1. 4/16/2017
Hydraulic Fracturing Short Course, Texas A&M University College Station Fracture Design Fracture Dimensions Fracture Modeling Peter P. Valkó
Fracture Design 2004

2. Source: Economides and Nolte: Reservoir Stimulation 3rd Ed.

3. Frac Design Goals

4. Well or Reservoir Stimulation?
Near wellbore region and/or bulk reservoir?
Acceleration versus increasing reserve?
Low permeability
Medium permeability
High permeability
Coupling of goals
Frac&pack

5. Hydraulic Fracturing Design and Evaluation
Why do we create a propped fracture?
How do we achieve our goals?
Data gathering
Design
Execution
Evaluation

6. Fractured Well Performance
Relation of morphology to performance
Streamline view
Flow regimes, Productivity Index, Pseudo-steady state Productivity Index, skin and equivalent wellbore radius

7. Well- Fracture Orientation
MATCH
Vertical well - Vertical fracture
Horizontal well – longitudinal fracture
MISMATCH (Choke effect)
Horizontal well with a transverse vertical fracture
Vertical well intersecting a horizontal fracture

8. Principle of least resistance
Least Principal Stress
Least Principal Stress
Horizontal fracture
Vertical fracture

9. Mismatch (Choked fracture)
Typical mismatch situations:
Horizontal well with a transverse vertical fracture
Vertical well intersecting a horizontal fracture

10. Vertical Fracture - Vertical well
Bypass damage
Original skin disappears
Change streamlines
Radial flow disappears
Wellbore radius is not a factor any more
Increased PI can be utilized
Dp or q

11. Longitudinal Vertical Fracture - Horizontal well
sH,max xf
sH,min
Can it be done?

12. Transverse Vertical Fractures - Horizontal Well
sH,max
Hydraulic Fracture D xf
sH,min
Radial converging flow in frac

13. Fracture Morphology
source: Economides at al.: Petroleum Well Construction

14. Main questions Which wellbore-fracture orientation is favorable?
Which can be done?
How large should the treatment be?
What part of the proppant will reach the pay?
Width and length (optimum dimensions)?
How can it be realized?

15. Prod Eng 101
Transient vs Pseudo-steady state
Productivity Index
Skin

16. Pseudo-steady state Productivity Index
4/16/2017
Pseudo-steady state Productivity Index
Production rate is proportional to drawdown, defined as average pressure in the reservoir minus wellbore flowing pressure
Drawdown
Circular:
Dimensionless Productivity Index
Fracture Design 2004

17. Hawkins formula
Damage penetration distance

18. Exercise 1 Calculate the skin factor due to radial damage if
0.5 ft
Damage penetration
Permeability impairment
0.328 ft
Wellbore radius
Solution of Exercise 1
Note that any "consistent" system of units is OK.

19. Exercise 2
Assume pseudo-steady state and drainage radius re = 2980 ft in Exercise 1. What portion of the pressure drawdown is lost in the skin zone? What is the damage ratio? What is the flow efficiency?
Solution 2
The fraction of pressure drawdown in the skin zone is given by (Since we deal only with ratios, we do not have to convert units.):
Therefore 31 % of the pressure drawdown is not utilized because of the near wellbore damage.
The damage ratio is DR = 31 %
The flow efficiency is FE = 69 %.

20. Exercise 3
Assume that the well of Exercise 2 has been matrix acidized and the original permeability has been restored in the skin zone.
What will be the folds of increase in the Productivity Index?
(What will be the folds of increase in production rate assuming the pressure drawdown is the same before and after the treatment?)
Solution 3
We can assume that the skin after the acidizing treatment becomes zero. Then the folds of increase is:
The Productivity Index increase is 44 % , therefore the production increase is 44 % .

21. Exercise 4
Assume that the well of Exercise 2 has been fracture treated and a negative pseudo skin factor has been created: sf = -5. What will be the folds of increase in the Productivity Index with respect to the damaged well?
Solution 4
The ratio of Productivity Indices after and before the treatment is
The Productivity Index will increase 260 % .

