a new approach to modeling hydraulic fractures in consolidated sands

7/18/2019 A New Approach to Modeling Hydraulic Fractures in Consolidated Sands

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Abstract Field data show that fracturing in poorly-consolidated rocks isnot adequately represented by traditional models for brittle,

linear-elastic rocks. This is not unexpected sinceunconsolidated sands do not exhibit brittle- elastic behavior.

In addition, sands have very low tensile and shear strengths.

A model is presented for the propagation of “fractures” in

unconsolidated sands. The model departs radically fromcurrent models in that brittle fracture mechanics is not used.

Instead the propagation of pore pressure is computed and the

porosity and permeability of the sand is specified as a functionof the effective stress. This results in the creation of an

anisotropic zone of increased porosity and permeability along

the plane of maximum in-situ stress (normal faulting stressregime) or at a certain angle to it (strike slip faulting regime).

This region of enhanced porosity defines a “fracture” in

unconsolidated sands. The physics of creation and propagation of this oriented, high permeability zone, is

modeled for the first time.

It is shown that in-situ stress anisotropy and shear failure play a very important role in determining the dimensions of

this fracture zone. In addition, the permeability anisotropy

generated due to the stress anisotropy in the sand is the criticaldriving force behind the creation of the oriented “fracture”.

During the hydraulic fracturing of an unconsolidatedformation, a high permeability zone (channel or fracture) willform in response to the difference in situ horizontal stresses

and the decrease in the net effective stress near wellbore. To

correctly model the fluid distribution, the fluid flow behavior

must be coupled to the mechanical behavior of the sands.Based on the coupled geo-mechanics and reservoir simulation

(model, iterative-coupled 2-D finite difference software isdeveloped to simulate the strain, stress change due to the

injection. Based on the constitutive relationship of

permeability and porosity, we modeled permeability and

porosity as a function of effective stress.

IntroductionSand control is a growing concern in most offshore wells in

the Gulf of Mexico, Western Canada, and Brazil. The

application of fracpacks in these poorly consolidatedreservoirs has been an effective method for preventing sanding

problems.In conventional hydraulic fracture simulations, to which

linear elastic fracture mechanics (LEFM) is applied [1]

fracture initiation and propagation is governed by in-situ

stresses, fracture toughness, tip dilatancy, and the processzone. Unlike competent formations, unconsolidated sand beds

have little or no tensile strength. LEFM is adequate for hardrocks, but the fracture geometry predictions fall short when

applied to fracturing soft rocks. For example, it has been

reported that millions of barrels of solid waste slurry can

easily be injected into soft formations over a period of severalyears [2]. To accommodate such a large volume of solids

fracture lengths of several miles would be required, even withfractures that are several centimeters wide when using

classical fracture models for simulating this process.

Some experimental and simulation work [2-9] has been done

to identify the mechanisms of fracture propagation and

initiation in unconsolidated sand formations. Khodaverdianand McElfresh’s experiments [2] show that fracture tip

propagation in unconsolidated sand is dominated by fluid

invasion and shear failure within a process zone ahead of the

tip. In addition, sub-parallel fractures form and contribute to

the post-stimulation skin since these fractures are not expectedto be propped open during frac-pack operations. Di Lullo and

Curtis [3] provide an alternative mechanism for the initiation

and propagation of the shear-failure zones based on theirexperiments. They postulate that fluid leakoff into the matrix

pressurizes and fluidizes the visco-plastic formation matrix

As the pressure surpasses the yield stress, the formation“parts” (or deforms) forming a channel, allowing sand-laden

slurry to penetrate and propagate. Wang and Sharma

[4

measured the mechanical properties of poorly consolidatedsands. Their data indicate that unconsolidated sands do not

show classic failure modes in compression. Instead, a region

of elasto-plasticity is observed as stresses are increasedresulting in ductile failure over an extended range of stresses.

Settari [5] [6] proposed a non-elastic injection model by

coupling fluid flow and soil mechanics behavior for

unconsolidated sands. The non-linearity of the compressibilityand shear failure were thought of as the principle mechanisms

controlling the injectivity in oil sands. In addition, dilatanfailure behavior increases porosity and permeability

Numerical methods for coupling fluid flow and gemechanic

SPE 96246

A New Approach to Modeling Hydraulic Fractures in Unconsolidated SandsZ. Zhai, SPE, and M.M. Sharma, SPE, U. of Texas at Austin



2 SPE 9624

were introduced. Chin and Montgomery [7] developed a model

for solids injection and compared their results with field data.

