1. Burst Exponents in Stochastic Modeling Experiments of Hydraulic Fracture
Bjørn Skjetne1 and Alex Hansen1
1Department of Physics
Norwegian University of Science and Technology (NTNU)
N-7491 Trondheim
Norway

2. Motivation
CCS – currently great interest in how one should go about in order to facilitate Carbon Capture and Storage. Politicians and environmentalists place a lot of emphasis on this. It is the storage of carbon that concerns us in the context of hydraulic fracture.
Hydraulic fracture is a well known method to enhance oil and gas production. Interest in this approach is increasing as the remaining oil becomes more difficult to extract. Also has an environmental side, since a lot of attention has been focused on pollution from oil extraction operations.
Physics. Hydraulic fracture also represents an interesting phenomenon from the academic point of view. Here we can test to what extent our physical understanding of fracture and flow tallies with experimental reality.

3. Our project
CCS, concerns geologic storage of CO2. A grant from the Norwegian Research Council via their CLIMIT program for the project ‘Efficient CO2 Absorption in Water-Saturated Porous Media through Hydraulic Fracture’. This is a joint project between SINTEF and NTNU.

4. Stochastic modeling of fracture
KEY FEATURES:
Assumption of a meso-scale, intermediate between the macroscopic and the microscopic scales.
Discretize the system on a regular lattice.
Define the forces between the nodes on the lattice.
Introduce a driving force in the system in the form of a potential gradient or a boundary condition.
Define a breaking rule which specifies when and where damage should occur.
Specify a consistent stopping criterion.

5. Biot’s theory of linear poroelasticity
In our problem we need to understand the behaviour of fluid-saturated porous
media. Hence we introduce the concept of a POROELASTIC ELEMENT. It
is a solid matrix which is permeated by a network of connected pores.
What is the mechanical behaviour of such an entity?
A well known relationship between stress and strain is
In analogy with this Maurice Biot (1941) introduced a similar quantity for the
fluid response, i.e.,

6. Biot’s theory of linear poroelasticity
In our problem we need to understand the behaviour of fluid-saturated porous
media. Hence we introduce the concept of a POROELASTIC ELEMENT. It
is a solid matrix which is permeated by a network of connected pores.
What is the mechanical behaviour of such an entity?
Maurice Biot coupled the two equations to obtain two constitutive equations:
Essentially, this is Biot theory in a nutshell!
From this assumption we can deduce governing equations of FLUID FLOW
and MECHANICAL EQUILIBRIUM.

7. Fluid flow in Biot theory
In hydraulic fracture fluid is injected at high pressure in a well-bore and due
to this fluid flows through the system. What balances the flow as the resulting
potential gradient is a frictional force between the fluid and the pore walls.
The average fluid velocity is then
Since this is the relative movement of the fluid with respect to the solid, this
can also be stated as

8. Fluid flow in Biot theory
We regard a small poroelastic element, or control volume, through which fluid flows
The volume of the fluid which enters on the left-hand side is
and that which exits on the right-hand side is
Consequently, the change in the fluid content is given by

9. Fluid flow in Biot theory
We regard a small poroelastic element, or control volume, through which fluid flows
We then get
where we have assumed that the porosity is the same throughout the control volume.
Next, we have for the right-hand side

10. Fluid flow in Biot theory
We regard a small poroelastic element, or control volume, through which fluid flows
This results in
Considering all three spatial directions and adding the contributions, we get
where our control volume is

11. Fluid flow in Biot theory
We regard a small poroelastic element, or control volume, through which fluid flows
Next we use the definition of Biot’s increment in fluid content
which now gives
Considering the time derivative of this we get

12. Fluid flow in Biot theory
We regard a small poroelastic element, or control volume, through which fluid flows
Now we make use of the expression above and obtain
We now use Darcy’s Law
to obtain

13. Fluid flow in Biot theory
If we now substitute for the increment in fluid content, given by one of Biot’s
constitutive equations, then we have
This is the GOVERNING EQUATION OF FLUID FLOW in our system.
Mechanical equilibrium in Biot theory
For the mechanical equilibrium the coupling of the expressions at the beginning
also result in the addition of an extra term. The standard expression in classical
elasticity becomes
This is the GOVERNING EQUATION OF MECHANICAL EQUILIBRIUM.