22. Fully penetrating vertical fracture: Relating Performance to Dimensionswp
2xf h
2Vfp

23. Dimensionless fracture conductivity
2 xf w
fracture conductivity
no name
Dimensionless fracture conductivity

24. Accounting for PI: sf and f and r’w
sf is pseudo skin factor used after the treatment to describe the productivity
JD is a function of what?
half-length,
dimensionless fracture conductivity
Drainage radius, re
sf is a function of what?
half-length,
dimensionless fracture conductivity
wellbore radius, rw

25. Pseudo-skin, equivalent radius, f-factor
Prats
Cinco-Ley

26. Notation But JD is the best rw wellbore radius, m (or ft) r'w
Prats’ equivalent wellbore radius due to fracture, m (or ft)
Cinco-Ley-Samanieggo factor, dimensionless sf
the pseudo skin factor due to fracture, dimensionless
Prats' dimensionless (equivalent) wellbore radius
But JD is the best

27. Example
Assume rw = 0.3 ft and A= 40 acre 7 -4 36

28. Dimensionless Productivity Index, sf and f and r’wor
Prats
Cinco-Ley

29. Penetration Ratio Dimensionless Fracture Conductivity Proppant Number
4/16/2017
Penetration Ratio Dimensionless Fracture Conductivity Proppant Number
2 xf
ye = xe xe
Fracture Design 2004

30. The following models, graphs and correlations are valid for low to moderate Proppant Number, Nprop
OK, so what IS the Proppant Number?
The weighted ratio of propped fracture volume to reservoir volume. The weight is 2kf/k .
A more rigorous definition will be given later.
The following models are valid for Nprop <=0.1 ! (The case when the boundaries do not distort the streamline structure (with respect to lower proppant numbers.)

31. Prats' Dimensionless Wellbore Radius
0.01
0.1
1.0 10
100

32. Cinco-Ley and Samaniego graph f (CfD)= sf + ln(xf/rw)1 2 3 4
0.1 10
100
1000
CfD f
use f = ln(2) for CfD > 1000

33. Infinite or finite conductivity fracture
Note that after CfD > 100 (or 30), nothing happens with f.
Infinite conductivity fracture.
Definition: finite conductivity fracture is a not infinite conductivity fracture (CfD < 100 or 30)
(Other concept: uniform flux fracture, we will learn later.)

34. Various ways to look at it
Proppant Number -
Various ways to look at it
Nprop= const means
fixed proppant volume

35. Fig 1: JD vs CfD (moderate Nprop)

36. Fig 2: JD vs CfD (large Nprop)

37. OPTIMIZATION

38. Optimal length and width
2Vfp = 2h wp xf
Struggle for propped volume: w and xf

39. The Key Parameter is the Proppant Number
Medium perm
High perm
Frac&Pack

40. The Key Parameter is the Proppant Number
Low perm
Massive HF
Medium perm

41. Let us read the optimum from the JD Figures
Let us read the optimum from the JD Figures! dimensionless fracture conductivity (for smaller Nprop) penetration ratio (for larger Nprop)

42. Optimum for low and moderate Proppant Number
CfDopt=1.6

43. Optimum for large Proppant Number

44. Tight Gas and Frac&Pack: the extremes
Tight gas k << 1 md (hard rock)
High permeability k >> 1 md (soft formation)

45. FracPi

46. Exercise No 1
Determine the "folds of increase" if 40,000 lbm proppant (pack porosity 0.35, specific gravity 2.6, permeability 60,000 md) is to be placed into a 65 ft thick formation of 0.5 md permeability. Assume all proppant goes to pay.
The drainage radius is re = 2100 ft, the well radius is
rw = ft, the skin factor before fracturing is spre = 5.
Determine the optimal fracture length and propped width.

47. 1: Proppant Number 2: Max Folds of Increase
40,000 lbm proppant, specific gravity 2.6, pack porosity 0.35
packed volume is 40,000/62.4/2.6/(1-0.35) = 380 ft3
Folds of Increase
FracPi
0.467
0.0768
FOI: 6.8 with respect to skin 5
FOI: 3.8 with respect to skin=0

48. Optimum frac dimensions
The volume of two propped wing is
2V1wp = 380 ft3
If the proppant number is not too large: the optimal fracture half-length is
The propped width is

49. Computer Exercise: High Perm
Determine the optimal fracture length and propped width if 40,000 lbm proppant (pack porosity 0.35, specific gravity 2.6, permeability 60,000 md) is to be placed into a 65 ft thick formation of 50 md permeability.
The drainage radius is re = 2100 ft, the well radius is
rw = ft, the skin factor before fracturing is spre = 5.
(Assume all proppant goes to pay.)