Many of the above studies rely on classical brittle fracturemechanics and in some cases do not account for shear failure.

Permeability anisotropy induced by in-situ stresses is notconsidered.

In this paper, we present a new approach to modeling the

mechanical behavior of unconsolidated sands that aresubjected to injection of water –based slurries.

Model for Stress Distribution around an Injection Well

To study fracture initiation and propagation, we need to obtain

the effective stress distribution around the well and then

update the permeability and porosity in accordance with theappropriate constitutive relationships.

The stresses around the wellbore can be divided into three

parts (Figure 1):1. Stresses induced by far-field, in-situ stresses.2. Stresses induced by the wellbore pressure.3. Flow induced stresses (poro-elastic stresses).Based on the stress distribution, we can determine the

anisotropic stress tensor as well as where and when tensile or

shear failure occurs. This approach has been widely applied to

wellbore stability problems for homogeneous, isotropic rocks

[4]. In this paper we couple the stress distribution with the pore

pressure and apply the model to hydraulic fracturing problems

in unconsolidated sands.

Stress Distribution due to In-situ Stresses. Thestresses around a wellbore in a cylindrical coordinate system

(r, θ, and z) due to the principal in-situ stresses are given by,

2

2

2 4

2 4

(1 )2

(1 4 3 ) cos 22

xx yywrr

xx yyw w

r

r

r r

r r

σ σ σ

σ σ θ

′ ′+′ = −

′ ′−+ − +

2

2

4

4

(1 )2

(1 3 ) cos 22

xx yyw

xx yyw

r

r

r

r

θθ

σ σ σ

σ σ θ

′ ′+′ = +

′ ′−− +

2 4

2 4(1 2 3 ) sin 2

2

xx yyw wr

r r

r r θ

σ σ σ θ

′ ′−′ = − + −

(1)

The Flow-induced Stresses. The flow-induced stresses can be

obtained by coupling the fluid flow equation and the geo-

mechanical equations as shown below.

Geomechanical Equations (Displacement Formulation)

The equilibrium relations in two-dimensions can be written as

0

2 10

r r r

r r

Pr r r r

P

r r r r

θ θ

θ θ θ

τ σ σ σ θ

τ σ τ

θ θ

⎧ ′ ′ ′′ ∂ −∂ ∂+ − + =⎪⎪ ∂ ∂ ∂⎨

′ ′ ′∂ ∂⎪ ∂+ + + =⎪ ∂ ∂ ∂⎩

(2)

The total stresses are related to the effective stresses by the

generalized Terzaghi principle (assuming Biot’s constant is

one),

/

/

,

,

r r r r P

P

θ θ

θθ θθ

σ σ σ σ

σ σ ′= + =

= +

(3)

The effective stresses are related to the strains by the

generalized form of Hooke’s law, as a first approximation. Fo

an isotropic material these relations are:

/ /

/

2 , 2

2

rr vol r r r

vol

G G

G

θ θ

θθ θθ

σ λε ε τ ε

σ λε ε

= − − = −

= − −

(4)

Here λ and G are the Lame constants. The volume strain εvo

in equation (4) is the sum of the two linear strains,

1( )vol rr

u vu

r r θθ ε ε ε

θ

∂ ∂= + = + +

∂ ∂

(5)

The strain components are related to the displacementcomponents by the compatibility equations,

1, ( ),

21

( )2

rr

r

u u v

r r r v v u

r r r

θ

θ

ε ε θ

ε θ

∂ ∂= = +

∂ ∂∂ ∂

= − +∂ ∂

(6)

The system of equations can be simplified considerably by

eliminating the stresses and strains, and finally expressing theequilibrium equations in terms of the displacements. For a

homogeneous material (where λ and G are constants) these

equations are:

fp−=σ

z y x uuu ,,



SPE 96246 3

2

2 2

2 2

2 2 2

1( 2 ) ( 2 ) ( 2 )

