Seabed nonlinear soil model theory

$\newcommand{\Kmax}{K_\mathrm{max}}$ $\newcommand{\Pu}{P_\mathrm{u}}$ $\newcommand{\Pusuc}{P_\textrm{u-suc}}$ $\newcommand{\EULz}{E_\mathrm{UL}\!(z)}$ $\newcommand{\ERPz}{E_\mathrm{RP}\!(z)}$

The nonlinear soil model has been developed in collaboration with Prof. Mark Randolph FRS (Centre for Offshore Foundation Systems, University of Western Australia). It is a development from earlier models that used a hyperbolic secant stiffness formulation, such as those proposed by Bridge et al and Aubeny et al.

The data used by the nonlinear soil model, and its suitability for different seabed types, are discussed under the seabed data topic.

Full details of the nonlinear soil model are given in Randolph and Quiggin (2009). The main aspects of the model are:

These features are described in more detail below.

Penetration modes

Figure: Soil model penetration modes

The penetration mode of a given penetrator (e.g. a node on a line, a vertex of a 6D buoy or the origin of a 3D buoy) is determined by its penetration, $z$, and (in the dynamic analysis) by whether the penetration has increased or decreased since the previous time step. The details are as follows.

Ultimate resistance limits

The resistance formulae are arranged so that as penetration $z$ increases (for penetration) or decreases (for uplift) then the resistance asymptotically approaches the ultimate penetration resistance $\Pu(z)$ (for penetration) or the ultimate suction resistance $\Pusuc(z)$ (for uplift). These ultimate penetration and suction asymptotic limits are given by \begin{align*} \Pu(z) &= N_\mathrm{c}(z/D)\ s_\mathrm{u}(z)\ D \\ \Pusuc(z) &= -f_\textrm{suc}\ \Pu(z) \end{align*}

where

$s_\mathrm{u}(z)=$ undrained shear strength at penetration $z$. This is given by $s_\mathrm{u}(z) = s_\mathrm{u0}+\rho z$, where $s_\mathrm{u0}$ is the undrained shear strength at the mudline and $\rho$ is the undrained shear strength gradient, both of which are specified by the seabed soil properties data.

$D=$ penetrator contact diameter. For 3D buoys and 6D buoys, the contact diameter is taken to be the square root of the contact area (see 3D buoy contact area and 6D buoy theory). For lines, the contact diameter is as specified by the line type contact data.

$N_\mathrm{c}(z/D)=$ bearing factor. For $z/D \ge 0.1$ this is modelled using the power law formula $N_\mathrm{c}(z/D) = a(z/D)^b$, where $a$ and $b$ are the non-dimensional penetration resistance parameters. For $z/D \lt 0.1$ the formula $N_\mathrm{c}(z/D) = N_\mathrm{c}(0.1)\ (10z/D)^{1/2}$ is used instead, which gives a good approximation to the theoretical bearing factor for shallow penetration.

$f_\textrm{suc}=$ non-dimensional suction resistance ratio parameter.

Penetration resistance formulae

In not in contact mode the penetration resistance $P(z)$ is zero.

In the other three modes the resistance $P(z)$ is modelled using formulae that involve the following variables:

$\zeta = \frac{z}{D/\Kmax}$. This is the penetration, but non-dimensionalised to be in units of $D/\Kmax$, where $\Kmax$ is the normalised maximum stiffness parameter.

$z_0$, the penetration $z$ at which the latest episode of this contact mode started, i.e. the value at the time the latest transition into this contact mode occurred.

$\zeta_0 = \frac{z_0}{D/\Kmax}$, the non-dimensionalised penetration at which the latest episode of this contact mode started.

$P_0$, the resistance $P(z)$ at which the latest episode of this contact mode started.

Initial penetration mode

The starting penetration and resistance values, $z_0$ and $P_0$, are both zero. The penetration resistance is then given by \begin{equation} \label{PzIP} P(z) = H_\mathrm{IP}\!(\zeta)\ \Pu(z) \end{equation} where

$H_\mathrm{IP}\!(\zeta) = \frac{\zeta}{1+\zeta}$.

