Asensio, Laura

Navarro, Vicente

Pintado Llurba, Xavier

Yustres, Ángel

English

UPCommons

Simulation of a pinhole test of an MX-80 bentonite 
 
 
Vicente Navarroa, Laura Asensioa, Xavier Pintadob, Ángel Yustresa 
 
 
a University of Castilla-La Mancha. Civil Engineering Department. Avda. Camilo José Cela s/n, 13071 Ciudad 
Real, Spain 
b B+Tech Oy. Laulukuja 4, 00420 Helsinki, Finland 
 
 
An approach is proposed for simulating the coupled processes of unsaturated swelling and mechanical 
erosion of MX-80 bentonites. The rate of bentonite surficial mass removal per unit area B is 
estimated using the expression: 
[1]  0,max CQeB   km  
where Q is the hydraulic shear stress produced by the water flow, C is the critical shear stress at 
which the mass removal starts, and the proportionality factor ke is the coefficient of bentonite erosion. 
This equation is often used in Soil Science (Hanson and Simon, 2001; Sanford and Maa, 2001) and in 
the analysis of piping in embankment dams (see, among others, Wan and Fell, 2004; Wahl, 2010). If, 
as happens on the pinhole tests analysed (Sane et al., 2013), slaking plays an important role (see Lim, 
2006, and Sane et al., 2013), a value of C=0 may be adopted, assuming an “instantaneous” activation 
of the erosion in the preferential flow path (PFP) analysed. 
If a quasi-steady laminar flow has fully developed in the PFP, Q is given by the equation (Wahl et al., 
2008): 
[2] 
3
t
W
Q
4
r
Q

 
  
where W is the dynamic viscosity of the water, Q is the water flow rate (0.1 l/min in the pinhole 
tests), and rt is the characteristic radius of the PFP at each time t. 
Therefore, the total erosion rate per unit length of pinhole, B, can be calculated as: 
[3] BtB 2 mrM     
which, using material coordinates, can be rewritten as 
[4] tdOtB 2 RRM     
where dO is the initial dry density of the bentonite, and Rt is the value of the material radial 
coordinate R at the erosion front at each time t (the soil at R < Rt has already been eroded at time t; see 
Figure 1 b). Therefore, once Rt and rtare known, the erosion rate t can be calculated from Eqs. (1) 
through (4) using the expression: 
[5] 
2
tt
e
dO
W
t
4
rR
Qk
R 


 
  
This ordinary differential equation allows for updating of the material domain by defining the rate at 
which the boundary of the active zone moves. Its time integration defines a relationship between the 
boundary of the active (not eroded) material domain (“material frame”) and a reference domain or 
“geometric frame”, equal to the initial material domain. The same finite element mesh is always used 
in the geometric frame. This is not the case in the material frame, changed by the erosion. We propose 
to apply a smoothing method in this frame to perturb the positions of the nodes of the finite-element 
mesh at every computer time step in order to continually adapt the mesh to the active domain, as the 
“Deformed Geometry” module of Comsol Multiphysics (COMSOL, 2013) does. Using the values of 
previous steps, the prediction of the values of the state variables are extrapolated in time and 
interpolated in space for the new nodes. Since the mesh used is of the same type as that defined in the 
geometric frame (the same mesh settings are used), a one-to-one relationship (controlled by t and by 
the smoothing method) can be defined between the new nodes of the material mesh and those of the 
geometric mesh, which is kept fixed. This one-to-one transform is used for assigning the information 
of the material mesh to the geometric mesh, being the latter the frame for all the calculations that are 
performed. Thus, using the geometric frame as the workspace, the material frame is continually 
updated. As it is done when a Lagrangian formulation is applied, the determinant and the deformation 
gradient tensor associated with the transform between reference and material frames have to be 
included in the weak formulation of the equilibrium and balance equations. This is automatically done 
by Comsol Multiphysics when the Deformed Geometry module is activated. 
The accuracy of the prediction of the erosion rate t depends on the accuracy of rt, that is, on the 
quality of the bentonite deformability model. Taking into account the usual bimodal pore-size 
distribution of active, a double-porosity model was used to describe their deformational behaviour 
(Alonso et al., 1999; Gens and Alonso, 1992; Musso et al., 2013). Using the Barcelona Expansive 
Model (Alonso et al., 1999; Gens and Alonso, 1992) as a reference, an additional volumetric strain 
was introduced to model the deformation induced in the macrostructure by the aggregate 
destructuration (Navarro et al., 2014). This deformation can be very high in the direction orthogonal 
to the PFP, where free deformation condition applies. Hence, a Lagrangian formulation was adopted 
to take into account that the deformation gradient tensor can be considerably different from the 
identity tensor. 
In Figure 1 experimental data and model results of mass loss per unit height are compared. A variety 
of pinhole tests on MX-80 samples with various configurations were analysed: sample diameters, 
sample heights and initial hole diameters: of (a) 100x100x6 mm, (b) 100x400x6 mm, (c) 100x100x12 
mm. The trend observed in the experiments is correctly reproduced. Thus, although the approach 
could be further improved (introducing, e.g., the effect of water salinity), the obtained results indicate 
that this method is a useful tool for describing and quantifying the coupled erosion and swelling of 
MX-80 bentonites. 
 
 
Figure 1 
REFERENCES 
 
Alonso EE, Vaunat J, Gens A (1999) Modelling the mechanical behaviour of expansive clays. Eng 
Geol 54(1-2):173-183. 
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Hanson GJ and Simon A (2001) Erodibility of cohesive streambeds in the loess area of the 
midwestern USA. Hydrol Process 15(1):23–38. 
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Hole Erosion Test and Jet Erosion Test. Technology Development Report DSO-08-05. U.S. 
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www.usbr.gov/pmts/hydraulics_lab/pubs/DSO/DSO-08-05.pdf, last accesed November 2014. 
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Simulation of a pinhole test of an MX-80 bentonite

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Simulation of a pinhole test of an MX-80 bentonite

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