Research Project Final

M.ENG RESEARCH PROJECT APRIL 2015 1
Investigation of seabed deformation using Plaxis
Kevin De Michelis supervised by Dr Mohammed Seaid
Abstract—This report investigates and develops a computa-
tionally inexpensive method to simulate wave propagation due to
sudden deformations in the seabed. The sudden deformation in
the seabed was modelled using the geotechnical software Plaxis,
the results of which were extracted and run by code written
in MATLAB to simulate the resultant wave propagation on the
free water surface. The soil deformation uses Linear Elastic and
Mohr-Coulomb constitutive models within the Plaxis package,
whereas the free water surface is modelled and tracked using
nonlinear shallow water equations in MATLAB. The coupling of
these two solvers will model three types of deformation as well
as a large seabed deformation in the Atlantic Ocean. The results
were analysed by examining the shape and magnitude of the soil
surface deformations, inspecting the wave propagation response
and inundation of dry land. The results showed that the Plaxis
- MATLAB combination handled various types of deformation
adequately, was quick and efficient at simulating accurate but
crude models and could be used to generate a very precise and
realistic model if desired. To produce the most realistic results a
better knowledge of soil modelling in Plaxis is necessary whereas
the nonlinear shallow water equations in MATLAB require a
little refinement to be as versatile as possible.
Index Terms—Plaxis 2D, MATLAB, nonlinear shallow water
equations, seabed deformation, free water response
I. INTRODUCTION
SUDDEN deformations in the earth’s crust cause a rise
or dip of the surface water level which, if large enough,
can propagate and become a tsunami. The 2004 tsunami in
the Indian Ocean is a prime example; over 230,000 people
had their lives taken, over a million and a half people were
displaced, and the damage and loss that the affected countries
suffered amounted to 8.71 billion US dollars [1], [2]. This
natural catastrophe is a repeating occurrence and the damage
from its consequences is one we wish to minimise or prevent
entirely. Therefore, a lot of research has gone into early
tsunami warning systems and tsunami simulations. This is
because accurately simulating tsunamis allows engineers to
properly plan evacuation routes and build appropriate coastal
defences, save lives and to minimise damage to infrastructure.
Consequently, research in tsunami modelling has increased
impressively over the last few decades and many hydrody-
namic models have been developed. A widely used method to
simulate tsunamis is the Method of Splitting Tsunami (MOST)
[3], developed by Dr. Vasily V. Titov of the Pacific Marine
environmental Laboratory and Synolakis of University of
Southern California. This model breaks the process down into
three stages; seabed deformation, wave propagation and the
inundation of dry land, and has proved to be successful when
compared to laboratory experiments and historical tsunamis.
A. Material deformation
The first phase in the MOST is the morphodynamics of the
soil. Modelling these deformations requires the use of Finite
Element Method (FEM). FEM is used to find an approximate
solution of boundary value problems with partial differential
equations using numerical analysis.
Typically, an earthquake with a magnitude of over 6.0 on
the Richter scale due to a dip-slip fault, has a rupture area
of 100 to 2000 km2
[4] while the height ranges from 0 -
15 m. Therefore, a two-dimensional model can be used when
analysing an earthquake mechanism. This was shown by an
early two-dimensional FEM model of a dip-slip fault that used
elastic dislocation theory developed by L. B. Freund and D.
M. Barnett in 1976. Freund and Barnett stated that "Although
the two-dimensional models may be over simplifications of
the physical system, they have been very useful in developing
insight into the relationships among the various fault parame-
ters." [5]. Furthermore, the two-dimensional model produced
results that were within an acceptable limit when compared to
data of a 1964 Alaskan earthquake. Thus proving that the use
of two-dimensional elastic dislocation theory is adequate for
analysis of deformations caused by dip-slip.
B. Wave propagation and inundation
The second and third phases of MOST involve hydro-
dynamics modelling the wave propagation. The propagation
of a wave induced by seabed deformation comes from its
starting profile, which originates from translating a vertical
displacement in the seabed onto the free surface of the water.
This is justified because generally, a seaquake area is large
with respect to the water depth, and the rupture velocity very
short with respect to the tsunami propagation velocity [6].
Once the wave profile is found, the wave propagation can
be calculated. This is done by using the nonlinear shallow
water equations, which are suitable when the water depth is
shallow when compared to the wavelength of the disturbance
[7]. These equations are derived from depth-integrating the
Navier-Stokes equations and are suitable for propagation of
waves in shallow water and inundation of non-breaking waves.
This report will focus on the deformation of the seabed as
opposed to the collapse of it. Additionally, large deformations
may not lead to tsunamis but can still cause flooding in
coastal zones from wave runup. Thus, this report investigates
three types of deformation using a new method to couple a
morphodynamic and a hydrodynamic solver, Plaxis 2D and
MATLAB respectively, to simulate wave propagation due to a
vertical displacement in the seabed. This follows the MOST,
though it is not limited to tsunami modeling.