The term $H_\mathrm{IP}\!(\zeta)$ is a hyperbolic factor that takes the value 0 when $\zeta{=}0$ (when initial penetration starts), 0.5 when $\zeta{=}1$, i.e. when $z = D/\Kmax$, and asymptotically approaches 1 as penetration becomes large compared to $D/\Kmax$. The purpose of this factor is to provide a high initial stiffness while ensuring that the penetration resistance $P(z)$ rises smoothly from zero when contact first starts (when $\zeta$ and $z$ are both 0) and asymptotically approaches the ultimate penetration resistance, $\Pu(z)$ as $\zeta$ becomes large (i.e. if $z$ becomes large compared to $D/\Kmax$). This is illustrated by the blue curve in the model characteristics diagram below.

Uplift mode

The penetration resistance is given by \begin{equation} \label{PzUplift} \tag{2a} P(z) = P_0 - H_\mathrm{UL}\!(\zeta_0-\zeta)\ (P_0-\Pusuc(z)) \end{equation} but (see below) subject to a suction limit. Here \begin{align*} H_\mathrm{UL}\!(\zeta_0-\zeta) &= \frac{\zeta_0-\zeta}{A_\mathrm{UL}\!(z)+\zeta_0-\zeta} \\ A_\mathrm{UL}\!(z) &= \frac{P_0-\Pusuc(z)}{\Pu(z_0)} \end{align*}

The term $H_\mathrm{UL}\!(\zeta_0-\zeta)$ is a hyperbolic factor that is 0 when $\zeta{=}\zeta_0$ at the start of this uplift, and asymptotically approaches 1 if the non-dimensional uplift $\zeta_0-\zeta$ becomes large compared to $A_\mathrm{UL}\!(z)$. So, in uplift mode, the resistance given by equation (\ref{PzUplift}) drops from the value $P_0$ it had when this uplift started, and asymptotically approaches the (negative) ultimate suction resistance $\Pusuc(z)$ as $\zeta_0-\zeta$ becomes large relative to $A_\mathrm{UL}\!(z)$. See, for example, the green curve in the model characteristics diagram below.

Suction limit

Experiments (Bridge et al) have found that suction resistance can only be sustained for a limited displacement beyond the point at which the net resistance becomes negative, and suction then decays as uplift continues further. To model this, the resistance given by equation (\ref{PzUplift}) is limited to be no less than (i.e. no more suction than) a negative lower bound $P_\textrm{min}(z)$, given by \begin{equation} \label{PzMin} \tag{2b} P_\textrm{min}(z) = \EULz\ \Pusuc(z) \end{equation}

in which

\begin{equation*} \EULz = \exp\left(\min\left\{0, \frac{z-z_\textrm{P=0}}{\lambda_\textrm{suc} z_\textrm{max}}\right\}\right) \end{equation*}

where

$z_\textrm{P=0}=$ largest penetration $z$ at which suction has started during any uplift

$z_\textrm{max}=$ largest-ever penetration $z$ for this penetrator

$\lambda_\textrm{suc}=$ non-dimensional normalised suction decay distance parameter.

The exponent in the expression for $\EULz$ is zero or negative, so $\EULz \leq 1$. The value of $\EULz$ is 1 when $z \geq z_\textrm{P=0}$, but decays towards zero if the penetration $z$ is less than the largest penetration, $z_\textrm{P=0}$, at which suction has ever occurred during uplift. The effect of this is that the term $P_\textrm{min}(z)$ limits suction to be no more than $\Pusuc(z)$ when the first uplift starts, but as the penetrator lifts up higher (relative to the maximum penetration at which suction has ever occurred during uplift) then the suction is limited more. This models the suction decay effect which has been seen in experiments.

