Hydraulic Fracturing
Short Course,
Texas A&M University
College Station
2005
Fracture Design
Fracture Dimensions
Fracture Modeling
Peter P. Valkó
Fracture Design
Fracture
Design
2
Source: Economides and Nolte: Reservoir Stimulation 3rd Ed.
Frac Design Goals
Fracture
Design
3
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
Fracture
Design
4
Hydraulic Fracturing Design and
Evaluation
Why do we create a propped fracture?
How do we achieve our goals?
Data gathering
Design
Execution
Evaluation
Fracture
Design
5
Fractured Well Performance
Relation of morphology to performance
Streamline view
Flow regimes, Productivity Index, Pseudosteady state Productivity Index, skin and
equivalent wellbore radius
Fracture
Design
6
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
Fracture
Design
7
Principle of least resistance
Least Principal Stress
Horizontal fracture
Fracture
Design
8
Least Principal Stress
Vertical fracture
Mismatch (Choked fracture)
Typical mismatch situations:
Horizontal well with a transverse vertical
fracture
Vertical well intersecting a horizontal
fracture
Fracture
Design
9
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
Fracture
Design
10
Dp or q
q  J post Dp
Longitudinal Vertical Fracture Horizontal well
Can it be done?
sH,min
sH,min
Fracture
Design
11
xf
sH,max
Transverse Vertical Fractures Horizontal Well
Hydraulic Fracture
sH,max
Radial
converging
flow in frac
Fracture
Design
12
sH,max
D
xf
sH,min
Fracture Morphology
source: Economides at al.: Petroleum Well Construction
Fracture
Design
13
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?
Fracture
Design
14
Prod Eng 101
Transient vs Pseudo-steady state
Productivity Index
Skin
Fracture
Design
15
Pseudo-steady state Productivity Index
q  JDp
Production rate is proportional to drawdown, defined as average
pressure in the reservoir minus wellbore flowing pressure
 2kh 
 J D Dp
q  
 B 
Drawdown
Circular:
1
JD 
 re  3
ln     s
 rw  4
Fracture
Design
16
Dimensionless
Productivity Index
Hawkins formula
k
 rs
s    1 ln
 k s  rw
k
ks
rw
Damage
penetration
distance
Fracture
Design
17
rs
Exercise 1
Calculate the skin factor due to radial damage if
rs
Wellbore radius
0.328 ft
Permeability impairment
k
 5 folds
ks
0.5 ft
Damage penetration
Solution of Exercise 1
k
 rs
s    1 ln
 k s  rw
Fracture
Design
18
rs  0.828 ft
0.828


s  5  1  ln[
]  3.7
Note that any "consistent" system of units is OK.
0.328
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.):
3.7
2980
ln[
]  0.75  3.7
0.328
 0.31
Therefore 31 % of the pressure drawdown is not utilized because of the near
wellbore damage.
The damage ratio is DR = 31 %
Fracture
Design
19
The flow efficiency is FE = 69 %.
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:
re
]  0.75  s
rw
FOI 
r
ln[ e ]  0.75
rw
ln[
 2980 
 0.75  ln 
 3.7

 0.328 
Folds of Increase :
 1.44
 2980 
 0.75  ln 
 0.328 
Fracture
Design
20
The Productivity Index increase is 44 % ,
therefore the production increase is 44 % .
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
2980
]  0.75  3.7
FOI  0.328
 3.6
2980
ln[
]  0.75  5
0.328
ln[
Fracture
Design
21
The Productivity Index will increase 260 % .
Fully penetrating vertical fracture:
Relating Performance to Dimensions
wp
h
2Vfp
Fracture
Design
22
2xf
Dimensionless fracture conductivity
2 xf
w
kf w
Dimensionless
fracture conductivity C fD  kx
f
Fracture
Design
23
fracture conductivity
no name
Accounting for PI: sf and f and r’w
q  JDp
sf is pseudo skin factor used after the treatment
to describe the productivity
 2kh 
 2kh 
1