II. GOVERNING EQUATIONS
This section describes the governing equations, in Plaxis
and the MATLAB code, used to solve the soil deformation
and wave propagation respectively. However, it is important
to note that the focus of this report is the free water response
to the deformation and not the mechanics within the soil. Thus,
the soil models are oversimplifications of real situations and
should not be taken as perfectly realistic as they are only a
tool to provide verification and qualitative evaluation of the
water response.
A. Linear Elastic and Mohr-Coulomb equations
The soil models have a simple two-dimensional setup: width
and depth, x and z respectively. The y direction can be ignored
because fault lengths can be taken as infinite [4], [8].
To solve the deformation within the soil models, this re-
port uses two constitutive models; Linear Elastic and Mohr-
Coulomb. This was done to investigate the adequacy and qual-
ity of the constitutive models. The Linear Elastic constitutive
model is described as suitable for large intact rock formations
but too crude for soil; however, it is computationally least
expensive [9]. Whereas the Mohr-Coulomb constitutive model
is described as a good first order approximation [9] and is more
suitable for soil modelling.
When Plaxis runs a simulation, all constitutive models
use the same calculation process, but differ when calculating
the plastic phase. The complete process can be viewed in
Appendix A [10]. This report investigates the deformation due
to a prescribed displacement at certain nodes, thus the first
calculation Plaxis solves is as follows:
{df } = [Kff ]−1
{{ff } − [Kfp]{dp}} (1)
Where {d} is the displacement vector, [K] is stiffness
matrix, {f} is load force vector, subscript f is free nodes and
subscript p is prescribed nodes. Thus the equation solves for
the displacement of the free nodes using the prescribed dis-
placement and the stiffness of the material. The displacement
of each element, in conjunction with the strain interpolation
matrix [B], is then used to calculate the change in strain {∆ε}.
{∆ε} = [B]{∆de
} (2)
The change in strain is then multiplied by the elastic
material stiffness matrix, [De
], and added to the previous stress
state, {σi−1
c } to solve for the trial stress {σtr
}.
{σtr
} = {σi−1
c } + [De
]{∆ε} (3)
For the Linear Elastic model the process described above
is repeated until the final deformation is reached. Whereas, if
there is a calculated plastic phase in the Mohr-Coulomb model,
a new stress state is found using a plastic stress corrector.
{σi
c} = {σtr
} −
< f({σtr
}) >
d
∂g
∂d
(4)
The plastic stress corrector, <f({σtr
})>
d
∂g
∂d , returns the stress
to the Mohr-Coulomb yield surface. The complexity of the
plastic stress corrector is beyond the scope of this report, thus
will not be discussed further.
B. Coupling
Coupling the soil deformation with water response is de-
scribed using domains; there is the soil domain, ΩS, where
the deformation occurs, and the water domain, ΩW , where
the wave propagation occurs. The two domains are treated as
separate entities with one interaction.
The first part of the process starts with the initial state of
the system, which has no deformation and thus no interaction
between the domains, as shown in Fig. 1 step 1. Consequently,
step 2 deforms the seabed statically; it can be considered static
because the time in which it occurs is small and there are no
further deformations.
The second part couples the domains and consists solely
of step 3. The coupling uses the work of Kajiura, where the
vertical displacement of the soil can be translated onto the free
water surface; this is known as the passive approach [8]. This
is done by extracting the seabed surface data to define the
starting wave profile. Additionally, it is important to note that
the deformation is uni-directional: from soil to water. This is
because both the soil and water are considered incompressible.
Thus, the effect of the water on the soil is very small and gives
similar results, whereas ignoring the water’s effect on the soil
saves a lot of computational effort with only a small loss in
precision.
The last part is the free water response using the initial
wave profile from the translocated seabed deformation. Step 4
shows the resultant wave propagation whereas step 5 shows the
final state when the free water surface has reached equilibrium
again.
Figure 1. Deformation and coupling process of the domains, steps 1 - 5.