14. Discretization of the model
In order to create a model of hydraulic fracture which the computer can understand
we assume the material properties to be embedded in a discrete manner and in a
way which allows any of these discrete points to have a consistent address.
An obvious choice would be a square lattice due to its simple geometry. However,
such a lattice does not have the proper macroscopic behaviour and we must choose
a lattice with TRIANGULAR TOPOLOGY.

15. Discretization of the model
In order to create a model of hydraulic fracture which the computer can understand
we assume the material properties to be embedded in a discrete manner and in a
way which allows any of these discrete points to have a consistent address.
An obvious choice would be a square lattice due to its simple geometry. However,
such a lattice does not have the proper macroscopic behaviour and we must choose
a lattice with TRIANGULAR TOPOLOGY.

16. Discretization of the model
In order to create a model of hydraulic fracture which the computer can understand
we assume the material properties to be embedded in a discrete manner and in a
way which allows any of these discrete points to have a consistent address.
An obvious choice would be a square lattice due to its simple geometry. However,
such a lattice does not have the proper macroscopic behaviour and we must choose
a lattice with TRIANGULAR TOPOLOGY.

17. Discretization of the model
In order to create a model of hydraulic fracture which the computer can understand
we assume the material properties to be embedded in a discrete manner and in a
way which allows any of these discrete points to have a consistent address.
An obvious choice would be a square lattice due to its simple geometry. However,
such a lattice does not have the proper macroscopic behaviour and we must choose
a lattice with TRIANGULAR TOPOLOGY.

18. Discretization of the model
In order to create a model of hydraulic fracture which the computer can understand
we assume the material properties to be embedded in a discrete manner and in a
way which allows any of these discrete points to have a consistent address.
An obvious choice would be a square lattice due to its simple geometry. However,
such a lattice does not have the proper macroscopic behaviour and we must choose
a lattice with TRIANGULAR TOPOLOGY.

19. Discretization of the model
In order to create a model of hydraulic fracture which the computer can understand
we assume the material properties to be embedded in a discrete manner and in a
way which allows any of these discrete points to have a consistent address.
In modeling hydraulic fracture we assume that our crack will propagate outward
from the center of the system towards the boundaries. We introduce therefore
introduce a ‘well-bore’ at the center of the lattice by removing a link.

20. Discretization of the model
In order to create a model of hydraulic fracture which the computer can understand
we assume the material properties to be embedded in a discrete manner and in a
way which allows any of these discrete points to have a consistent address.
Due to the radial symmetry of the problem, we would ideally like to have a more
or less circular geometry. As an approximation to this we choose a HEXAGONAL
shape for the outer boundary.

21. Discretization of the model
In order to create a model of hydraulic fracture which the computer can understand
we assume the material properties to be embedded in a discrete manner and in a
way which allows any of these discrete points to have a consistent address.
Finally, the underlying structure where our hydraulic fracture modeling takes place
looks like this.

22. Specifying forces between nodes
The discretization in our model is in terms of a DEFORMABLE lattice. Hence the
forces between the nodes must be defined in terms of displacements.
On a triangular lattice there are six neighbours to each node. These are numbered
from one to six, beginning with the horizontal beam which extends towards the
left-hand side.

23. Specifying forces between nodes
The discretization in our model is in terms of a DEFORMABLE lattice. Hence the
forces between the nodes must be defined in terms of displacements.
Forces between the nodes are defined IN ANALOGY with ELASTIC BEAMS. The
beams are fastened onto each other in such a way that the angle between them
(60 degrees) is preserved, even when there is a rotation at the node.

24. Specifying forces between nodes
The discretization in our model is in terms of a DEFORMABLE lattice. Hence the
forces between the nodes must be defined in terms of displacements.
We place a COORDINATE SYSTEM on each node in order to keep track of the
displacements.