Repenetration mode

Here, the penetration resistance is given by \begin{equation} \label{PzRepenetration} \tag{3a} P(z) = P_0 + H_\mathrm{RP}\!(\zeta-\zeta_0)\ (\Pu(z)-P_0) \end{equation} but (see below) subject to a repenetration resistance upper bound. Here \begin{align*} H_\mathrm{RP}\!(\zeta-\zeta_0) &= \frac{\zeta-\zeta_0}{A_\mathrm{RP}\!(z)+\zeta-\zeta_0} \\ A_\mathrm{RP}\!(z) &= \frac{\Pu(z)-P_0}{P_\mathrm{u*}} \\ \end{align*} \begin{equation*} P_\mathrm{u*} = \begin{cases} \Pu(z) & \text{if $P_0\leq0$, i.e. if this repenetration started from a zero or negative resistance} \\ \Pu(z^*) & \text{if $P_0\gt0$, where $z^*$ is the penetration when the preceding episode of uplift started} \end{cases} \end{equation*}

and $\zeta_0$ and $P_0$ are the non-dimensional penetration and resistance at the start of this repenetration.

The term $H_\mathrm{RP}\!(\zeta-\zeta_0)$ in equation (\ref{PzRepenetration}) is a hyperbolic factor that has the value 0 when $\zeta{=}\zeta_0$ at the start of this repenetration, and asymptotically approaches 1 if the non-dimensional repenetration $\zeta{-}\zeta_0$ becomes large compared to $A_\mathrm{RP}\!(z)$. So the repenetration mode resistance given by equation (\ref{PzRepenetration}) rises from its value $P_0$ when this repenetration starts, and asymptotically approaches the ultimate penetration resistance $\Pu(z)$ if $\zeta{-}\zeta_0$ grows large compared to $A_\mathrm{RP}\!(z)$. See the purple curve in the model characteristics diagram below.

Repenetration resistance reduction after uplift

Experiments (Bridge et al) have found that when repenetration occurs following large uplift movement, the repenetration resistance is reduced until the previous maximum penetration is approached. To model this behaviour, the repenetration resistance given by equation (3a) is limited to be no greater than an upper limit $P_\textrm{max}(z)$ given by \begin{equation} \label{PzMax} \tag{3b} P_\textrm{max}(z) = \ERPz\ P_\mathrm{IP}(z) \end{equation} in which \begin{equation*} \ERPz = \exp\left(\min\left\{0, -\lambda_\textrm{rep} + \frac{z-z_\textrm{P=0}}{\lambda_\textrm{suc} z_\textrm{max}}\right\}\right) \end{equation*}

and

$P_\mathrm{IP}(z)=$ penetration resistance that initial penetration mode would give at this penetration, as given by equation (\ref{PzIP})

$z_\textrm{max}=$ largest-ever penetration $z$ for this penetrator

$z_\textrm{P=0}=$ largest penetration $z$ at which suction has started during any uplift

$\lambda_\textrm{suc}=$ non-dimensional normalised suction decay distance parameter

$\lambda_\textrm{rep}=$ non-dimensional repenetration offset after uplift parameter.

The exponent in the expression for $\ERPz$ is zero or negative, so $\ERPz\leq1$. The value of $\ERPz$ will be strictly less than 1, so limiting the repenetration resistance to be less than the ultimate penetration resistance $\Pu(z)$, until the penetration $z$ exceeds $z_\textrm{P=0}$ by a certain amount quantified by $\lambda_\textrm{rep}$. This models the effect that repenetration following large uplift movement shows reduced resistance until the previous maximum penetration is approached.

Model characteristics

The following diagram illustrates the effect of the above equations, for all three modes, as penetration changes for a catenary line moving up and down on the seabed.

Figure:Soil model characteristics

The model starts in initial penetration mode and gives a resistance (blue curve, see note (1) in diagram) that increases as the pipe sinks into the seabed, and asymptotically approaches the ultimate penetration resistance $\Pu$ (upper dashed grey curve).

Then, when the pipe starts to lift up again, the model enters uplift mode and the resistance falls (green curve, see note (2) in diagram) and asymptotically approaches the ultimate suction resistance $\Pusuc$ (lower dashed grey curve). In this case the uplift is enough that the resistance becomes negative – i.e. suction (note (3) in diagram).

If the uplift continues and the pipe lifts off the seabed, then the model stays in uplift mode and continues along the green curve (note (4) in diagram). The suction reduces as the uplift continues, and drops to zero when the penetration drops to zero.

If, however, the uplift ends and repenetration starts, then the model enters repenetration mode (purple curve, note (5) in diagram). The suction rapidly falls and instead soon becomes positive resistance. As repenetration continues the resistance continues to rise (note (6) in diagram) and again asymptotically approaches the ultimate penetration resistance.