C. Nonlinear shallow water equations
The nonlinear shallow water equations are used for the
calculation of the free water response. This code was written
using the basic principle that the water depth is small in
comparison to the wavelength. These equations are derived
from depth-integrating the Navier-Stokes equations and result
in a computationally faster solution. This is because the
Navier-Stokes equations require continuous remeshing to track
the wave propagation whereas the nonlinear shallow water
equations do not. Due to the two-dimensional nature of the
soil model, the nonlinear shallow water equations use the one-
dimensional equations for the conservation of mass and con-
servation of momentum. They also take into account gravity,
shear with the seabed surface and can take into consideration
more factors such as wind on the water surface if desired. The
equation below is for the conservation of mass, where h is the
water height from the seabed, t is time and u is the water
velocity.
∂h
∂t
+
∂(hu)
∂x
= 0 (5)
The change in water height due to the translocation of the
seabed deformation generates a wave propagation, hu, which
can then be substituted in the equation for conservation of
momentum below:
∂(hu)
∂t
+
∂
∂x
(hu2
+
1
2
gh2
) = −gh
∂B
∂x
+ τ (6)
Where g is gravitational acceleration, B is the seabed
bathymetry and τ is the seabed shear effect. At first there
will be no shear effect as u = 0, but after the first step shear
can be solved using (7).
τ = −ghcf
u|u|
h
5
3
(7)
Where cf is a coefficient of friction. This process is looped
until the free water surface comes to rest. Fig. 2 is an
illustration of the more important parameters for the nonlinear
shallow water equations.
Figure 2. Illustration of nonlinear shallow water equation parameters.
The inundation phase is taken into account with the nonlin-
ear shallow water equations [11] but requires a slope of less
than 70◦
and applies only for non-breaking waves.
The full MATLAB code that was written for this report can
be found in Appendix B.
III. METHODOLOGY AND MODELS
This section describes the process which was used to ensure
valid results and the models’ setup and parameters.
A. Method
The method with which the models were solved used a
process which started by designing the model. First, the
bathymetry of the model had to be determined, this was either
arbitrarily chosen or a database of sea and ocean bathymetries
could be accessed. Then, the nature of the deformation was
chosen; this was either done by choosing a prescribed force
or displacement as well as choosing a point or line on which
to perform the action on. The models in this report use only a
prescribed displacement as it is more suitable for the problem
at hand.
The next step is meshing the model, using a triangulation
procedure, so that Plaxis can solve it. This involves choosing
the refinement level from preset options or manually choosing
the relative element size. It is also possible to select local
refinement if there is a specific area of the model that requires
higher precision. However, Plaxis automatically refines more
around the point or line displacements as well as thin soil
layers. After the desired mesh has been achieved Plaxis can
run the simulation.
When the simulation is completed the first data processing
phase can begin. This involves extracting all the required data,
specifically the displacement of all the nodes, and organising
it in Microsoft Excel. The data is organised so that it can
be read by MATLAB to generate the mesh before and after
deformation, and a series of surface points that are used for the
translocation onto the free water surface. This was done mostly
by using Excel formulas and creating .txt files for MATLAB.
Within MATLAB there were then two steps; one was
generating the meshes and the other was performing the water
flow calculation. Firstly, the meshes were checked, and if there
was an error the process returned to the data processing phase.
The flow was then checked and followed the same check for
the mesh. Lastly, if both parts were outputting valid results
the model was assessed for its adequacy and value.
This method was performed on all the models herein, and
underwent vigorous checks until no errors were found in the
final results.
B. Models
The models described in this section are a selected few that,
together, were deemed best suited to encompassing the various
types of possible soil deformations.
The first three models used a simple rectangular geometry
of soil that had a width of 1000 m and a depth of 500 m; all
three of which had a Linear Elastic bedrock for the bottom
100 m, as shown in Fig. 3. The bedrock was always modelled
as Linear Elastic because a negligible amount of deformation
would occur. The fourth model used real seabed data from
New York to Lisbon, as shown in Fig. 4, and used a bedrock
that followed the geometry of the top surface, 400 m below
it.

Figure 3. Schematic of first three models before deformation.
Figure 4. Bathymetry of Atlantic Ocean from New York to Lisbon.
All the models used two material; hard clay and bedrock.
However, the Soil Layers model also used loose sand, dense
sand and soft clay. The parameters for which are listed in Table
I [12], [13].
Table I
MATERIAL PARAMETERS
Material
Young’s
modulus,
E, MPa
Poisson’s
ratio, ν
Cohesion,
c, kPa
Friction
Angle,
ϕ
Saturated
unit weight,
γsat, kN/m3
Loose
Sand
15 0.2 50 0◦ -
Dense
Sand
35 0.3 50 0◦ -
Soft
Clay
10 0.35 11 0◦ -
Hard
Clay
35 0.35 11 0◦ 20
Bedrock
(Basalt)
45x103 0.17 - - -
The two principal parameters are Young’s modulus, E, and
Poisson’s ratio, ν. However, more parameters are required
when using the Mohr-Coulomb constitutive model. These
parameters are cohesion, c, friction angle, ϕ, and dilantancy
angle, ψ.
The models are for soils under a large amount of water, so
a setting in Plaxis was chosen that applies a large bulk of stiff
water. In this case cohesion becomes undrained shear strength
and the friction angle can be set to ϕ = 0 reducing it to the
Tresca criterion. When this is the case the dilatancy angle is
automatically set to ψ = 0 in the Plaxis software [9].
The saturated unit weight was required for a dynamic
simulation to better model the compression wave velocity and
shear wave velocity. This was done uniquely for a model using
only hard clay above the bedrock.
The first deformation type was the Rise and Dip model.
The rise and dip refers to the two point displacements that
were located within the soil, one displacing 50 m upwards
and the other 50 m downwards. The prescribed displacements
were set 25 m below the soil surface and 75 m either side
of the centreline. Therefore, their respective coordinates were
(425, 475) and (575, 475). This model was chosen because
it inspected the soil deforming in tension and compression as
well as inducing a rise and a dip in the water level above.
The second model, Rupture, used a rupture mechanism that
caused a large dip in the water level due to a horizontal
deformation in the soil. This was achieved by creating outward
facing horizontal displacements originating from two near
vertical lines in the soil. The displacements at the bottom of
the lines started at 0 m and increased linearly until the top,
reaching a displacement of 50 m. The coordinates of which
are (490, 110) to (480, 500) for the first line and (510, 110)
to (520, 500) for the second line.
Thirdly, is the dynamic displacement model, Seaquake,
which was modelled to simulate the vibrations in the soil
that would be caused by a real seaquake. This model used
a single point displacement, with coordinates (500, 475), and
a table with dynamic multiplier against time. The data from
this table was in an .smc format which is a common format
for earthquake data. The file used can be viewed as a graph,
shown in Fig. 5. However, Plaxis requires the user to input
a displacement for the dynamic movement: a displacement of
25 m was chosen.
-1.5
-1
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8
Dynamicmultiplier
Time (s)
Variation of dynamic displacement with time
Figure 5. Graph of smc data.
Lastly, the New York - Lisbon model was generated. This
is the only model with different dimensions and geometry;
using real seabed data for the bathymetry [14]. The width of
the model spans approximately 5600 km (distance from New
York to Lisbon) with a depth of 300 km. The depth was chosen
to better represent the mesh of the model, this is because, due
to the greatest depth of the Atlantic Ocean only being 6 km,
the model would appear very thin with an unobservable mesh.
The prescribed displacement was a 500 m uplift applied on a