1 1 1( 3 ) ( )

u u uG G G

r r r r

v v u PG G G

r r r r r

λ λ λ

λ λ θ θ θ

∂ ∂+ + + − +

∂ ∂

∂ ∂ ∂ ∂− + + + + =

∂ ∂ ∂ ∂ ∂

2

2

2 2

2 2 2 2

1 1

( ) ( 3 )

1 1( 2 ) ( )

u u

G Gr r r

v v v v PG G

r r r r r r

λ λ θ θ

λ θ θ

∂ ∂+ + +

∂ ∂ ∂∂ ∂ ∂ ∂

+ + + + − =∂ ∂ ∂ ∂

(7)

The boundary conditions used are:

w w= P at r = r ;

u = 0 at r = .

r σ

∞

(8)We obtain the displacements and the effective stresses at

every point in the reservoir by solving (4), (6), (7), and (8).

Fluid Flow Equation (Storage Equation). To calculate thestresses around the wellbore due to poro-elastic effects, it is

necessary to calculate the pressure profile around the wellbore.

The mass conservation equation describes the change in pore pressure with time and position (note that the porosity

and permeability are not constant):

2

[ ] [ ]

1 1[ ] [ ]

vol

r

cP k Pt t

P Pk k

r r r r θ

ε φµ µ

θ θ

∂∂+ = ∇ ⋅ ∇

∂ ∂

∂ ∂ ∂ ∂= +

∂ ∂ ∂ ∂

(9)

0

w w w

IC: t = 0, P(r,0) = P ;

BC: r = r , P(r ,t)=P .

(10)

From the above equations (4), (6), (7), (8), (9), and (10), we

can obtain the flow-induced effective stresses and the stressesdue to the wellbore pressure σrr

', σr ө' , σөө

'.

The stress distribution can be obtained by superimposing the

flow induced effective stresses and effective stresses induced by the in-situ stresses. By assuming that the thickness of the

pay zone is much smaller than the drainage radius, we can justify a plane stress approximation. The stress in the zdirection is approximated by:

p z z −=′ σ σ

(11)

Constitutive Relations for Sand

The porosity and permeability at every location in the

reservoir are a function of the effective stress. This

dependence is specified either by a material balance on the

solids (for porosity) or by empirically derived constitutive

relationships for the stress-dependent permeability.

Porosity. Based on the solid mass balance equation and thedefinition of porosity, we derived a relationship between

porosity and volumetric strain as follows (Ref 14):

( ) ( )ε φ φ −⋅−−= exp11 0 (12)

Permeability. The stress-dependent permeability is given by

the empirical relationship:

Before failure.

[ ]{ }[ ]{ }

),,,,(,)/(1ln

)/(1ln

2/

0

*

2/*

0

zr k jik

k m

jk

m

jk

ii

ii θ σ σ

σ σ =

+

+=

(13)

Where σ

and m are two coefficients to be determined fromfitting laboratory data and σ jk ' is the net confining stress.

2

//

/ kk jj

jk

σ σ σ

+=

(14)

After failure. If the rocks fail in shear, the sand grains are

expected to roll past each other resulting in dilation, i.e. a

significant increase in porosity and permeability (Ref 5).

zr iV k

k i

ii

ii ,,,cos10

θ α =+=

(15)

Where, cosαi is the unit vector perpendicular to the failure

plane. Figure 2 shows how the permeability changes with theaverage effective stress.

Modified Coulomb Failure CriterionWe use a modified Coulomb failure criterion (Figure 3) to

check for formation failure.

( ) ( ) 02

12

32

12

1 2]1[]1[ S =++′−−+′ µ µ σ µ µ σ

]4/1[ 2

00001 S T C C −>′σ

(16)And

2

3 0 1 0 0 0 0, [1 / 4 ]T C C T S σ σ ′ ′= − = −

Where,

])1[(2 2

1

2

00 µ µ ++= S C

Computer ImplementationBased on the above model, an iterative coupled program has

been developed to calculate the pore pressure, stress / strain

distribution, porosity and permeability in the formation. The



4 SPE 9624

flow chart for the computer implementation is shown in Figure

4.