25. Specifying forces between nodes
The discretization in our model is in terms of a DEFORMABLE lattice. Hence the
forces between the nodes must be defined in terms of displacements.
The beams are assumed to be brittle-elastic and the elastic response is a combination
of AXIAL, SHEAR and FLEXURAL forces. Thick beams have been assumed in the
expressions derived.

26. Specifying forces between nodes
The discretization in our model is in terms of a DEFORMABLE lattice. Hence the
forces between the nodes must be defined in terms of displacements.
Depending on the combination of displacements at any given point in time, the
forces are calculated accordingly..

27. Poroelastic beams
To include Biot’s theory in our model of hydraulic fracture we include in our
beams the assumption that they are porous.
Flow through the beams are along the axis between the nodes that it connects.

28. Poroelastic beams
To include Biot’s theory in our model of hydraulic fracture we include in our
beams the assumption that they are porous.
Flow through the beams are along the axis between the nodes that it connects.
To say a few words about the Biot-Willis parameter which couples in the pore
pressure, this expresses how much of the bulk strain is taken up by the change
in the pore volume and how much is taken up by the change in the solid volume.

29. Biot-Willis parameter
The governing equation of mechanical equilibrium was obtained from the two
constitutive equation resulting from Biot’s coupling of volumetric strain-stress
with increment in fluid content-pore pressure.
For the relationship between the independent variables we have
The expression for the parameter which obtains in the derivation of mechanical
equilibrium in the equation at the top is

30. Biot-Willis parameter
PHYSICAL INTERPRETATION
The Biot-Willis parameter is an expression of the increment in fluid content
with respect to changes in volumetric strain.
A number between zero and one having the following interpretation:
INCOMPRESSIBLE LIQUID
INCOMPRESSIBLE SOLID
All the bulk strain is taken
up by the change in solid
volume
All the bulk strain is taken
up by the change in pore
volume
0.79
(Berea sandstone)

31. Beam failure criterion
In order for our system to fracture we need to define a fracture criterion. The one
we use is
Here F is the axial force and M is the bending moment. Random thresholds are
generated for each beam on the lattice, one for the amount of axial force that the
beam can withstand, and one for the amount of bending it can withstand. These
are then combined as shown. When a beam breaks it is removed irreversibly
from the lattice, only retaining its axial force where relative displacements
indicate local compression. Axial contribution to beam breaking is assumed
to occur only when beams are loaded in tension.
Disorder
Thresholds are generated by raising a random number to a power D, a positive
power indicates that some thresholds deviate towards weak strength. Likewise,
A negative D indicates that some of the thresholds deviate towards stronger
strength. The magnitude of D controls how many of the thresholds deviate in this
way, i.e., how strong the disorder is.

32. L=90 lattice, longitudinal
Crack growth and disorder – uniaxial
loading of a sheet with a central crack
Fracture criterion:
Crack tip: high
stress intensity
STRETCHED
No disorder: fracture is stress
dominated, and crack growth
localized to a single existing
crack. Unstable.
Disorder: if the local variation
in material strength is strong
enough, new cracks will appear
randomly. Fracture is disorder
dominated. Stable. ?
COMPRESSED
L=90 lattice, longitudinal
axial stresses shown

33. Brittle fracture in the presence of disorder
Disorder is imposed on the beams in the form of random breaking
thresholds. Elastic properties are assumed to be identical from one
beam to the next.
Initially, cracks appear on random locations.
Some of the smaller cracks grow before being arrested by the surroundings.
A cross-over occurs from a disorder-dominated process to a stress-dominated process, where smaller cracks merge into a macroscopic crack.
Localized fracture, catastrophic crack growth, sets in.

34. Numerical calculation
Fracture in our model is driven by a potential gradient which is set up by injecting
fluid at the center of the system. Here the pressure is kept constant throughout the
fracturing process.

35. Numerical calculation
Fracture in our model is driven by a potential gradient which is set up by injecting
fluid at the center of the system. Here the pressure is kept constant throughout the
fracturing process.
CONJUGATE GRADIENTS
Since the expressions which we derive for the poroelastic forces in the system
are all linear, the resulting elastic energy expression is quadratic.