Table II
MESH CONVERGENCE
Mesh Displacement (m) Time (s)
Preset
Relative
Element Size
No. of
Elements
No. of
Nodes
Max. Rise Dip Mesh Calculation Total
Very Coarse 2 100 227 50.72 45.83 45.95 1.2 4 5.2
Coarse 1.333 162 359 50.11 45.18 45.72 1.3 4.1 5.4
Medium 1 318 681 50.55 43.34 40.54 1.5 4.6 6.1
Fine 0.667 620 1307 50.09 37.24 36.68 1.6 4.7 6.3
Very Fine 0.5 1214 2519 50.2 37.32 34.06 1.9 7.9 9.8
50 km horizontal line with coordinates, in km, (3950, -2.18)
to (4000, -2.18). The displacement was 0 m at both ends of
the line and increased linearly up to 500 m in the middle.
It is important to note that care must be taken when setting
the prescribed displacement. This is because the prescribed
displacement is located on a node and when it has fully
displaced the calculation will stop. The elements and nodes
around it will deform differently according to the constitutive
model. However, if the prescribed displacements are too large
the mesh will distort and elements will overlap. Therefore,
because this is an issue that is not well understood and
produces highly unrealistic deformations, this is avoided by
selecting displacements appropriate to the model.
IV. RESULTS & DISCUSSION
This section discusses and investigates the results of the
various models, particularly looking at the wave profiles and
validity of the soil models. The Rise and Dip model was used
for the mesh convergence test, comparing realistic with unreal-
istic deformations and to investigate the effect of different soil
layers. Otherwise, all models will only have two simulations:
comparing Linear Elastic against Mohr-Coulomb. Comparing
the two constitutive models checks for major discrepancies
as well as the importance of the necessary and adequate
constitutive model.
A. Mesh convergence test
The mesh convergence test is a necessary step in ensuring a
correct level of precision while also maintaining a reasonable
computational time for the solution. Therefore, using the
Mohr-Coulomb Rise and Dip model, five different mesh sizes
ranging from very coarse to very fine were tested. Three
displacements were recorded; the maximum displacement in
the soil, the largest displacement of the resulting rise and the
largest displacement of the resulting dip.
The results in Table II show that the maximum displacement
ranges from 50.09 m to 50.72 m without any real sign of
converging. Looking at the displacement of the rise and the
dip it is apparent that they are converging to about 37.3 m and
34.0 m respectively. The value given by the rise displacement
of the Fine mesh is slightly lower than expected; however, it
can be considered as an acceptable error due to its value being
very similar to that of the Very Fine mesh. This is because the
surface shape varies relatively little, as shown in Fig. 6, and
makes little difference when looking at the overall time taken.
The total times taken for the mesh generation and the
calculation are not significant at this level of simulation.
450
460
470
480
490
500
510
520
530
540
550
0 200 400 600 800 1000
z(m)
x (m)
Deformed soil surface plot for various meshes
Very Coarse
Coarse
Medium
Fine
Very Fine
Figure 6. Graph of various mesh refinements for deformed soil surfaces of
Rise and Dip model.
Therefore, the Very Fine mesh was chosen because it kept
a high level of precision and used an acceptable amount of
CPU time.
B. Rise and Dip model
As mentioned previously, there is a Linear Elastic and a
Mohr-Coulomb version for every model; however, this does
not change the geometry or mesh of the model. Therefore,
the undeformed models appear the same for the two cases, as
shown for the Rise and Dip model in Fig. 7. Fig. 7 shows
the completed model as it appears in MATLAB with a chosen
water depth of 150 m. The choice of water depth depends
on the type of deformation because the fluid solver does not
operate correctly if the water surface makes contact with the
soil surface. Thus the appropriate water height must be chosen
for each model to work correctly.
Figure 7. Meshed Rise and Dip model with water layer.