Results and DiscussionThe model and equations presented in the previous section

have been solved using a fully-implicit, finite-difference

variable-grid simulation. Results are presented for a typical

base case to demonstrate the main results obtained from thesimulations. The formation of “fractures” or zones of failure

(as postulated in this paper) in unconsolidated sands dependsvery much on the in-situ state of stress. Under stress regimes

that favor normal faulting, the vertical stress is the maximum

principal stress and the minimum and maximum horizontal

stresses are 0.6 to 0.9 of the vertical stress. An example of thisstress regime is the Gulf of Mexico (Ref 17). Strike-slip faults

form when the vertical stress is smaller than the maximum

horizontal stress. Examples of this stress regime include the

North Sea and Western Canada (Ref 18). Both cases are

considered below.

Fracture Growth in a Normal Faulting Stress Regime. InCase 1 and Case 2, we will discuss how the fracture forms and

propagates under a normal faulting stress regime.

Base Case. An injection well is placed in the middle of

homogeneous reservoir on 40-acre spacing. All formation

properties are assumed to be isotropic. The vertical stress isassumed to be 10,000 psi. The minimum and maximum in-situ

stresses are 5,000 psi and 6,000 psi respectively. Water is

injected into the well at a bottom-hole pressure of 9,500 psi.

The initial reservoir pressure is assumed to be 1,500 psi.Details of the input data used are provided in Table 1.

As clearly seen in Figure 5 (map view of the formation), the

injection of water results in the formation of a high permeability zone oriented in the direction of intermediate

stress (maximum horizontal in-situ stress). This high permeability zone is created primarily as a result of shear

failure occurring in the direction perpendicular to the

minimum horizontal stress.Figure 6 shows how the pore pressure increases due to fluid

injection. The increase in pore pressure is anisotropic because

the in-situ effective stresses are anisotropic, giving rise to ananisotropic permeability distribution. This effect becomes

more pronounced when shear failure occurs. At failure, a

significant increase in the permeability along the failure planeoccurs due to dilation (Figure 2). The permeability

perpendicular to the failure plane does not change appreciably.

This causes the pore pressure profile to become increasingly

anisotropic. Pore pressure increases are observed to propagatefaster in the direction of the maximum horizontal stress(perpendicular to the direction of the minimum horizontal

stress). This is also the direction of the plane of shear failure.

The change in vertical, radial and tangential stress

distributions with an increase in pore pressure is shown inFigure 7. In this example (Base Case), the initial effective

vertical stress is 8,500 psi. As the injection is initiated, the

hoop stress increases from negative 2,000 psi at the wellboreto a constant 3,500 psi away from wellbore and the radial

stress decreases from 8,000 psi at the wellbore to a constant

4,500 psi away from the wellbore.

The injection of the fluid increases the pore pressure

resulting in a decrease in all the effective stresses in the near-

wellbore region. The vertical effective stress remains themaximum principle stress (Figure 7). This ensures the

propagation of a failure zone in the direction of the maximum

horizontal in-situ stress.

Figure 8 shows the maximum and minimum principle

stresses (σ3 and σ1). In this figure, the dark-shaded regionindicates the shear failure zone and the lightly-shaded region

indicates the tensile failure zone. The arrows indicate thedirection of increasing radius away from the wellbore starting

at 0.4 ft and going to 7 ft away from the wellbore.

From the changes in the stress distribution shown in the

figure, it is clear that the injection process is dominated byshear failure. The shear failure zone expands from about 1 foo

to several tens of feet from the wellbore over the first few

minutes of injection (Figure 8). The shear failure results in an

increase in permeability in this zone resulting in the creation

of a high permeability zone perpendicular to the minimumhorizontal stress.

In summary, a high permeability shear failure zone formsand propagates perpendicular to the direction of the minimum

horizontal stress, due to fluid injection. The creation of this

high permeability zone is caused by shear failure and a

reduction in the effective stress due to injection. The

anisotropic propagation of this high permeability zone iscaused by differences in the in-situ horizontal stresses.

Case 2 (The effect of in-situ stresses). Clearly the behavior

reported above is very sensitive to the anisotropy in the in-situ

stresses. In Case 2 we present results for a simulation in whichthe difference in the maximum and minimum horizontal

stresses is increased. The maximum in-situ stress is increased

from 6,000 psi to 8,000 psi. The minimum in-situ stress is keptthe same.

