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DISROC Examples |
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| DISROC 5 - Example: |
Other Examples |
Fracture propagation in heterogeneous materials
The example of heterogeneous material considered is inspired of shale formations containing heterogeneity inclusions which are parallel bedded fractures filled with calcites called “beef” (Figure 1).
Figure 1 : (a) Calcite bed inclusions (“beef”) surrounded by mudstone in a shale formation, (b) details of the zone selected for the numerical model
Numerical determination of the equivalent
or homogenized toughness of these materials needs modeling fracture
propagation under external forces. To this purpose a Finite Element model
for a representative volume is created (Figure 2). A central part where the
fracture can propagate has been more finely meshed.
Figure 2: Initial mesh of the model
Then, the mesh has been enriched with joint
elements placed around all the elements in the central part where the
fracture can propagate and also at the interface between the two materials
(Figure 3). Cohesive zone models (the 21400 model of the Disroc catalog)
have been assigned to the joint elements with different parameters values
for the joints in the mudstone, in beef and at the interface between the two materials (see mesh
details on Figure 4).
Figure
3 : Mesh enriched with joint elements
Figure 4 : Details of mesh enrichement with joint elements
The load applied on the model consists in displacements ux imposed on two points (Figure
3):
ux(A)= -1, ux(B)=
1
when the condition uy = 0
is imposed on all the bottom side of the model.
This loading makes a fracture propagate through the model. The Figure 5 presents the ux
displacement field of the model at some stage of fracturing. The figure 6
shows the stress sxx
and the Figure 7 the normal stress sn on joint elements. Theses stresses
are maximum at the fracture tip.
Figure 5 : The displacement field ux in the fractured model
Figure 6: The normal stress sxx is maximum at the fracture tip
By clicking on the result one can see the video-animation which shows an interesting phenomenon: an acceletation
of the propagation and even instabilities (jumps) when passing from the harder medium
(beef) to the softer one (mudstone).
Simulations of this type make it possible to identify the equivalent toughness or
energy release rate for the homogenized medium by calculating the dissipation as
the difference between the external work applied on the domain's boundary and the
elastic energy stored in the domain. The following figure displays the force-displacement
for this loading. The Ux displacement is applied on four nodes on the boundary at the
right and four nodes at the left of the crack intersection point on the boundary.
The nodal forces Fx for these nodes is given by dahsed-lines on Fig. 7a, and their
by the thicker line. The curve shows well the instable propagation intervals
leading to sudden force drops making the curve discontinuous. The positions of the crack-tip
at these discontinuity instants is given on the Fig. 7b. The instable propagation of
the crack at these instants can be well seen on the video of Fig.6.
Fig.7a: Force-Displacement curve for crack propagation with different instability instants
Fig.7b: Crack positions at differents instants
Figure 8: The normal stress sn on the joint elements is maximum at the
fracture tip
Figure 9: The fracture propagation on the
beef-mudstone interface
This shows also the capability of DISROC to
simulate fracture propagation in already fractured media. As a matter of fact, the strength of the
joint elements on the beef-mudstone interface could be decreased to zero in
order to represent an existing fracture.
The videos showing the fracture propagation
can be found by clicking on images.