The model above was also used to compare realistic de-
formations with unrealistic deformations; the realistic defor-
mations used stress as the limiting factor in its maximum
displacement, whereas the unrealistic deformations did not.
This led to large differences in the soil deformations, as shown
in Fig. 8.
460
470
480
490
500
510
520
530
540
0 200 400 600 800 1000
z(m)
x (m)
Deformed soil surface plot of realistic and
unrealistic deformations
Unrealistic - Mohr-Coulomb
Unrealistic - Linear Elastic
Realistic - Linear Elastic
Realistic - Mohr-Coulomb
Figure 8. Rise and Dip soil surface of Mohr-Coulomb and Linear Elastic
models for both realistic and unrealistic deformations.
The most notable difference in Fig. 8 is between the unre-
alistic models versus the realistic ones. The realistic models
have much smaller displacements in the soil: the greatest
displacement is less than 9 m as opposed to 38 m from
the unrealistic deformation. This difference occurred because
the realistic deformations were obtained by having the mesh
update itself during the calculation to output the correct stress
values shown in Fig. 9. During this method Plaxis ran the
calculation checking the stresses until they exceeded their limit
at which point it stopped at the corresponding deformation
step. Meanwhile, the unrealistic models ignored the stress
limits and completed the simulation reaching the prescribed
displacement in the soil.
Figure 9. Distribution of principal effective stress in soil of realistic defor-
mation for Linear Elastic Rise and Dip model
The stress distribution shown in Fig. 9 is for the principal
effective stress of the final step of the Plaxis calculation,
demonstrating that Plaxis can be used for real deformations
in soils. However, this report is focused on the versatility
and resilience of coupling Plaxis with MATLAB, so using
larger deformations is more useful to test how robust the
model is. Therefore, now that a real deformation has been
proved to work, this report will continue with only unrealistic
deformations as they are more interesting for the investigation
of this MOST. Thus all further models will no longer need to
be referred to as unrealistic.
The second difference to note in Fig. 8 is that between the
Linear Elastic and Mohr-Coulomb models. For the realistic
set the Linear Elastic model has a greater displacement.
This is because the Mohr-Coulomb model reaches its stress
limit sooner than its Linear Elastic counterpart. Therefore,
the displacement is smaller. The unrealistic deformations do
not behave the same way because both models reach their
full prescribed displacement. In this case the Mohr-Coulomb
model deforms more because its plastic deformation is a
better representation of how the soil would behave and thus
is less inclined to distort the mesh; though, the Linear Elastic
constitutive model behaves more similarly under compression
(dip) than in tension (rise). The reason is better explained at
the end of Section III.
Consequently, the model chosen to investigate the water re-
sponse was the unrealistic Mohr-Coulomb model as it behaved
better than the Linear Elastic and was more informative than
the realistic. The free water response and its propagation to
the 50 m prescribed displacements for the Rise and Dip model
is shown in Fig. 10 below.
Figure 10. Deformation of Mohr-Coulomb Rise and Dip model and its free
water response at (a) t = 0.01 s, (b) t = 8 s and (c) t = 15 s.
The first step, Fig. 10 (a), demonstrates how the MATLAB
code uses Kajiura’s principle of translocation to define the
starting wave profile. The second step shows the free water
surface profile after 7.99 s of propagation; the initial wave

Figure 11. Deformation of Mohr-Coulomb Rupture model and its free water response at (a) t = 0.01 s, (b) t = 6 s and (c) t = 15 s.
dissipated in both directions leading with the rise as it prop-
agated left and lead with a depression to the right. These are
the two major waves produced by the deformation and will be
the main cause of inundation when reaching land. After 7 s
of dissipation there are no more apparent waves and the water
level has reached equilibrium again.
Lastly, the Rise and Dip model investigated the effect of
having different soil layers for the topmost 15 m. The first
model used loose sand for the top 5 m of soil, the second added
another 5 m layer of dense sand and the third model used soft
clay as the third 5 m layer. The material parameters for these
soils can be found in Table I. The soil surface deformations
of the three models were compared to the purely hard clay
model as shown in Fig. 12.
460
470
480
490
500
510
520
530
540
0 200 400 600 800 1000
z(m)
x (m)
Deformed soil surface plot of soil layers
Hard Clay (HC)
Loose Sand (LS) + HC
Dense Sand (DS) + HC + LS
Soft Clay + HC + LS + DS
Figure 12. Rise and Dip soil surface of Mohr-Coulomb and Linear Elastic
models for soil layers.
The curves in Fig. 12 follow a very similar path, suggesting
that the effect of soil layers has little effect on the overall
deformation. The main contributing factor leading to these
results was due to the soils needing to reach their full pre-
scribed displacement. Thus including various soil layers in the
Plaxis model is unnecessary unless an accurate and realistic
soil model is desired.
C. Rupture model
The Rupture model’s initial state was similar to that of
the Rise and Dip model in Fig. 7. However, the mesh was
refined differently due to location and size of the prescribed
displacements, and the water level was set to 400 m above the
soil surface. The horizontal displacement in the soil caused a
100 m wide rift, narrowing down for almost 400 m, as shown
in Fig. 13.
100
150
200
250
300
350
400
450
500
550
0 200 400 600 800 1000
z(m)
x (m)
Deformed soil surface plot of Rupture model
Mohr-Coulomb
Linear Elastic
Figure 13. Rupture soil surface of Mohr-Coulomb and Linear Elastic models
for soil layers.
However, this model shows little difference between Linear
Elastic and Mohr-Coulomb. This is because the deformation is
a compressive force which, as previously established, induces
a smaller difference between the two constitutive models. Ad-
ditionally, the prescribed displacement acts on a set of nodes
for the Rupture model with a linearly increasing magnitude.
Therefore, the deformation is applied more evenly throughout
the soil leading to smaller and less concentrated displacements
of the surrounding nodes.
The free water response for the rupture deformation is
large and as expected; Fig. 11 (a) shows just how great the
resultant water displacement is due to the translocation of the
soil surface. The second step shows the water response 5.99
seconds after the initial wave profile, at this stage the water dip
has dissipated in both directions causing two large waves with
leading depressions. The height of the waves, from the bottom
of the depression to the next peak is 90 m. After the main wave