As in the base case, a high-permeability zone forms and propagates in the direction of the maximum horizontal stress

However, the high permeability zone in this case is narrower

and shorter than that in the base case (Figure 9).Figure 10 shows how the pore pressure propagates along the

high-permeability zone resulting in the formation of an elliptic

region of high pore pressure. When comparing to the basecase, this high-pressure region is narrower and shorter.

The changes in the stresses are shown in Figure 11. The

initial vertical stress is a constant 8,500 psi; the hoop stressincreases from negative 4,000 psi at the wellbore to a constan

3,000 psi away from wellbore region; and the radial stress

decreases from 8,000 psi at the wellbore to a constant 6,000

psi away from wellbore region.As time increases, all the effective stresses decrease in the

near wellbore region. This causes the failure zone to expand

from 0.8 ft to 20 ft over a period of 3 minutes (Figure 12).

Under normal faulting in-situ stress regime conditions in

unconsolidated sands, shear failure is the dominant or maybethe only failure mechanism when a fluid is injected. Tensile

failure may happen when the injection pressure is close to or

higher than the vertical stress (rare). The length and width ofthe high permeability zone will be decreased if the maximum

horizontal stress is increased. An increase in the maximum

horizontal stress has no effect on the failure (intermediate



SPE 96246 5

stress) but will cause the effective stresses to increase and the

permeability to decrease.

Fracture Growth in a Strike-Slip Stress Regime. In Case 3

and Case 4, we discuss how a zone of high permeability forms

and propagates under a strike-slip stress regime.Case 3. The vertical stress is decreased from 10,000 psi to

5,500 psi from the Base Case. The maximum and minimumhorizontal stresses are kept the same.

As seen in Figure 13, at early time (less than 1 min), a high permeability region caused by tensile failure forms and

propagates in the maximum horizontal stress direction. Then

(after several minutes) the high permeability region propagates

at a certain angle to the maximum horizontal stress due to theshear failure.

Figure 14 shows how the pore pressure increases due to

fluid injection. The high pore pressure zone propagates in

direction of the maximum horizontal stress at the beginning,

and then propagates in a certain angle to the maximumhorizontal stress.

The changes in the stresses are shown in Figure 15. Theinitial effective vertical stress is 4,000 psi. As the injection isinitiated, the hoop stress increases from negative 2,000 psi at

the wellbore to a constant 3,500 psi away from wellbore and

the radial stress decreases from 8,000 psi at the wellbore to aconstant 4,500 psi away from the wellbore. The injection of

the fluid increases the pore pressure resulting in a decrease in

all the effective stresses in the near-wellbore region. The

radial stress remains the maximum principal stress and thehoop stress remains the minimum principal stress (Figure 15).

This ensures the propagation of a failure zone at a certain

angle to the maximum horizontal in-situ stress (if only shear

failure happens).

Figure 16 shows how the maximum and minimum principlestresses (σ3 and σ1) change with time. In this figure, we can see

that in the near wellbore region, only tensile failure occurs

(lightly shaded region), which explains the propagation in thedirection of the maximum horizontal stress. Away from the

wellbore, shear failure occurs and the failure zone extends

from 0.6ft to 5.3ft within the first minute.Case 4. In Case 4 we present results for a simulation in

which the difference in the maximum and minimum horizontal

stresses is increased from Case 3. The maximum in-situ stress

is increased from 6,000 psi to 7,000 psi. The minimum in-situ

stress is kept the same.As in Case 3, a high-permeability zone forms and

propagates at a certain angle to the maximum horizontal stress

direction. However, the high permeability zone in this case islonger and narrower than that in the Case 3 (Figure 17). The

shear failure zone increases. Furthermore, same as in Case 3,

the high permeability zone propagates in direction of the

maximum horizontal stress at the very beginning (less than 1

min) and then propagates at a certain angle to the maximumhorizontal stress.

From Figure 18, we can see that the pore pressure

propagates along the high permeability zone and the high-

pressure zone is longer and narrower than Case 3.Figure 19 shows the stresses change with time and location.

