0$ because the outside penetration resistance in the clay layers is greater than the weight of the anchor and the clay part of the soil plug. Resistance is therefore calculated to be zero when using Fig. 2 for initial penetration into the sand layer. When looking at the measurement data from the installation in Fig. 10, it is clear that the big increase in resistance predicted by the traditional Flow model does not match the observation. Even though the extended Flow model does not provide a perfect match, it seems to capture the observation better. In order to fit the observation, a relatively large ratio between inside and outside permeabilities was used. This change in permeability cannot be explained only by dilation in the sand but must also be related to the soil plug lift and seepage flow that may force additional change in the relative density of the sand layer.

The plug lift estimate in this example accumulates during penetration in the permeable layer. At a penetration depth of $z=D$, the plug lift is found to be relatively small and acceptable for a design case.

Panayides et al. (2017) presented another case of a suction anchor installation in layered soils. The available information on soil conditions for the actual site was limited, but the authors presented a $q_c$ profile and a back-analysis of nearby anchor installations, on which the layering and soil parameters used here are based. The parameters used in the model are shown in Table 2. The soil parameters for the clay are based on a back-analysis and therefore represent the remolded average strength. A sensitivity factor of 1 and strength anisotropy ratios of $s_u^{DSS}/s_u^C=1$ and $s_u^E/s_u^C=1$ are assumed. In the sand layer, an earth pressure coefficient of $K=0.8$ was used together with $k_i/k_o=3.5$ and with an effective friction angle derived from the measured tip resistance ($q_c$) of the CPT and the database of sands presented in Andersen and Schjetne (2013). An oedometer modulus of $M=\mathrm{60,000}\mathrm{kPa}$ together with an outside permeability of $k_o={10}^{-4}\mathrm{m}/\mathrm{s}$ was used for the plug lift analysis. The diameter of the anchor was $D=7\mathrm{m}$, with an assumed skirt wall thickness of $t=0.04\mathrm{m}$ and a skirt length of $s=6.5\mathrm{m}$. The following parameters were not presented in Panayides et al. (2017) and were estimated for this analysis: $k_i/k_o$, $M$, $k_o$, and $t$.

Fig. 11 shows a comparison between recorded measurements from the installation, predictions using the No Flow model, and predictions using the extended Flow model. Similar observations as for Example 1 are seen in Example 2. In the clay layer, the extended Flow and No Flow models provide identical results. However, the moment the skirt tip starts to penetrate into the sand layer, the two models begin to deviate. The No Flow model assumes that the clay above the sand layer makes a seal that prevents any flow around the skirt tip, whereas the extended Flow model assumes that the clay plug is lifted and generates seepage flow in the underlying sand layer. Again, the responses from the two models are fundamentally different. The proposed extended Flow model accurately predicts the behavior observed in data measured during installation for the given $k_i/k_o$ ratio, and again it can be seen that the plug lift is acceptable.