several smaller ripples propagate until the simulation reaches
equilibrium again, as shown at t = 15 seconds in Fig. 11.
The results from the Rupture model show that the Plaxis
- MATLAB coupling can handle large horizontal displace-
ments in the soil causing devastatingly tall waves as well as
reasserting that Linear Elastic constitutive model behaves more
similarly when under compression.
D. Seaquake model
The Seaquake model used a dynamic displacement to de-
form the water surface using a prescribed displacement of 25
m and the .smc file, shown in Fig. 5, on a single node. The
dynamic displacement generated a vibration that was made
of forty deformations over an 8 second period, simulating an
effect similar to that of a real seaquake. Fig. 14 shows two
seabed deformations with a 0.2 s gap between them, with the
first wave becoming indistinct after the second displacement.
Figure 14. Deformation of Mohr-Coulomb Seaquake model and its free water
response at (a) t = 2.2 s and (b) t = 2.4 s.
The lack of a clear wave shape is due to the continuous
translocation of the soil surfaces to the water surface. This
disrupts the wave that was previously formed because a rise in
the soil surface is usually led with a dip, thus greatly reducing
the height of the wave. The repetitive sequence of the process
produced many small waves, as shown in Fig. 15.
Fig. 15 is directly comparable to the .smc file in Fig. 5, but
in addition, shows water height decaying between deforma-
tions as the wave propagates in either direction. The decays
shown represent the small disrupted waves that propagate
outwards from the intial wave profile seen in Fig. 14.
The difference between the Linear Elastic and Mohr-
Coulomb models was minimal and displayed the same be-
haviour as previously: more similar in compression than in
tension.
580
600
620
640
660
680
700
0 1 2 3 4 5 6 7 8 9 10
Waterheight,z,(m)
Time (s)
Variation of water height at x = 500 m
Figure 15. Vertical water elevation at x = 500 for the first 10 seconds.
E. NY - Lisbon model
The New York - Lisbon model applied a single rise defor-
mation in its seabed near the edge of the Mid-Atlantic ridge.
Its aim was to investigate the effect of a deformation in deep
sea and any resulting inundation. However, due to the scale
of the model only part of the model is shown at one time, the
complete meshed model can be found in Appendix C.
Figure 16. Initial wave profile of Mohr-Coulomb New York - Lisbon model
at t = 0.01 s.
Fig. 16 shows that Kajiura’s translocation is applied regard-
less of the depth at which the soil deforms. The height of the
initial wave profile is approximately 500 m and propagates in
both directions. However, after the wave has travelled almost
1600 km to the coast of Lisbon in two and a half hours, the
wave reduced to a height of 25 m as shown in Fig. 17 (b).
Fig. 17 (c) shows how the arriving wave runs up onto the dry
land, further reducing the height of the wave as it overcomes
the gradient and the shear between the water and the land. At
the stage where the water is inundating the dry land the water
depth is approximately 10 m; consequently, the effect of the
shear increases, reducing the speed of the wave. That is why
it takes the peak of the wave 1565 s to move 25 km from Fig.
17 (b) to (c) whereas it takes 4300 s for the inundation of 3
km of dry land from (c) to (d). However, the 22 m wave in
(c) successfully inundates 7 km of dry land with a 2 m layer
of water, though after 10,000 s the inundated area drains back

into the sea and the model returns to its state in Fig. 17 (a).
Figure 17. Inundation phase of Lisbon’s coast at (a) t = 0.01 s, (b) t = 8135
s, (c) t = 9700 s, (d) t = 14000 s.
Another observation to be made from Fig. 17 is that of the
trailing water height. When the wave arrives at the coast it
slows down, reduces in height and has a more gradual rise,
but it also reflects some of the energy back and a smaller wave
propagates back in the direction of New York. This returning
wave is more evident in Fig. 18 for x = 4500 km. This curve
shows the passing of the initial wave at t = 2980 s with a height
of 32 m but shows the peak of another wave at t = 13,000 s.
The same wave can be seen later for x = 3975 km and x =
4100 km. The reflected energy from the ocean boundary is
representative of real hydrodynamics and is attributed to the
nonlinear shallow water equations used to solve it.
Closer inspection of the water height at locations of x close
to the coast (5500 km and 5520 km) show how the wave
builds and inundates. At x = 5500 km, the peak water height
is higher than its previous peak at x = 5333 km, demonstrating
how the MATLAB code transfers energy from the wavelength
into wave height. This follows the same principle that causes
tsunamigenic waves to increase in height in coastal areas
[15]. At x = 5520 km the gradient of increasing wave height
is gentler than its predecessors and is demonstrative of the
slowed speed of the water.
5995
6005
6015
6025
6035
6045
6055
6065
6075
0 3600 7200 10800 14400 18000 21600 25200
z(m)
Time (s)
x = 3975 km x = 4100 km
x = 4500 km x = 5333 km
x = 5500km x = 5520 km
Variation of water height at certain locations of x
Figure 18. Change in water level at various locations of x.
The New York - Lisbon model successfully showed that
the hydrodynamic solver behaves correctly for inundation
and propagation of long distance waves with real seabed
bathymetry. However, a 500 m wave is a highly unrealistic
initial wave profile yet results in only 20 m wave height
2 km from the dry land. A tsunami, on the other hand, is
caused by many small (0.2 m - 3 m) displacements in the
seabed that dissipates large amounts of energy to the water.
The water then propagates in the form of high velocity waves
with a very long wavelength and undergoes the same transfer
of energy from wavelength and speed to wave height as the
New York - Lisbon model did. Though in the case a tsunami,
the wave height goes from 2 m at sea to 15 m at the coast.
Therefore, even if the New York - Lisbon model shows that
the MATLAB code handles wave propagation and inundation
correctly, it needs to be refined to be able to model tsunamis
correctly; more specifically, the energy transferred from the
seabed displacement to the water.
V. CONCLUSION
Plaxis 2D and MATLAB were coupled to simulate seabed
deformations and their resultant wave propagation using the
Method of Splitting Tsunamis (MOST), from which four
successful models were developed. The geotechnical software,
Plaxis, was used to model the morphodynamics whereas
MATLAB used nonlinear shallow water equations to solve
the hydrodynamics part. However, because the generation,
propagation and inundation of the water is dependent on initial
conditions, each model investigated different initial conditions
to determine the robustness of the Plaxis - MATLAB coupling.
The first model showed that a small scale model, 500 m
x 1000 m, could easily handle a very fine mesh with low
computational cost. The model was then used to analyse the

