Hydraulic Fracture

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Burst Exponents in Stochastic

Modeling Experiments of Hydraulic

Fracture

Bjrn Skjetne1 and Alex Hansen1

1Department of Physics

Norwegian University of Science and Technology (NTNU)

N-7491 Trondheim

Norway


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Motivation

CCScurrently great interest in how one should go about inorder to facilitate Carbon Capture and Storage. Politicians andenvironmentalists place a lot of emphasis on this. It is thestorage of carbon that concerns us in the context of hydraulic

fracture.

Hydraulic fracture is a well known method to enhance oil andgas production. Interest in this approach is increasing as theremaining oil becomes more difficult to extract. Also has anenvironmental side, since a lot of attention has been focused onpollution from oil extraction operations.

Physics. Hydraulic fracture also represents an interestingphenomenon from the academic point of view. Here we can testto what extent our physical understanding of fracture and flowtallies with experimental reality.


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Our project

CCS, concerns geologic storage of CO2. A grant from the Norwegian Research

Council via theirCLIMIT program for the project Efficient CO2 Absorption inWater-Saturated Porous Media through Hydraulic Fracture. This is a jointproject between SINTEF and NTNU.


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Stochastic modeling of fracture

Assumption of a meso-scale, intermediate between the macroscopic and themicroscopic 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 aboundary condition.

Define a breaking rule which specifies when and where damage should occur.

Specify a consistent stopping criterion.

KEY FEATURES:


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Biots 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 the

fluid response, i.e.,

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 behaviourof such an entity?


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Biots 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 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 behaviourof such an entity?

Essentially, this is Biot theory in a nutshell!

From this assumption we can deduce governing equations ofFLUID FLOW

and MECHANICAL EQUILIBRIUM.


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Fluid flow in Biot theoryIn 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


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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


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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 Biots 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 Darcys Law

to obtain


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If we now substitute for the increment in fluid content, given by one of Biots

constitutive equations, then we have

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

Mechanical equi l ibr ium 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.


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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 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 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.


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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.


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Finally, the underlying structure where our hydraulic fracture modeling takes place

looks like this.


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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.


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Forces between the nodes are definedIN ANALOGYwith 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.


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

displacements.


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The beams are assumed to be brittle-elastic and the elastic response is a combination

ofAXIAL, SHEARand FLEXURAL forces. Thick beams have been assumed in the

expressions derived.


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Depending on the combination of displacements at any given point in time, the

forces are calculated accordingly..


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Poroelastic beams

To include Biots 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.


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Poroelastic beams

To include Biots 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 parameterwhich couples in the pore

pressure, this expresses how much of the bulk strain is taken up by the change

in thepore volume and how much is taken up by the change in thesolid volume.


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Biot-Willis parameter

The governing equation of mechanical equilibrium was obtained from the two

constitutive equation resulting from Biots coupling ofvolumetric 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


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Biot-Willis parameter

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

PHYSICAL INTERPRETATION

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)


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Beam failure criterionIn 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.

DisorderThresholds are generated by raising a random number to a powerD, a positive

power indicates that some thresholds deviate towards weak strength. Likewise,

A negativeD indicates that some of the thresholds deviate towards stronger

strength. The magnitude ofD controls how many of the thresholds deviate in this

way, i.e., how strong the disorder is.


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Crack growth and disorderuniaxial

loading of a sheet with a central crack

?

Crack tip: high

stress intensity

Fracture criterion:

L=90 lattice, longitudinal

axial stresses shown

STRETCHED

COMPRESSED

No disorder: fracture isstress

dominated, and crack growthlocalized to a single existing

crack. Unstable.

Disorder: if the local variation

in material strength is strongenough, 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 bythe 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 breaking

thresholds. Elastic properties are assumed to be identical from one

beam to the next.


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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.


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fracturing process.

CONJUGATE GRADIENTS

Since the expressions which we derive for the poroelastic forces in the system

are all linear, the resulting elastic energy expressionis quadratic.


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fracturing process.

CONJUGATE GRADIENTS

Since the expressions which we derive for the poroelastic forces in the system

are all linear, the resulting elastic energy expressionis 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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fracturing process.

The process is continued until one of the cracks which connect to the injection hole

reaches 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 aCLUSTER MAPPING

algorithm which makes sure that the pressure distribution is consistent with our

assumption on the permeability.

The permeability in the system is set to be FINITE wherever the system is intact

and INFINITE within cracks.


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Stages 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 Samples

DisorderD=1.4


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Crack Samples

DisorderD=2


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Crack Samples

DisorderD=0.4


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Crack Samples

DisorderD=-4


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Crack Samples

DisorderD=-6


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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.


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Force and pressure

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

D > 0


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Force and pressure

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 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.


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Burst exponents

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.

Z=-1.50(3)


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Burst exponents



Agrees fairly well with other results obtained for catastrophic rupture.

Z=-1.58(2)


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Burst exponents



No longer agrees with expected Z=-1.50 result for catastrophic rupture.

Z=-1.66(2)


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Burst exponents



Agrees even less with expected Z=-1.50 result for catastrophic rupture.

Z=-1.68(2)


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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.


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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.


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Burst exponents

In the case ofD>0 disorder the burst exponent converges on the valueD=-2 as

the disorder magnitude increases.


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Burst exponents

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Hydraulic Fracture

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