36. Numerical calculation
Fracture in our model is driven by a potential gradient which is set up by injecting
fluid at the center of the system. Here the pressure is kept constant throughout the
fracturing process.
CONJUGATE GRADIENTS
Since the expressions which we derive for the poroelastic forces in the system
are all linear, the resulting elastic energy expression is quadratic.
An efficient way to minimize a quadratic form is to use conjugate gradients.
The minimum in our system corresponds to that situation where on each node the
sum of forces and moments is zero. Physically, this is the required situation for
a system in mechanical equilibrium.

37. Numerical calculation
Fracture in our model is driven by a potential gradient which is set up by injecting
fluid at the center of the system. Here the pressure is kept constant throughout the
fracturing process.
The permeability in the system is set to be FINITE wherever the system is intact
and INFINITE within cracks.
As cracks grow on the lattice the pressure is updated using a CLUSTER MAPPING
algorithm which makes sure that the pressure distribution is consistent with our
assumption on the permeability.
Each time a crack touches the boundary the pressure within that crack is ‘vented’,
that is, it is set to zero.
The process is continued until one of the cracks which connect to the injection hole
reaches the boundary. This is a consistent stopping criterion.

38. Stages of hydraulic fracture
N= 0 beams broken

39. Stages of hydraulic fracture
N= 200 beams broken

40. Stages of hydraulic fracture
N= 1000 beams broken

41. Stages of hydraulic fracture
N= 1745 beams broken

42. Stages of hydraulic fracture
N= 1746 beams broken

43. Stages of hydraulic fracture
N= 2408 beams broken

44. Stages of hydraulic fracture
N= 2409 beams broken

45. Crack Samples
Disorder D=1.4

47. Crack Samples
Disorder D=2

49. Crack Samples
Disorder D=0.4

51. Crack Samples
Disorder D=-4

56. Crack Samples
Disorder D=-6

62. Animation of beam failure criterion
In order for our system to fracture we need to define a fracture criterion. The one
we use is
Essentially, this is the ratio of stress to strenght. Its intensity increases as
the stress increases and it also increases as the strength decreases.

64. Force and pressure D > 0
Here we show the average force-pressure responce for 1000 samples each of
systems with disorders ranging from D=0.4 (A) to D=2 (H) in steps of 0.2

65. Force and pressure D < 0
Force-pressure responce for 1000 samples each of systems with disorders
D=-2 (A), D=-3 (B) D=-4 (C) and D=-5 (D)

66. Force and pressure
Force-pressure responce for 1000 samples of D=-3 showing the average
position of broken beams N=1, N=2, N=3, and N=4.

67. Burst exponents
Z=-1.50(3)
Average over 1000 samples for the burst-size distribution obtained for the
injection pressure monitored as a function of the number of breaks.
This result agrees very well with other results obtained for catastrophic
rupture.

68. Burst exponents
Z=-1.58(2)
Average over 1000 samples for the burst-size distribution obtained for the
injection pressure monitored as a function of the number of breaks.
Agrees fairly well with other results obtained for catastrophic rupture.

69. Burst exponents
Z=-1.66(2)
Average over 1000 samples for the burst-size distribution obtained for the
injection pressure monitored as a function of the number of breaks.
No longer agrees with expected Z=-1.50 result for catastrophic rupture.

70. Burst exponents
Z=-1.68(2)
Average over 1000 samples for the burst-size distribution obtained for the
injection pressure monitored as a function of the number of breaks.
Agrees even less with expected Z=-1.50 result for catastrophic rupture.

71. Burst exponents
If we concentrate on those events that occur closer to the catastrophic point
then we see that the result improves. In fact, the improvement gets better
the closer we are to the catastrophic failure point.

72. Burst exponents
If we concentrate on those events that occur closer to the catastrophic point
then we see that the result improves. In fact, the improvement gets better
the closer we are to the catastrophic failure point.

73. Burst exponents
In the case of D>0 disorder the burst exponent converges on the value D=-2 as
the disorder magnitude increases.

74. Burst exponents