difference between realistic and unrealistic models, demon-
strating that the user is free to choose whether to model a
highly accurate soil model with different soil layers and real
stresses, or to make a crude model to quickly generate and
investigate the free water response.
The second and third models tested the method against
different types of deformations, using a horizontal displace-
ment to cause a dip in the free water surface and a dynamic
movement to generate vibrations in the soil similar to that of
a real seaquake. The Plaxis - MATLAB coupling withstood
the very large deformation but did not behave so well with
the dynamic one as only small, disorderly and distorted waves
were formed.
Lastly, a model was created using real seabed data of the
Atlantic Ocean from New York to Lisbon. This model was
made with real seabed geometries, though care should be taken
with the scale of the model, and a coast to be inundated. The
model proved successful in both areas showing correct wave
propagation and decay as well as the inundation of 7 km of
dry land.
All the models described above were modelled twice, using
a Mohr-Coulomb constitutive model and a Linear Elastic
constitutive model. As expected, it was found that the Mohr-
Coulomb models behaved more realistically than its counter-
part due to the geophysics in the Plaxis software, therefore
the Linear Elastic constitutive model should only be used for
crude models.
The complexity of the initial soil model is user defined,
and therefore allows for highly accurate and precise results.
These are achieved by selecting different soil compositions,
level of mesh refinement, and appropriate constitutive models.
Nonetheless, if only an approximate model is required then a
simple model can be constructed in a short amount of time.
MATLAB then produces a free water response that is accurate
for all problems tested except soil vibrations. Therefore, the
code should be developed further to better simulate water
response to seaquake displacements. Overall, the Plaxis -
MATLAB MOST is a robust and versatile method that shows
promise for the investigation of free water response to seabed
deformations, but requires some knowledge of Plaxis and a
few improvements in MATLAB.
ACKNOWLEDGEMENT
The author would like to thank Dr. Mohammed Seaid for
overseeing the project and the positive encouragement. Further
thanks are given Dr. Will Coombs for helping understand the
soil mechanics in Plaxis 2D. Furthermore, the author’s family
and friends must be thanked for their support and optimism
over the last four years at Durham University and abroad.
REFERENCES
[1] D. A. Joseph, “Tsunamis; detection, monitoring and early-warning
technologies,” Elsevier, 2011.
[2] A. Cessna. (2009) Tsunami 2004. [Online]. Available:
http://www.universetoday.com/49007/tsunami-2004/
[3] "N.O.A.A Administration", “Tsunami modelling and research,” 2006.
[Online]. Available: http://nctr.pmel.noaa.gov/model.html
[4] B. C. Papazachos, E. M. Scordilis, D. G. Panagiotopoulos, C. B. P. C. B.,
and G. F. Karakaisis, “Global relations between seismic fault parameters
and moment magnitude of earthquakes,” in Bulletin of the Geological
Society of Greece, vol. 36, 2004, proceedings of the 10th International
Congress.
[5] L. B. Freund and D. M. Barnett, “A two-dimensional analysis of surface
deformation due to dip-slip faulting,” Bulletin of the Seismological
Society of America, vol. 66, no. 3, pp. 667–675, 1976.
[6] N. Shuto, “Numerical simulation of tsunamis - Its present and near
future,” in Tsunami Hazard, E. N. Bernard, Ed. Kluwer Acedemic,
1991, vol. 4, pp. 171–191.
[7] C. Moler, Experiments with MATLAB. MathWorks, Inc., 2011.
[8] F. Dias and D. Dutykh, Tsunami generation by dynamic displacement
of seabed due to dip-slip faulting. Elsevier, 2008.
[9] PLAXIS Material Models Manual, 2014.
[10] PLAXIS Scientific Manual, 2014.
[11] S. Tadepalli and C. E. Synolakis, “Model for the leading waves of
tsunamis,” Physical Review Letters, 1996.
[12] Geotechdata.info. (2013) Cohesion. [Online]. Available:
http://geotechdata.info/parameter/cohesion
[13] K. Hadzic. (2014) Some useful numbers on the
engineering properties of materials. [Online]. Avail-
able: http://www.scribd.com/doc/214376674/Some-Useful-Numbers-
docscribd
[14] U. of California San Diego. (2014) Extract XYZ data - topography or
gravity. [Online]. Available: http://topex.ucsd.edu/cgi-bin/get_data.cgi
[15] B. Levin and M. Nosov, Physics of Tsunamis. Springer, 2009.

