Qingfeng Tan

Two Dimensional Hydraulic Fracture Simulations Using FRANC2D

Vapor extraction well intersecting horizontal hydraulic fracture, from Bradner (2002)

kfrx/ k10 100 1000 10000

1

10

Flow Index

Importance of 2-D

Objective

Develop and apply a model for predicting the forms of curving hydraulic fractures in two dimensions

Overview• Previous work

– Vertical and horizontal fracture– Analytical models

• Theoretical Analysis– Coupling mechanical and fluid flow analysis

• Code Development– Automatic propagation (EXC_AUTO_DRIVER_FLOW)– Fracture form calculation routines– Fluid flow simulation routines

• Application– Shallow soil model– Effects of layering and lateral residual compression

Hydraulic Fracture DesignVertical Fractures

a

Qh

X

Y

Z Horizontal Fractures

(a)

(d)(c)

(b)

a

Qd

Z

r

Q

a

d

Z

r

Y

Z

Q

h

X

a

Previous ModelsPressure

Length

Aperture

time

time

time

1

1

CtfP

2

2

Ctfa

3

3

Ctf

)],,,([ 3,2,1 QKEff

)2.05.0( 1 C

)44.025.0( 2 C

)5.011.0( 3 C

Simulate Hydraulic Fracture

• Fracture aperture—analyze as elastic displacements due to fluid pressure

• Fluid pressure—analyze as flow in deforming fracture

• Propagation—require stress intensity to equal critical value

Problem with Analysis in 2-D

• Fracture curves-- numerical methods for stress analysis required

• Fracture propagation-- analyze as a series of quasi static models. Requires many analyses to be conducted.

Need FEM method with automatic regridding around fracture

FRANC2D• 2-D stress and displacement• Developed for structural

fracture mechanics applications

• Auto regrid around fracture• Fluid flow within

fracture not included

Fracture with Fluid Flow-Coupled Approach

• Modify FRANC2D to perform mechanical analysis, then calculate geometry of fracture, caused by fluid pressure, and other loadings

• Fluid flow analysis adjust fluid pressure due to the shape changes of fracture, coupled with mechanical analysis

• Propagation criterion: is decided by fracture geometry and fluid pressure

ICI KK IK

Flow and Deformation CouplingP

ress

ure

Ape

rtur

e

From 1-D implicit solution; flow bc at well, head bc at tip

From FEM elasticity solutionx

x

Propagation

• KI =Stress intensity factor

• KI=KIc for propagation

• KIC is material property, called fracture toughness.

How to ensure KI=KIc?P

ress

ure

Ptip

KI

Ptip

KIc

x

Code Development• Fracture propagation control routine

-EXC_AUTO_DRIVER_FLOW

• Fracture geometry calculation routines-EXC_LENGTH_FLOW-EXC_APER_FLOW-EXC_VOLU_FLOW

• Fluid flow simulation routines-FLUID_FLOW_INIT

-FLUID_FLOW_CALC

Automatic Propagation Subroutine

ICI KK

• Fluid flow and mechanical analysis coupling to decide pressure and geometry

•Propagation criterion: KI=KIC

•Auto-remesh around fracture tip

Fracture Form Calculation• Length – EXC_LENGTH_FLOW• Aperture – EXC_APER_FLOW• Volume – EXC_VOLU_FLOW• Obtain Crack node info• Calculation in each segment, then integral

Fluid Flow and Aperture Subroutine

• Calculate new heads using initial aperture• Calculate aperture using new head• Calculate heads using new aperture• Repeat and compare heads and apertures between

successive iterations• Converge when change is less than tolerance,

usually less than 7 iterations

Propagation Subroutine•Calculate KI for pressure at tip

•Adjust pressure at tip slightly, redo fluid pressure calculations, and calculate new KI

•Use two values of KI and pressure tip to interpolate new value of pressure tip that should give KI=KIc

•Check KI and revise pressure tip as needed until KI is within tolerance of KIc

VerificationUniform Pressure: Model Setting

P•Infinite elastic media

•Uniform pressure

•Radial symmetric

a

z

r

Verification-Driving Pressure

5

10

0 5 10 15Time(min)

Pre

ss

ure

(KP

a)

Verification (II): Fracture Length

1

3

5

0 5 10 15Time(min)

Le

ng

th(m

)

Verification (III): Fracture Aperture

0.5

1.0

1.5

0 5 10 15Time(min)

Ap

ert

ure

(mm

)

Error Analysis

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

1 2 3 4 5

Length (m)

Re

lati

ve

Err

or

Error PError aError d

Applications

• Hydraulic fracture in shallow soil:- Gravity

- Fluid injection

• Soil with under-lying softer material

• Soil with high lateral residual stress

Forms of Hydraulic Fractures in the Field

Field Data Adoption

• Four cross-section selection

• Each cross-section starts from center of fracture to the edge of it, perpendicular with each other

• Fracture path, uplift, and sand extent data are adopted

0 5 10 15 feet

0.1

0.3

0.5

0.7

N

0.9

Cross 1

Cross 4Cross 3

Cross 2

General case-Model Setting

Depth

0 m

-2 m

12 m

-5 m

Distance from well0 m

frx-1.6 m

Vertical Stress During Propagation

Pressure Log

0

10

20

30

40

50

60

0 2 4 6 8

Time (minutes)

Pre

ss

ure

(p

si)

Measured

Simulated

Fracture Form

-1.8

-1.5

-1.2

-0.9

0 1 2

Distance from center of fracture (m)

De

pth

be

low

gro

un

d s

urf

ac

e (

m)

simulatedWell H Cross-s 1Well H Cross-s 2Well H Cross-s 3Well H Cross-s 4

Aperture and Uplift

0.00

0.02

0 1 2 3 4

distance from center (m)

Up

lift

fro

m f

ield

, or

sim

ula

ted

ap

ertu

re

simulated

Well H cross 1

Well H cross 2

Well H cross 3

Well H cross 4

Average radial extent of sand

(m)

Effects of Layeringob

serv

ed

-2

-1.5

-1

-0.5

0

0 1 2 3 4 5

E2=2000psi, E1 = 5000psi

E1=E2=5000psi

E2=3000psi, E1=5000psi

E2=4000psi, E1=5000psi

Sim

ulat

ed

Richardson

Effects of Lateral compression

-1.8

-1.2

-0.6

0

0 1 2 3 4 5 6 7

Distance from Wellbore (m)

Dep

th (

m)

Fracture Path from lowresidual area

Fracture path from highresidual compressionregion

v

hv

Conclusions

•FRANC2D has been modified to simulate hydro-mechanical coupling conditions during hydraulic fracturing.

•A new simulation tool, HFRANC2D?, is available

• The model has been verified using analytical solutions, error within a few percent

Conclusions, applications

• Gentle bowl-like forms of hydraulic fractures in shallow soils can be predicted.

• Effects of state of stress and material properties can be predicted and results resemble field observations.