WISPP
Finite Element Model

Wave Induced Stresses and Pore Pressures

About WISPP »

**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.**

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.**

If the equations do not display properly please use an updated version of your web browser.

If you are using google chrome, please install the MathJax Extension

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))` |

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.

Default values have been provided to get you started with WISPP

Title
Text

Wavelength (L)
m

Wave Period (T)
s

Drained Shear Modulus of Soil (G)
kPa

Drained Bulk Modulus of Soil (K)
kPa

Seabed Thickness (D)
m

Pressure Wave Amplitude on the Seabed (P_{0})
kPa

Compressibility of Pore Fluid (c_{f})
kPa^{-1}

Soil Porosity (∅)
decimal

Weight Density of Pore Fluid (γ_{f})
kNm^{-3}

Horizontal Coefficient of Permeability (k_{x})
m s^{-1}

Vertical Coefficient of Permeability (k_{z})
m s^{-1}

Choose either Permeable or Impermeable

The base of the sediment is modelled as rough

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.

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**

OGI's Software is for teaching and research purposes only

Software used on the OGI website is provided 'as is' without warranty of any kind, either express or implied, including, but not limited to, the implied warranties of fitness for a purpose, or the warranty of non-infringement. Without limiting the foregoing, OGI makes no warranty that:

**i).**the software will meet your requirements

**ii).**the software will be uninterrupted, timely, secure or error-free

**iii).**the results that may be obtained from the use of the software will be effective, accurate or reliable

**iv).**the quality of the software will meet your expectations

**v).**any errors in the software obtained from the OGI web site will be corrected.

Software and its documentation made available on the OGI web site:
**vi).**could include technical or other mistakes, inaccuracies or typographical errors. OGI may make changes to the software or documentation made available on its web site.

**vii).**may be out of date, and OGI makes no commitment to update such materials.

OGI assumes no responsibility for errors or omissions in the software or documentation available from its web site.

In no event shall OGI be liable to you or any third parties for any special, punitive, incidental, indirect or consequential damages of any kind, or any damages whatsoever, including, without limitation, those resulting from loss of use, data or profits, whether or not the OGI has been advised of the possibility of such damages, and on any theory of liability, arising out of or in connection with the use of this software.

The use of the software on the OGI web site is done at your own discretion and risk and with agreement that you will be solely responsible for any damage to your computer system or loss of data that results from such activities. No advice or information, whether oral or written, obtained by you from OGI or from the OGI web site shall create any warranty for the software.

© 2022 - OGI Groundwater Specialists Limited 2015

Tel: 0191 378 7010

Email: admin@ogi.co.uk

Company Reg. No 2448675

OGI Groundwater Specialists Ltd

One St John's Court

St John's Road

Meadowfield

Durham

DH7 8TP