 J D
J  
 
 B  ln[ re ]  0.75  s
 B 
f
rw
Fracture
Design
24
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
Pseudo-skin, equivalent radius, f-factor
J
2kh


re
B ln 0.472  s f 
rw


J
or
2kh

re 
B ln 0.472 
r 'w 

Prats
f (C fD )
J
Fracture
Design
25
2πkh
 0.472re 
x f 

Bμ ln
  s f  ln
xf
rw 



2πkh
 0.472re

Bμ ln
 f
xf


Cinco-Ley
Notation
rw
wellbore radius, m (or ft)
r'w
Prats’ equivalent wellbore radius due to fracture,
m (or ft)
f  s f  ln
Fracture
Design
26
xf
rw
Cinco-Ley-Samanieggo factor, dimensionless
sf
the pseudo skin factor due to fracture,
dimensionless
rw
xf
Prats' dimensionless (equivalent) wellbore
radius
But JD is the best
Example
Assume rw = 0.3 ft and A= 40 acre
s
,
w
r , ft
7
-4
36
Fracture
Design
27
Dimensionless Productivity Index, sf
and f and r’w
JD 
1
re
ln 0.472  s f
rw
or
JD 
1
re
ln 0.472
r 'w
Prats
f (C fD )
1
1
JD 

0.472re
xf 
0.472re 
f
 ln
ln
  s f  ln
xf
xf
rw 

Cinco-Ley
Fracture
Design
28
Penetration Ratio
Dimensionless Fracture Conductivity
Proppant Number
Ix 
2x f
C fD 
xe
y e = xe
kf w
2 xf
kx f
xe
N prop 
Fracture
Design
29
4k f V f,prop,1 wing
kVres

2k f V f,prop,2 wing
kVres
 (I x )2C fD
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.
Fracture
Design
30
 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.)
Prats' Dimensionless Wellbore Radius
1.0
rw'
 0 .5
xf
'
w
r
x f 0.1
rw'
 0.25  C fD
xf
0.01
0.1
Fracture
Design
31
1.0
C fD 
kf w
kx f
10
100
Cinco-Ley and Samaniego graph
f (CfD)= sf + ln(xf/rw)
4
1.65-0.328u+0.116u 2
f (C fD ) 
1+0.18u+0.064u 2+0.005u 3
where u  ln C fD
f
3
2
1
use f = ln(2) for CfD > 1000
0
Fracture
Design
32
0.1
1
10
CfD
100
1000
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.)
Fracture
Design
33
Proppant Number Various ways to look at it
N prop  I C fD
2
x
Nprop= const means
fixed proppant
volume
Fracture
Design
34

4k f x f w

4k f V1 wing , propped
N prop 
2k f V2 wing , propped
2
e
kx
2
e
kx h
kVreservoir
Fig 1: JD vs CfD (moderate Nprop)
Fracture
Design
35
Fig 2: JD vs CfD (large Nprop)
Fracture
Design
36
OPTIMIZATION
Fracture
Design
37
Optimal length and width
Struggle for propped volume: w and xf
2Vfp = 2h wp xf
V fp  hw p x f
C fD 
Fracture
Design
38
k f wp
kx f
2Vfp = 2h wp xf
 V fp k f
xf  
 C hk
 fD
1/ 2




