1. Two Dimensional Hydraulic Fracture Simulations Using FRANC2D
Thank you for the introduction and thank you all for coming to hear what I have to say.
I’m going to describe work that I did to simulate hydraulic fractures.
Here is a hydraulic fracture that has been excavated.
It was created by injecting a sand slurry into a pipe that was right about here. The injection pressure increased and the soil around the pipe fractured. The sand slurry filled the fracture and after injection the sand remained as a layer in the subsurface. You can see that these layers are roughly flat lying, but they curve upward as you go along the trench away from the borehole.
Understanding how these fractures form is an important part of my research.
Qingfeng Tan

2. Vapor extraction well intersecting horizontal hydraulic fracture, from Bradner (2002)k
frx / 10
100
1000
10000 1
Flow Index
One example of how hydraulic fractures are used is to increase performance of vapor extraction wells
Here are some theoretical analyses and field data showing the performance of a vapor extraction well intersecting a hydraulic fracture shaped like a horizontal, circular disk (Picture)
The lines on the graph show how much a hydraulic fracture will increase the discharge of a vapor extraction well—the different lines are for fractures of different volumes. The symbols are field results.
It is clear from these results that flat-lying hydraulic fractures can increase the discharge of vapor extraction wells by 10 times or more.
This is a method for improving the remediation of low permeability formations, but it is based on fractures that are shaped like a horizontal disk.

3. Importance of 2-D This is important
We know that not all fractures are shaped like flat-lying disks. Here a fracture and a well are shown in cross-section, and I am assuming the fracture has axial symmetry so only one half is shown.
So this is a fracture shaped like a flat disk. Some fractures look like this, but others dip slightly—the fractures shown on the previous slide would look something like this.
Other fractures dip even more steeply
And some fractures curve downward slightly.
In some cases fractures curve sharply upward and reach the ground surface.
Clearly fractures that curve upward and are essentially vertical will have a much different effect on vapor extraction, and on any other application.
The extent to which a fracture curves upward or downward plays a key role in how the fracture performs during remediation. Predicting this process requires analyzing fracture propagation in two dimensions.

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

5. Overview Previous work Theoretical Analysis Code Development
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

6. Hydraulic Fracture Design
Vertical Fractures Y Z Q h X a a Q h X Y Z
(a)
(b)
Horizontal Fractures a Q d Z r Q a d Z r
One dimensional fractures, growth in a single plane
Vertical fractures growing horizontally—this happens when fracture is between confining layers—oil reservoir
Horizontal fractures—deep fractures that compress overburden, shallow fractures that lift overburden
Analyses of idealized forms predict length, aperture, pressure as functions of time
Good results for fractures that have the assumed geometry—planar, horizontal growth
Limitations: Unable to predict propagation that is out of plane, with vertical component
(c)
(d)

7. Previous Models
Pressure
time
Length
time
Aperture
time

8. 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

9. 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

10. FRANC2D 2-D stress and displacement
Developed for structural fracture mechanics applications
Auto regrid around
fracture
Fluid flow within
fracture not included

11. 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
This is a complicated process to implement. Describing all the details would take a long time and would cause everyone to go to sleep. Instead, what I want to do is highlight some of the principles behind the methods that are implemented in the code.

12. Flow and Deformation Coupling
From 1-D implicit solution; flow bc at well, head bc at tip
From FEM elasticity solution
Pressure x
Start with Pressure and aperture at one frx length
Increase fracture length
Assume pressure=same as previous step
Calculate aperture. Ap increases because pressure is the same and the frx is longer
Calculate pressure with new aperture. Pressure less because aperture greater (effect of continuity)
Calculate aperture with new pressure. Aperture smaller because pressure smaller
Calc P. P larger because ap smaller
Calc ap. Ap larger because P larger.
Results for ap and P converge. Stop iteration when change between successive iterations is less than tolerance
Aperture x

13. Propagation KI =Stress intensity factor KI=KIc for propagation
KIC is material property, called fracture toughness.
Assumed to be valid in soils in cohesive soils

14. How to ensure KI=KIc?
Pressure
Ptip x KI
KIc
Ptip

15. 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

16. Automatic Propagation Subroutine
Fluid flow and mechanical analysis coupling to decide pressure and geometry
Propagation criterion: KI=KIC
Auto-remesh around fracture tip

17. 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

18. 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

19. 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

20. Verification Uniform Pressure: Model Setting
Infinite elastic media
Uniform pressure
Radial symmetric a

21. Verification-Driving Pressure

22. Verification (II): Fracture Length

23. Verification (III): Fracture Aperture

24. Error Analysis

25. Applications Hydraulic fracture in shallow soil:
Gravity
Fluid injection
Soil with under-lying softer material
Soil with high lateral residual stress

26. Forms of Hydraulic Fractures in the Field

27. N 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
Cross 3
0.9
0.7
0.5
0.3
Cross 2
0.1 N
Cross 1 5 10 15
feet

28. General case-Model Setting
Depth
0 m
-1.6 m
-2 m
frx
-5 m
0 m
12 m
Distance from well

29. Vertical Stress During Propagation

30. Pressure Log

31. Fracture Form

32. Average radial extent of sand
Aperture and Uplift
(m)
Average radial extent of sand

33. Effects of Layering
observed
Richardson
Simulated

34. Effects of Lateral compression

35. 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

36. 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.