Burst Exponents in Stochastic Modeling Experiments of Hydraulic Fracture

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Burst Exponents in Stochastic Modeling Experiments of Hydraulic

Fracture

Bjørn Skjetne1 and Alex Hansen1

1Department of PhysicsNorwegian University of Science and Technology (NTNU)

N-7491 Trondheim Norway

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MotivationMotivation

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.

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Our projectOur 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.

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Stochastic modeling of fractureStochastic modeling of fracture

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.

KEY FEATURES:KEY FEATURES:

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Biot’s theory of linear poroelasticityBiot’s theory of linear poroelasticity

A well known relationship between stress and strain is

In analogy with this Maurice Biot (1941) introduced a similar quantity for thefluid response, i.e.,

In our problem we need to understand the behaviour of fluid-saturated porousmedia. Hence we introduce the concept of a POROELASTIC ELEMENT. Itis a solid matrix which is permeated by a network of connected pores.

What is the mechanical behaviour of such an entity?

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Biot’s theory of linear poroelasticityBiot’s theory of linear poroelasticity

Maurice Biot coupled the two equations to obtain two constitutive equations:

In our problem we need to understand the behaviour of fluid-saturated porousmedia. Hence we introduce the concept of a POROELASTIC ELEMENT. Itis a solid matrix which is permeated by a network of connected pores.

What is the mechanical behaviour of such an entity?

Essentially, this is Biot theory in a nutshell!

From this assumption we can deduce governing equations of FLUID FLOWand MECHANICAL EQUILIBRIUM.

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Fluid flow in Biot theoryFluid flow in Biot theoryIn hydraulic fracture fluid is injected at high pressure in a well-bore and dueto this fluid flows through the system. What balances the flow as the resultingpotential 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, thiscan also be stated as

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Fluid flow in Biot theoryFluid flow in Biot theoryWe 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

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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

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This results in

Considering all three spatial directions and adding the contributions, we get

where our control volume is

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Next we use the definition of Biot’s increment in fluid content

which now gives

Considering the time derivative of this we get

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Now we make use of the expression above and obtain

We now use Darcy’s Law

to obtain

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Fluid flow in Biot theoryFluid flow in Biot theoryIf we now substitute for the increment in fluid content, given by one of Biot’sconstitutive equations, then we have

This is the GOVERNING EQUATION OF FLUID FLOW in our system.

Mechanical equilibrium in Biot theoryMechanical equilibrium in Biot theoryFor the mechanical equilibrium the coupling of the expressions at the beginningalso result in the addition of an extra term. The standard expression in classicalelasticity becomes

This is the GOVERNING EQUATION OF MECHANICAL EQUILIBRIUM.

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Discretization of the modelDiscretization of the modelIn order to create a model of hydraulic fracture which the computer can understandwe assume the material properties to be embedded in a discrete manner and in away 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.

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In modeling hydraulic fracture we assume that our crack will propagate outwardfrom 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.

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Due to the radial symmetry of the problem, we would ideally like to have a moreor less circular geometry. As an approximation to this we choose a HEXAGONALshape for the outer boundary.

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Finally, the underlying structure where our hydraulic fracture modeling takes placelooks like this.

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Specifying forces between nodesSpecifying forces between nodesThe discretization in our model is in terms of a DEFORMABLE lattice. Hence theforces between the nodes must be defined in terms of displacements.

On a triangular lattice there are six neighbours to each node. These are numberedfrom one to six, beginning with the horizontal beam which extends towards theleft-hand side.

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Forces between the nodes are defined IN ANALOGY with ELASTIC BEAMS. Thebeams 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.

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We place a COORDINATE SYSTEM on each node in order to keep track of thedisplacements.

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The beams are assumed to be brittle-elastic and the elastic response is a combinationof AXIAL, SHEAR and FLEXURAL forces. Thick beams have been assumed in theexpressions derived.

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Depending on the combination of displacements at any given point in time, theforces are calculated accordingly..

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Poroelastic beamsPoroelastic beamsTo include Biot’s theory in our model of hydraulic fracture we include in ourbeams the assumption that they are porous.

Flow through the beams are along the axis between the nodes that it connects.

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Poroelastic beamsPoroelastic beamsTo include Biot’s theory in our model of hydraulic fracture we include in ourbeams 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 porepressure, this expresses how much of the bulk strain is taken up by the changein the pore volume and how much is taken up by the change in the solid volume.