The initial effective vertical stress is a constant 4,000 psi,

while the initial effective hoop stress and radial stress change

with the location. The initial effective hoop stress increases

from negative 3,000 psi at the wellbore to a constant 3,500 psaway from wellbore and the initial effective radial stress

decreases from 8,000 psi at the wellbore to a constant 5,500

psi away from the wellbore. The injection of the fluid

increases the pore pressure resulting in a decrease in all the

effective stresses in the near wellbore region. As in Case 3, theradial stress remains the maximum principal stress and the

hoop stress remains the minimum principal stress.From Figure 17, we can see that tensile failure (lightly

shaded region) is dominant in the near wellbore region. While

away from the wellbore region, shear failure occurs. And the

failure zone extends initially from 0.6ft to 2 ft and 5.5ft at 8and 24 seconds respectively.

Under a strike-slip stress state in unconsolidated sands

shear failure is the dominant failure mechanism when the

fluids are injected. Tensile failure happens only in the near

wellbore region where the pore pressure is very large. Thewidth of the high permeability zone decreases while the length

increases with the increase of the maximum horizontal stressThe Increase in the maximum horizontal stress (principa

stress) will increase the shear failure zone, so the permeability

along the shear failure zone will be larger than the other

directions.

Conclusion

Classical models for linear-elastic, brittle fracture mechanicsthat have been traditionally applied to hard rock fracturing are

not applicable for unconsolidated sands.

A new model is presented to describe “fracture” propagation

in unconsolidated sands. It is shown that shear failure is the

dominant failure mechanism when fluids are injected intounconsolidated sands. This is consistent with experimenta

observations reported in the past. Tensile failure happens only

at the near wellbore region under strike-slip stress conditions.

Under normal faulting conditions, as the pore pressureincreases a region of high permeability forms in a direction

perpendicular to the minimum horizontal stress. The width and

length of this region is controlled by the minimum andmaximum horizontal stresses. The high permeability zone is

narrower and shorter if the difference of the in-situ stresses

(increase the maximum horizontal stress) is larger.

Under strike-slip faulting stress condition, as the pore

pressure increases, a high permeability zones forms and propagates in the maximum horizontal stress direction firs

because of tensile failure in the near wellbore region and then

propagates at a certain angle to the maximum horizontal stresdirection because of shear failure. The high permeability zone

is narrower but longer if the difference of the in-situ stresses is

larger.

Acknowledgements

The authors would like to acknowledge valuable comments provided by Dr. Jon Olson and Dr. Mark Mear.

Nomenclature

P = pore pressure



6 SPE 9624

σ = stress

σ1, σ3 = principle stress

σ' = effective stress

λ = Lame’s constants

G = shear modulusS0 = cohesion

T0 = tensile strength

u = displacement at r directionv = displacement at ө direction

k = permeability

ф = porosity

ε = strain

pw = injection pressureµ = viscosity

c = fluid compressibility

E = Young’s modulus

v = Poisson’s ratioSubscripts

r, ө = coordinates in r, ө direction

H,h = horizontal direction

References

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Mechanics, John Wiley & Sons Ltd., West Sussex,

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SPE Journal.

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paper SPE 90507 presented in Houston, Texas,U.S.A,

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paper SPE/IADC 79084 presented at the 2003 SPE/IADC

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International Journal of Rock Mechanics and Mining

Sciences.V.37, 317-336.



SPE 96246 7

Table 1 Base Case Parameters

Internal Friction 0.6

Internal Shear Strength, S0 (psi) 100

Tensile strength, T0 (psi) 100

Max horizontal Insitu Stress, σHmax (psi) (East-West) 6000

Min horizontal insitu stress, σHmin (psi)(North-South) 5000

Vertical stress, σz (psi) 10000

Young’s modulus, E (psi) 10^5

Well radius, r w (ft) 0.4

Viscosity, µ (cp) 1

Compressibility, Cp (1/psi) 3*10^(-4)