50. Computer Exercise: Tight gas
Determine the optimal fracture length and propped width if 40,000 lbm proppant (pack porosity 0.35, specific gravity 2.6, permeability 60,000 md) is to be placed into a 65 ft thick formation of 0.01 md permeability.
The drainage radius is re = 2100 ft, the well radius is
rw = ft, the skin factor before fracturing is spre = 5.
(Assume all proppant goes to pay.)

51. Economic optimization
Production forecast
Transient regime
Stabilized
Economics: Converting additional production into value
Time value of money
Discounted revenue
NPV

52. Costs and Benefits
The more proppant (larger proppant number) the higher Productivity Index, if the given proppant volume is placed according to the optimal dimensionless fracture conductivity
The more proppant, the larger costs
How large should be the treatment?
NPV optimization

53. Treatment Sizing

54. Pre-Treatment Data Gathering

55. Design Input Data Petroleum Engineering Data Rock Properties
Hydrocarbon in Place, Drainage area, Thickness, Permeability
Rock Properties
Young’s modulus, Poisson ratio,
Fracture toughness, poroelastic const
Stress State
Leakoff
Proppant and Other Fluid properties
Operational constraints

56. Rock Properties Linear Elasticity Poroelasticity Fracture Mechanics
4/16/2017
Rock Properties
Linear Elasticity
Poroelasticity
Fracture Mechanics
Fracture Design 2004

57. Young's modulus and Poisson ratio Uniaxial test
DD/2 l
D l F A s xx F A =
Linear stress-strain relations

58. Other elasticity constants

59. Formation Classification
Two types
Consolidated and tight E = psi
Unconsolidated and soft E = psi

60. Poroelasticity and Biot’s constant
Total Stress = Effective Stress + a[Pore Pressure]

61. Total Stress = Effective Stress + a[Pore Pressure]
Who Carries the Load?
Total Stress = Effective Stress + a[Pore Pressure]
Grains
Force
Pore Fluid
Biot’s constant
a ~ 0.7

62. Stress State in Formations Far Field and Induced Stresses, Fracture Initiation and Orientation
Stress versus Depth
Minimum Horizontal Stress
Magnitude and Direction

63. Total (absolute) horizontal stress
The simplest model:
1) Poisson ratio changes from layer to layer
2) Pore pressure changes in time

64. Crossover of Minimum Stress
80x106
20x106
40x106
60x106
Stress, Pa
Depth from original ground surface, m
Original Vertical Stress
True Vertical Stress
Minimum Horizontal Stress
Critical Depth 977 m
-3000
-2500
-2000
-1500
-1000
-500
Current Depth , m
Ground Surface

65. Stress Gradients Overburden gradient gradient Frac gradient
Slope of the Vertical Stress line  1.1 psi/ft
Frac gradient
Basically the slope of the minimum horizontal stress line psi/ft
Extreme value: 1.1 psi/ft or more

66. Fracture width

67. Linear Elasticity + Fractures
The force opening the fracture comes from net pressure
Net pressure = fluid pressure - minimum principal stress
pn = p min
The net pressure distribution determines the width profile
Plane strain modulus and characteristic half length

68. Ideal Crack Shapes (Plane strain)
Half length c
pn(x)
Deformation (distribution)
net pressure (distribution)
Plane - strain modulus: w
Plane strain: Infinite repetition of the same picture (2D)

69. Shape of a pressurized crack, pn=cons
Width
pn : net pressure
c : half length
“characteristic dimension”
Max Width w c
linearity preserved

70. Height and Width in Layered Formation
Questions:
Far-field Stress
Upper tip
Contained?
Breakthrough?
Run-away?
Up or Down?
Width?
Hydrostatic
pressure?
Height
control?
What can be
measured?
Pinch point
Lower tip