1/ 2
 C fDV fp k 

wp  
 hk

f


The Key Parameter is the
Proppant Number
0.5
Dimensionless Productivity Index, J
D
X e=Y e
Ye
0.4
Ix =1
2Xf
Medium perm
0.1
Xe
0.06
0.03
0.3
0.01
0.006
0.003
High perm
Frac&Pack
0.001
0.2
0.0006
0.0003
N prop =0.0001
-4
10
Fracture
Design
39
-3
10
-2
-1
0
10
10
10
Dimensionless Fracture Conductivity, CfD
1
10
2
10
The Key Parameter is the
Proppant Number
2.0
Dimensionless Productivity Index, J
D
X e=Y e
Ye
Ix =1
100
2Xf
1.5
60
Xe
30
10
6
1.0
Low perm
Massive HF
3
1
0.6
0.3
0.5
Medium perm
N prop =0.1
0.1
Fracture
Design
40
1
10
100
Dimensionless Fracture Conductivity, CfD
1000
Let us read the optimum from the JD
Figures!
dimensionless fracture conductivity
(for smaller Nprop)
penetration ratio
(for larger Nprop)
Fracture
Design
41
Optimum for low and moderate
Proppant Number
0.5
Dimensionless Productivity Index, J
D
X e=Y e
Ye
0.4
Ix =1
2Xf
0.1
Xe
0.06
0.03
0.3
0.01
0.006
0.003
0.001
0.2
0.0006
0.0003
N prop =0.0001
-4
10
Fracture
Design
42
-3
10
-2
-1
0
10
10
10
Dimensionless Fracture Conductivity, CfD
1
10
2
10
CfDopt=1.6
Optimum for large Proppant Number
100
30
Dimensionless Productivity Index, J
D
1.8
1.6
X e=Y e
Ye
10
2Xf
6
Xe
3
1.4
1.2
1.0
1
0.8
0.6
0.6
0.3
0.4
0.01
Fracture
Design
43
N prop =0.1
0.1
Penetration Rate, IX
1
Tight Gas and Frac&Pack:
the extremes
Tight gas k << 1 md (hard rock)
1/ 2
 V fp k f 

xf  
C

hk
fDopt


1/ 2
 C fDoptV fp k 

wp  


hk
f


High permeability k >> 1 md (soft formation)
1/ 2
 V fp k f 

x f  
 1.6hk 
Fracture
Design
44
1/ 2
 1.6V fp k 

wp  
 hk

f


FracPi
Fracture
Design
45
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 = 0.328 ft, the skin factor before fracturing is spre = 5.
Determine the optimal fracture length and propped width.
Fracture
Design
46
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
N prop

 


2  60103 md  380 ft 3

 0.1
2
2
0.5 md  2100 ft    65 ft 

1
 0.467
0.99  0.5 ln 0.1
Folds of Increase
J post
J pre
Fracture
Design
47
J D ,opt ( N prop )

1
2100
ln[
]  0.75  5
0.328
FOI: 6.8 with respect to skin 5
FOI: 3.8 with respect to skin=0
FracPi
0.467
0.0768
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
V1wp
1/2
 380 ft 3


(60,000 md) 

xf   2
 1.6  (65 ft)(0.5 md) 




The propped width is
wp 
Fracture
Design
48
V1wp
h xf
 468 ft
 0.075 in. (1.8 mm)
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 = 0.328 ft, the skin factor before fracturing is spre = 5.
(Assume all proppant goes to pay.)
Fracture
Design
49
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 = 0.328 ft, the skin factor before fracturing is spre = 5.
(Assume all proppant goes to pay.)
Fracture
Design
50
Economic optimization
 Production forecast
 Transient regime
 Stabilized
 Economics: Converting additional production into
value
 Time value of money
 Discounted revenue
NPV
Fracture
Design
51
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
Fracture
Design
52
Treatment Sizing
Δ Rev n
- Cost
n
n 1 (1  i)
N
NPV  
Fracture
Design
53
Pre-Treatment Data Gathering
Fracture
Design
54
Design Input Data
 Petroleum Engineering Data
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
Fracture
Design
55
Rock Properties
Linear Elasticity
Poroelasticity
Fracture Mechanics
Fracture
Design
56
Young's modulus and Poisson ratio
Uniaxial test
F
A
 xx  Dl
l
 yy  DD
D
Dl
l
D
s xx  F
A
Fracture
Design
57
s xx
E
xx
 yy
 
xx
DD/2
Linear stress-strain relations
Other elasticity constants
Required
\
Known
Shear modulus, G
E
21   
Young's modulus, E
E
Poisson ratio, 
Plane strain modulus, E'
Fracture
Design
58
E, 