Further cycles of uplift and repenetration would give further episodes of uplift and repenetration modes and so lead to hysteresis loops of seabed resistance.

Soil extra buoyancy force

The seabed resistance formulae above model the resistance $P(z)$ due to the soil shear strength. In addition, there is an extra buoyancy force due to the fact that the penetrator displaces soil that has a higher saturated density than the water. To model this contribution, an extra buoyancy force is applied, vertically upwards, in addition to the resistance $P(z)$. This force has magnitude $b$ given by \begin{equation*} b = f_\mathrm{b}\ V_\textrm{disp}(\rho_\textrm{soil}-\rho_\textrm{sea})\ g \end{equation*}

where

$f_\mathrm{b}$ is a non-dimensional soil buoyancy factor.

$V_\textrm{disp}=$ displacement volume = volume of the part of the penetrating object that is below the seabed tangent plane

$\rho_\textrm{soil}=$ saturated soil density.

$\rho_\textrm{sea}=$ sea water density at the seabed origin.

$g=$ acceleration due to gravity for the units being used.

The factor $f_\mathrm{b}$ is normally greater than 1. This reflects the fact that when seabed soil is displaced it does not disperse thinly across the seabed plane, but instead tends to heave locally around the penetrating object. The effect of this is that the extra buoyancy is greater than the standard theoretical buoyancy force $V_\textrm{disp}(\rho_\textrm{soil}-\rho_\textrm{sea})\ g$ that would apply if the soil was fully fluid.

Soil model parameters

Several non-dimensional constants are used in the formulae given above for the seabed normal reaction force. These are parameters that control how the soil response is modelled by the nonlinear soil model. Their values can be edited on the seabed soil model page of the environment data form, but we recommend that these parameters are normally left at their default values.

Penetration resistance parameters

The parameters $a$ and $b$ control how the bearing factor $N_\mathrm{c}(z/D)$, and hence the ultimate penetration and suction resistance limits $\Pu(z)$ and $\Pusuc(z)$, vary with penetration $z$. See ultimate resistance limits above.

Soil buoyancy factor

The factor $f_\mathrm{b}$ that controls the modelling of the extra buoyancy effect that occurs when a penetrating object displaces soil, as described by soil extra buoyancy force above. The buoyancy factor should normally be greater than 1, to model the fact that the displaced soil tends to heave locally around the penetrating object.

Normalised maximum stiffness

The factor $K_\textrm{max}$ that determines the reference penetration $D/K_\textrm{max}$ used to calculate the non-dimensional penetration values, $\zeta$ and $\zeta_0$, used in the hyperbolic factors in the formulae. It controls the maximum stiffness during initial penetration or on reversal of motion, and the speed at which the penetration resistance asymptotically approaches its limiting value. A higher value means the resistance more rapidly approaches the limit as penetration changes, and so gives a stiffer seabed model.

Suction resistance ratio

The factor $f_\textrm{suc}$ that controls the ultimate suction resistance $\Pusuc(z)$. See ultimate resistance limits above. A lower value gives less suction, a higher value gives more.

Normalised suction decay distance

The factor $\lambda_\textrm{suc}$ that controls the suction decay limit term $P_\textrm{min}(z)$ in equation (\ref{PzMin}) in uplift mode. A lower value gives less suction effect, by causing suction to decay after less uplift. A higher value causes suction to persist over greater uplift distances. This parameter also affects the repenetration limit term $P_\textrm{max}(z)$ in equation (\ref{PzMax}) in repenetration mode.

Repenetration offset after uplift

The parameter $\lambda_\textrm{rep}$ that controls the penetration at which the repenetration resistance limit, $P_\textrm{max}(z)$ in equation (\ref{PzMax}) in repenetration mode, merges with the bounding curve for initial penetration resistance $P_\mathrm{IP}(z)$. A smaller value results in less penetration past $z_\textrm{P=0}$ before the repenetration resistance after uplift merges with the bounding curve of initial penetration resistance. A higher value leads to greater penetration before the bounding curve is reached.