WISPP: Wave Induced Stresses and Pore Pressures in the Seabed

WISPP Finite Element Model

Wave Induced Stresses and Pore Pressures

Wave Induced Stresses and Pore Pressures (WISPP) is a one dimensional finite element model!
Written by Dr. Stephen Thomas at Oxford Geotechnica International. Based on work at the Soil Mechanics Group, Oxford University, Department of Engneering Science.

What is WISPP?

The numerical model WISPP (Wave Induced Stresses and Pore Pressures) is a hybrid Finite Element – Analytical Model developed to simulate the time and spatially dependent stresses, pore pressures and soil displacements in a seabed when subject to sinusoidal wave loading.

The model simulates a sinusoidal wave loading on a seabed soil of finite thickness, with the vertical domain discretized in to 300 elements. The variability in the direction of wave travel, and the variability with time is simulated using analytical methods using complex number mathematics, i.e. real and imaginary conditions.

WISPP Governing Equations

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The model simulates the two-dimensional plane strain Biot equations of soil displacement and pore pressure in the vertical (`z`) and horizontal (`x`) directions, together with variability in time (`t`). The governing equations solved are as follows:

from equilibrium in the `x` direction,

`(lambda + 2G)(del^2w_x)/(del x^2) + G (del^2w_x)/(delz^2) + lambda (del^2 w_z)/(delxdelz) + G(del^2w_z)/(delzdelx) - (delu_w)/(delx) = 0 `

from equilibrium in the `z` direction,

`(lambda + 2G)(del^2w_z)/(del z^2) + G (del^2w_z)/(delx^2) + lambda (del^2 w_x)/(delzdelx) + G(del^2w_x)/(delxdelz) - (delu_w)/(delz) = 0 `

and from continuity principles incorporating Darcy's Law of fluid flow through a deformable porous medium

` del/(delx)[(k_(x x))/(gamma_w)(delu_w)/(delx) + (k_(xz))/(gamma_w)(delu_w)/(delz)] + del/(delz)[(k_(z x))/(gamma_w)(delu_w)/(delx) + (k_(zz))/(gamma_w)(delu_w)/(delz)] = C(delu_w)/(delt) + del/(delt)[(delw_x)/(delx) + (delw_z)/(delz)]`

The variation in time (`t`) and horizontal distance (`x`) is simulated in real and imaginary space:

`w_x(x, z, t,) = barw_x(z) exp[i(ax - omegat)]`

`w_z(x, z, t,) = barw_z(z) exp[i(ax - omegat)]`

`u_w(x, z, t,) = baru_w(z) exp[i(ax - omegat)]`

Resulting in the governing equation in real and imaginary conditions:

from equilibrium in the `x` direction,

`-a^2(lambda+2G)w_x + G(del^2w_x)/(delz^2) + ia(lambda + G) (delw_z)/(delz) - iau_w = 0`

From equilibrium in the `z` direction,

`(lambda +2G)(del^2w_z)/(delz^2) - a^2Gw_z + ia(lambda +2G)(delw_x)/(delz) - (delu_w)/(delz) = 0`

and from the continuity equation and Darcy's Law.

`-a^2(k_(x x))/(gamma_w)u_w + (k_(zz))/(gamma_w)(del^2u_w)/(delz^2) + ia(k_(xz))/(gamma_w)(delu_w)/(delz) + ia(k_(zx))/(gamma_w)(delu_w)/(delz) = - iomegaCu_w + aomegaw_x - iomega(delw_z)/(delz)`

`x`	horizontal axis (m)
`z`	depth into the seabed (m)
`t`	time (s)
`T`	wave period (s)
`L`	wavelength (m)
`lambda`	Lame's elastic constant (kPa)
`G`	elastic shear modulus (kPa)
`K`	drained bulk modulus (kPa)

`C`	compressibility coefficient (kPa^-1)
`k`	coefficient of permeability (m s^-1)
`gamma_w`	specific weight of water (kNm^-3)
`w_x`	total displacement in the horizontal direction (m)
`w_z`	total displacement in the vertical direction (m)
`u_w`	pore water pressure (kPa)

`P_0`	pressure wave amplitude (kPa)
`a`	wavenumber `((2pi)/L)` (1/m)
`omega`	angular velocity `((2pi)/T)` (1/s)
`i`	imaginary unit `(sqrt(-1))`

Background

Dynamic wave loading on the seabed originates as a surface water wave that propagates along the water surface. A certain proportion of the energy held by this wave will be transferred down through the water and will be felt on the seabed as a dynamic load as the wave passes overhead. The proportion of wave energy that reaches the seabed will be controlled by the wavelength and the water depth. To the designer of the foundation of an offshore structure, it is essential that the behaviour of the seabed under wave loading is understood before a safe design can be made.

Two main conditions must be considered in regard to the wave loading. The first, most obvious, is to evaluate the maximum stress envelope throughout the soil domain. To obtain this, the maximum wave induced stresses must be added to those stresses caused by the deployment of the proposed structure. The second condition that must be be considered is the reduction in soil strength due to the generation of pore water pressures, which occurs due to repeated cyclic loading on a soil. The magnitude of the pore water pressures generated will depend upon a number of factors, including the amplitude of the soil stress and the frequency and duration of loading. In addition, the soil permeability is important as this controls the rate at which the generated pore water pressures dissipate.

For the above conditions, it is often necessary to evaluate the wave induced stresses, displacements and pore pressures in a saturated or unsaturated seabed with the variable material properties and initial conditions.

WISPP Inputs

Default values have been provided to get you started with WISPP

Title

Text

Plase enter a Title for your model.

Wavelength (L)

Plase enter a value for the Wavelength.

Wave Period (T)

Plase enter a value for the Wave Period.

Drained Shear Modulus of Soil (G)

kPa

Plase enter a value for the Shear Modulus.

Drained Bulk Modulus of Soil (K)

kPa

Plase enter a value for the Drained Bulk Modulus.

Seabed Thickness (D)

Plase enter a value for the Seabed Thickness.

Pressure Wave Amplitude on the Seabed (P₀)

kPa

Plase enter a value for the Pressure Wave.

Compressibility of Pore Fluid (c_f)

kPa^-1

Plase enter a value for the Fluid Compressibility.

Soil Porosity (∅)

decimal

Plase enter a value for the Soil Porosity.

Weight Density of Pore Fluid (γ_f)

kNm^-3

Plase enter a value for the Fluid Unit Weight.

Horizontal Coefficient of Permeability (k_x)

m s^-1

Plase enter a value for the Horizontal Permeability.

Vertical Coefficient of Permeability (k_z)

m s^-1

Plase enter a value for the Vertical Permeability.

Seafloor Characteristics

Choose either Permeable or Impermeable

Permeable Impermeable

Sediment Base Characteristics

The base of the sediment is modelled as rough

Smooth Rough

Run Wispp!

Wispp Output

Wispp outputs 20 columns of data for 300 elements.
You can view the results from WISPP in the graphs below.
Alternativley you can download the data in an excel template. This will give you more control over the graphs.
The data column headings are described below.
You will need to enable active content in the excel workbook to allow easy navigation between the graphs.

Graphs

View the graphs for Maximum Diplacement, Horizontal Displacement and Vertical Displacement

View the graphs for Deviator Stress, Horizontal Effective Stress, Vertical Effective Stress, and Vertical minus Horizontal Effective Stress

View the graphs for Pressure Modulus and Pore Water Pressure

Disclamier

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