E
1  2
G, 
E ,G
G
G
2G 1   
E

E  2G
2G
2G
1 
4G 2
4G  E
Formation Classification
Two types
Consolidated and tight E = 106 + psi
Unconsolidated and soft E = 105 - psi
Fracture
Design
59
Poroelasticity and Biot’s constant
σ  σ   αp
Total Stress = Effective Stress + a[Pore Pressure]
Fracture
Design
60
Who Carries the Load?
Total Stress = Effective Stress + a[Pore Pressure]
Grains
Force
Biot’s constant
Fracture
Design
61
Pore Fluid
a ~ 0.7
Stress State in Formations
Far Field and Induced Stresses, Fracture
Initiation and Orientation
Stress versus Depth
Minimum Horizontal Stress
Magnitude and Direction
Fracture
Design
62
Total (absolute) horizontal stress
The simplest model:
D
s v  g  dz
0
s v  s v  ap
sh
'
sh 
Fracture
Design
63

1 

1 
s v  ap 
s v  ap   ap
1) Poisson ratio changes from layer to layer
2) Pore pressure changes in time
Crossover of Minimum Stress
Fracture
Design
64
Ground Surface
-500
0
Critical Depth
977 m
-1000
-500
-1500
-1000
-2000
-1500
-2500
-2000
-3000
-2500
80x106
0
20x106
40x106
Stress, Pa
60x106
Current Depth , m
Depth from original ground surface, m
0
Stress Gradients
Overburden gradient gradient
Slope of the Vertical Stress line
 1.1 psi/ft
Frac gradient
Basically the slope of the minimum
horizontal stress line 0.4 - 0.9 psi/ft
Extreme value: 1.1 psi/ft or more
Fracture
Design
65
Fracture width
Fracture
Design
66
Linear Elasticity + Fractures
The force opening the fracture comes from net pressure
Net pressure = fluid pressure - minimum principal stress
pn
=
p
smin
The net pressure distribution determines the width profile
Plane strain modulus and characteristic half length
Fracture
Design
67
Ideal Crack Shapes (Plane strain)
Plane strain: Infinite repetition of the same picture (2D)
E

Plane - strain modulus: E 
1  2
Half length c
w
pn(x)
Deformation (distribution)
net pressure (distribution)
Fracture
Design
68
Shape of a pressurized crack, pn=cons
Width
4 pn
w( x) 
E'
c2  x2
pn : net pressure
c : half length
“characteristic dimension”
Max Width
4c
w0 
pn
E'
linearity preserved
Fracture
Design
69
c
w
Height and Width in Layered Formation
Far-field Stress
Upper tip
Pinch point
Lower tip
Fracture
Design
70
Questions:
Contained?
Breakthrough?
Run-away?
Up or Down?
Width?
Hydrostatic
pressure?
Height
control?
What can be
measured?
From Fracture Mechanics to Fracture
Height
Fracture
Design
71
Stress Intensity Factor
weighted pressure at tip
Pa · m1/2
psi - in.1/2
c
K I  2c 
Weighting function: the
nearer to tip, the more
important the pressure
value
Fracture
Design
72
stress distribution
at tip
c
pn ( x )
c x
2
2
dx

1
cx
x
c
KI : proportionality const
Stability of Crack, Propagation
Critical value of stress intensity factor:
Fracture Toughness KIC
Propagation: when stress intensity factor
is larger than fracture toughness
Fracture
Design
73
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)
Fracture
Design
74
Calculation Based on Equilibrium
Fracture Height Theory
far field stress
fluid pressure
Fracture
Design
75
p
r
o
f
i
l
e
Stress Intensity Factor at the Tips (calc)
= Fracture Toughness of the Layer (given)
KI,top =
KI,bottom =
hp
  yu  yd 
hp
1
  pn  y 
  yu  yd 
-1
1
  pn  y 
-1
1 y
dy
1 y
1 y
dy
1 y
Two equations, two unknowns
Fracture
Design
76
Penetration Into Upper and Lower
Layers
y
Klc,2
1
s2
Dhu
yu
hp
0
s1
yd
-1
Dhd
s3
Fracture
Design
77
Klc,3
Notation
2Dhu
yu  1 
hp  Dhu  Dhd
Dhd  Dhu
k 00  pcp  g
2
2Dhd
yd  1 
hp  Dhu  Dhd
k1   g
p( y)  k00  k1 y
pn ( y )  p( y )  s ( y )
Fracture
Design
78
2h p
y u  yd
Input to a Height Map Calculation
hp
s1
s2
s3
KIC,2
KIC,3