M. ENG RESEARCH PROJECT APRIL 2015 1
Appendix A
Plaxis Procedure

Appendix B
MatLab Code
close all
clear all functions
addpath('Plaxis')
load Nodes.txt;
TRI=[Nodes(:,1) Nodes(:,2) Nodes(:,3)];
k = 0;
sfilename = strcat('SurfaceNodes',int2str(k),'.txt');
sData = importdata(sfilename);
nfilename = strcat('newmeshdata',int2str(k),'.txt');
nData = importdata(nfilename);
Def = [10]';
xleft = sData(1,1); xright = sData(end,1); h0 = 0;
n = 100; dx = (xright-xleft)/(n+1);
g = 9.81; alpha = 1.; h_crit = 1e-9;
cfl = 0.5; tend = 0.01;
% Set the type of interpolation
%interptype = 'linear';
%interptype = 'spline';
%interptype = 'cubic';
interptype = 'pchip';
% Generate the gridpoints
for i=1:n+2
x(i) = xleft+(i-1)*dx;
end
x0(1:n+1) = 0.5*(x(1:n+1)+x(2:n+2));
%
% Set the initial topography
%
t = 0;
z = interp1(sData(:,1),sData(:,2),x,'cubic');
% Set initial conditions
uo = zeros(size(x));

ho = h0*ones(size(x))-z;
% Start the time loop
ntplot = 1;
nb = 0;
while (t<tend)
nb = nb+1;
if ismember(nb,Def) == 1
k = k+1
sfilename = strcat('SurfaceNodes',int2str(k),'.txt');
sData = importdata(sfilename);
nfilename = strcat('newmeshdata',int2str(k),'.txt');
nData = importdata(nfilename);
z = interp1(sData(:,1),sData(:,2),x,interptype);
end
% Compute the time step
maxc = max(abs(uo) + sqrt(g*ho));
dt = cfl*dx/maxc;
if (t+dt > tend)
dt = tend-t;
end
nu = dt/dx;
% Compute the depature points
deriv = interp1(x,uo,x0,interptype);
x0car = x0-dt*alpha*deriv;
% Interpolation procedure
ui = interp1(x,uo,x0car,interptype);
hi = interp1(x,ho,x0car,interptype);
for i=1:n
% Predictor stage
hm = hi(i)-alpha*nu*hi(i)*(uo(i+1)-uo(i));
um = ui(i)-alpha*nu*g*(ho(i+1)-ho(i)+z(i+1)-z(i));
hp = hi(i+1)-alpha*nu*hi(i+1)*(uo(i+2)-uo(i+1));
up = ui(i+1)-alpha*nu*g*(ho(i+2)-ho(i+1)+z(i+2)-z(i+1));
% Corrector stage
hn(i+1) = ho(i+1)-nu*(up*hp-um*hm);

fp = hp*up*up+0.5*g*hp*hp;
fm = hm*um*um+0.5*g*hm*hm;
q = ho(i+1)*uo(i+1)-nu*(fp-fm);
q = q-0.125*g*nu*(ho(i)+2.0*ho(i+1)+ho(i+2))*(z(i+2)-z(i));
% Treatment of dry areas
if(hn(i+1) <= h_crit)
hn(i+1) = h_crit;
un(i+1) = 0.0;
else
un(i+1) = q/hn(i+1);
end
end
% Overwrite the old solution
uo(2:n+1) = un(2:n+1);
ho(2:n+1) = hn(2:n+1);
% Boundary conditions
ho(1) = ho(2);
ho(n+2) = ho(n+1);
uo(1) = uo(2);
uo(n+2) = uo(n+1);
t = t+dt;
% Display the results
figure(1)
if (mod(n,ntplot) == 0),
clf
fill([x fliplr(x)],[ho+z fliplr(z)],'b','Edgecolor','none');
hold on
fill([x fliplr(x)],[z fliplr(0*z)],[0.5 0.5 0.5],'Edgecolor','none');
hold on
plot(sData(:,1),sData(:,2),'k-','LineWidth',3)
triplot(TRI,nData(:,2),nData(:,3),'k');
xlabel('$x$ (m)','Interpreter','Latex')
ylabel('$eta$ (m)','Interpreter','Latex')
title(['time = ',num2str(t),' seconds'])
axis([xleft xright 0 h0+70])
set(gca,'DataAspectRatio',[1 1 1]);
pause(0.01)
end
end

Appendix C
New York – Lisbon Data
Complete mesh of New York - Lisbon model
Atlantic Ocean bathymetry
Coordinates and map view of New York to Lisbon

Research Project Final

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