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Biot-Willis parameterBiot-Willis parameter

The governing equation of mechanical equilibrium was obtained from the twoconstitutive equation resulting from Biot’s coupling of volumetric strain-stresswith 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 mechanicalequilibrium in the equation at the top is

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Biot-Willis parameterBiot-Willis parameter

The Biot-Willis parameter is an expression of the increment in fluid contentwith respect to changes in volumetric strain.

PHYSICAL INTERPRETATIONPHYSICAL INTERPRETATION

A number between zero and one having the following interpretation:

INCOMPRESSIBLE LIQUID INCOMPRESSIBLE SOLID

All the bulk strain is takenup by the change in solidvolume

All the bulk strain is takenup by the change in porevolume

0.79(Berea sandstone)

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Beam failure criterionBeam failure criterionIn order for our system to fracture we need to define a fracture criterion. The onewe use is

Here F is the axial force and M is the bending moment. Random thresholds aregenerated for each beam on the lattice, one for the amount of axial force that thebeam 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 irreversiblyfrom the lattice, only retaining its axial force where relative displacementsindicate local compression. Axial contribution to beam breaking is assumedto occur only when beams are loaded in tension.

DisorderDisorderThresholds are generated by raising a random number to a power D, a positivepower indicates that some thresholds deviate towards weak strength. Likewise,A negative D indicates that some of the thresholds deviate towards strongerstrength. The magnitude of D controls how many of the thresholds deviate in thisway, i.e., how strong the disorder is.

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Crack growth and disorder – uniaxial loading of a sheet with a central crack

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Crack tip: high stress intensity

Fracture criterion:

L=90 lattice, longitudinalaxial stresses shown

STRETCHED

COMPRESSED

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.

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Initially, cracks appear on random locations.

Brittle fracture in the presence of disorder

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.

Disorder is imposed on the beams in the form of random breakingthresholds. Elastic properties are assumed to be identical from onebeam to the next.

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Numerical calculationNumerical calculation

Fracture in our model is driven by a potential gradient which is set up by injectingfluid at the center of the system. Here the pressure is kept constant throughout thefracturing process.

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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.

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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 thesum of forces and moments is zero. Physically, this is the required situation for a system in mechanical equilibrium.

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The process is continued until one of the cracks which connect to the injection holereaches the boundary. This is a consistent stopping criterion.

Each time a crack touches the boundary the pressure within that crack is ‘vented’,that is, it is set to zero.

As cracks grow on the lattice the pressure is updated using a CLUSTER MAPPINGalgorithm which makes sure that the pressure distribution is consistent with ourassumption on the permeability.

The permeability in the system is set to be FINITE wherever the system is intactand INFINITE within cracks.

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Stages of hydraulic fractureStages of hydraulic fracture

N= 0 beams broken

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N= 200 beams broken

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N= 1000 beams broken

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Crack SamplesCrack Samples

Disorder D=1.4

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Disorder D=2

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Disorder D=0.4

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Disorder D=-4

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Disorder D=-6

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Animation of beam failure criterionAnimation of beam failure criterion

In order for our system to fracture we need to define a fracture criterion. The onewe 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.

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Force and pressure Force and pressure

Here we show the average force-pressure responce for 1000 samples each ofsystems with disorders ranging from D=0.4 (A) to D=2 (H) in steps of 0.2

D > 0

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

D < 0

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

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Burst exponents Burst exponents

Average over 1000 samples for the burst-size distribution obtained for theinjection pressure monitored as a function of the number of breaks. This result agrees very well with other results obtained for catastrophicrupture.

Z=-1.50(3)

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Average over 1000 samples for the burst-size distribution obtained for theinjection pressure monitored as a function of the number of breaks. Agrees fairly well with other results obtained for catastrophic rupture.

Z=-1.58(2)

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Average over 1000 samples for the burst-size distribution obtained for theinjection pressure monitored as a function of the number of breaks. No longer agrees with expected Z=-1.50 result for catastrophic rupture.

Z=-1.66(2)

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Average over 1000 samples for the burst-size distribution obtained for theinjection pressure monitored as a function of the number of breaks. Agrees even less with expected Z=-1.50 result for catastrophic rupture.

Z=-1.68(2)

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If we concentrate on those events that occur closer to the catastrophic pointthen we see that the result improves. In fact, the improvement gets betterthe closer we are to the catastrophic failure point.

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If we concentrate on those events that occur closer to the catastrophic pointthen we see that the result improves. In fact, the improvement gets betterthe closer we are to the catastrophic failure point.

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In the case of D>0 disorder the burst exponent converges on the value D=-2 asthe disorder magnitude increases.

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Burst Exponents in Stochastic Modeling Experiments of Hydraulic Fracture

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