Permeability in the r direction, Kr0 (darcy) 0.5

Permeability in the ө direction, K ө0 (darcy) 0.5Porosity 0.25

Initial pore pressure, P0 (psi) 1500

Wellbore pressure, Pw(psi) 9500

Poison’s ratio 0.25

Figure 1. Stress distribution around well bore



8 SPE 9624

Figure 2. Postulated model for change inpermeability with stress

Figure 3. Modified coulomb failure criteria

Figure 4. Flow Chart for solution of geomechanicaland flow equations

Figure 5. Base case: Permeability Change with Time



SPE 96246 9

Figure 6. Base case: Pore Pressure Change withTime

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

-0.5 0.5 1.5 2.5 3.5 4.5 5.5 6.5

r (ft)

s t r e

s s e

s

P

s i

t = 0s, σrr t = 0s, σθθt = 0s, σzz t = 6s, σrr t = 6s, σθθ t = 6s, σzzt = 10s, σrr t = 10s, σθ

t = 10s, σzz

Figure 7. Base case: Stress distribution changewith time

-8000

-6000

-4000

-2000

0

2000

4000

-1000 0 1000 2000 3000 4000 5000 6000 7000 8000

Maximum principal stress σ1 (Psi)

M

i n i m

u m

p r i n c i p a l s t r e s s σ

3 (

P s

i )

t=0s

t=10s

t=178s

shear failue

t=20s

t=6s

Shear Failure zone

t=0s,r=0.8Failure zone:

6s, r = 6ft

10s, r = 8ft

20s, r = 9.9 ft

178s, r = 21 ft

r increase direction from rw to 7 ft

Figure 8. Base case: Stress state change withtime



10 SPE 9624

Figure 9. Case 2: Permeability change withtime

Fig 10. Case 2: Pore pressure change with time



SPE 96246 11

-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

-0.5 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5

r (ft)

s t r e s s e s

( P s i )

t = 0s, σrr t = 0s, σθθt = 0s, σzz t = 6s, σrr

t = 6s, σθθ t = 6s, σzzt = 10s, σrr t = 10s, σθθ

t = 10s,σ

zz

Fig 11. Case 2: stress distribution changewith time

-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Maximum principal stress σ1 (Psi)

t=0s

t=10s

t=178s

shear failue line

t=20s

t=6s

Failure zone:

0s, r = 0.8ft

6s, r = 5.5 ft

10s, r = 7ft

20s, r = 8.9 ft

178s, r = 20 ft

Fig 12. Case 2: Stress state change with timeFig 13. Case 3: Permeability change with time



12 SPE 9624

Fig 14. Case 3: Pressure change with time

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

0 5 10 15 20 25

r (ft)

S

t r e s s e s

s

i

t = 0s, σzz

t = 0s, σrr

t = 0s, σөө

t = 8s, σzz

t = 8s, σrr

t = 8s, σөө

Fig 15. Case 3: Stress distribution change with time

-9000

-7000

-5000

-3000

-1000

1000

3000

-2000 0 2000 4000 6000 8000

Maximum principal stressσ1 (psi)

m

i n

i m

u

m

r i n

c

i

a

l s

t r e

s

s σ

3

s

i

t = 0s

Shear failure line

t = 8s

t = 24s

t = 744s

Shear failure zone t e

n

s

i l e

f a

i l u

r e

z

o

n

e

r increase direction from 0.7 ft to 23 ft

Fig 16. Case 3: Stress state change with time



SPE 96246 13

Fig 17. Case 3: Permeability change with timeFig 18. Case 3: Pressure change with time



14 SPE 9624

-8500

-6500

-4500

-2500

-500

1500

3500

5500

7500

0 5 10 15 20 25

r (ft)

S t r e s s e s

( p s i )

t = 0s, σzz

t = 0s, σrr

t = 0s,σөө

t = 8s, σzz

t = 8s, σrr

t = 8s, σөө

Fig19. Case 3: Pressure change with time

-9000

-7000

-5000

-3000

-1000

1000

3000

-2000 0 2000 4000 6000 8000

Maximum principal stress σ1 (psi)

m i n i m u m p

r i n c i p a l s t r e s s σ 3 ( p s i )

t = 0s

Shear failure line

t = 8s

t = 24s

t = 744s

Shear failure zone t e n s i l e f

a i l u r e

z o n e

r increase direction from 0.7 ft to 23 ft

Fig 20. Case 3: Pressure change with time

a new approach to modeling hydraulic fractures in consolidated sands

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