71. From Fracture Mechanics to Fracture Height

72. Stress Intensity Factor
weighted pressure at tip
Pa · m1/2
psi - in.1/2
Weighting function: the nearer to tip, the more important the pressure value
stress distribution
at tip x c
KI : proportionality const

73. Stability of Crack, Propagation
Critical value of stress intensity factor:
Fracture Toughness KIC
Propagation: when stress intensity factor is larger than fracture toughness

74. Application: Fracture Height Prediction
Height containment: why is it critical?
Fracturing to water or gas
Wasting proppant and fluid
Can it be controlled?
Passive: safety limit on injection pressure
Active: proppant (light and heavy)

75. Calculation Based on Equilibrium Fracture Height Theory
fluid pressure
far field stress
profile

76. Stress Intensity Factor at the Tips (calc) = Fracture Toughness of the Layer (given)
Two equations, two unknowns

77. Penetration Into Upper and Lower Layers
Klc,2 1 s2
Dhu yu s1 hp yd
Dhd -1 s3
Klc,3

78. Notation

79. Input to a Height Map Calculation

80. Calculated Height Map (after HFM) Tip Location [ft] Tip Location [m]
-1200
-1000
-800
-600
-400
-200
200
400
600
800
1000
3000
3100
3200
3300
3400
3500
3600
3700
3800
300
-300 21 26
psi
MPa
100
-100
Tip Location [m]
Tip Location [ft]
Treating Pressure

81. How to Use a Height Map? 1 Off-line:
Assume a height, make a 2D design,
Calculate net pressure (averaged in time)
Read-off a better estimate of height
2 In-line:
P3D design (3D),
Calculate net pressure at a location
Adjust height to equilibrium

82. Fluid loss: the property of both the rock and the fluid
1 Leak-off
2 Spurt loss

83. AL Fluid Loss in Lab 2CL Sp y = 0.0024 + 0.000069x 10 20 30 40 50 6010 20 30 40 50 60
Square root time, t1/2 (s1/2)
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Lost volume per unit surface, m
2CL Sp AL

84. Fluid Loss in the Formation: Ct
Flow through filtercake covered wall
filtercake build-up and filtercake integrity
Flow through polymer invaded zone
“viscosity” of polymer in formation
Flow in bulk of formation
compressibility, permeability, viscosity of original reservoir fluid

85. Description of leakoff through flow in porous media and/or filtercake build-up
4/16/2017
Concept of leakoff coefficient
Integrated leakoff volume:
Leakoff Width
Where are those “twos” coming from?
What is the physical meaning?
Fracture Design 2004

86. Step rate test
Bottomhole pressure
Injection rate
Time

87. Step rate test Propagation pressure Two straight lines
Injection rate
Bottomhole pressure
Propagation pressure
Two straight lines

88. Fall-off (minifrac) Bottomhole pressure Injection rate Injection rate
3 ISIP
4 Closure
5 Reopening
6 Forced closure
7 Pseudo steady state
8 Rebound 1 5 2 3 4 8 6
Injection rate
Bottomhole pressure
Injection rate
2nD injection
cycle
1st injection cycle 7
shut-in
flow-back
Time

89. Pressure fall-off analysis (Nolte)

90. g-function where F[a, b; c; z] is the Hypergeometric function,
dimensionless
shut-in time
area-growth
exponent
where F[a, b; c; z] is the Hypergeometric function,
available in the form of tables and computing algorithms

91. g-function

92. Pressure fall-off
Fracture stiffness

93. Fracture Stiffness (reciprocal compliance)
Pa/m

94. Shlyapobersky assumption
No spurt-loss bN mN
Ae from intercept pw g

95. Nolte-Shlyapobersky ( ) C m E t h - ' 4 p x 2 R 3 8
PKN
a=4/5
KGD
a=2/3
Radial
a=8/9
Leakoff
coefficient, C L ( ) N e f m E t h - ' 4 p x 2 R 3 8
Fracture
Extent i b V =
Width w
830 .
956
754
Fluid
Efficiency
Vi: injected into one wing