Fracture
Design
79
50 ft
3000 psi
3500 psi
4000 psi
1000 psiin.1/2
1000 psiin.1/2
3
62.4 lbm/ft
15.24 m
20.68 MPa
24.13 MPa
27.58 MPa
1.01 MPam1/2
1.01 MPam1/2
3
1000 kg/m
Calculated Height Map
(after HFM)
Tip
Location
[ft]
Tip
Location
[m]
1000
300
800
200
600
400
100
200
0
0
-200
-100
-400
-600
-200
-800
-300
-1000
-1200
3000
Fracture
Design
80
3100
3200
3300
3400
3500
3600
3700
3800 psi
26 MPa
21
Treating Pressure
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
Fracture
Design
81
Fluid loss:
the property of both the rock and the fluid
1
2
Fracture
Design
82
Leak-off
Spurt loss
Fluid Loss in Lab
AL
CL
uL 
t
VLost
= S p  2CL t
AL
Fracture
Design
83
units :
m  mm
Lost volume per unit surface, m
0.007
0.006
0.005
0.004
0.003
y = 0.0024 + 0.000069x
0.002
Sp
0.001
0
0
10
CL
Sp
2CL
20
30
40
50
1/2
1/2
Square root time, t (s )
m
unit :
s
unit : m
60
m3
or
m2 s
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
1
1
1
1



Ct CW Cv Cc
Fracture
Design
84
Description of leakoff through flow in
porous media and/or filtercake build-up
 Concept of leakoff coefficient
Where are those “twos” coming from?
 Integrated leakoff volume:
 Leakoff Width
What is the physical meaning?
Fracture
Design
85
CL
uL 
t
m m / s1/ 2
 1/ 2
s
s
VL  2 ACL t
VL
wL 
 2C L t
AL
m mm
Fracture
Design
86
Injection rate
Bottomhole pressure
Step rate test
Time
Bottomhole pressure
Step rate test
Propagation pressure
Two straight lines
Fracture
Design
87
Injection rate
3 ISIP
Fall-off (minifrac)
4 Closure
5 Reopening
6 Forced closure
1
5
2
7 Pseudo steady state
8 Rebound
3
2nD injection
cycle
7
shut-in
Fracture
Design
88
flow-back
Time
8
Injection rate
6
Injection rate
1st injection
cycle
Bottomhole pressure
4
Pressure fall-off analysis
(Nolte)
Dt D  Dt / te
Vte  Dt = Vi  2Ae S p  g Dt D ,a 2Ae C L te
wte  Dt
Fracture
Design
89
Vi

- 2 S p  g Dt D , a  2C L te
Ae
g-function
1 Dt D

1
g Dt D ,a     
dt D dAD
1/a

0 
 A1D/ a t D  AD
1
dimensionless
shut-in time
area-growth
exponent

4a Dt D  2 1  Dt D  F 1 / 2,a ;1  a ;1  Dt D 
g Dt D ,a  
1  2a
Fracture
Design
90
where F[a, b; c; z] is the Hypergeometric function,
available in the form of tables and computing algorithms
1

g-function
Approximation of the g-function for various exponents a (d = DtD)
4
1.41495 + 79.4125 d + 632.457 d 2 + 1293.07 d 3 + 763.19 d 4 + 94.0367 d 5

g  d ,a   
5  1. + 54.8534 d + 383.11 d 2 + 540.342 d 3 + 167.741 d 4 + 6.49129 d 5  0.0765693 d 6

2
1.47835 + 81.9445 d + 635.354 d 2 + 1251.53 d 3 + 717.71 d 4 + 86.843 d 5

g d , a   

3  1. + 54.2865 d + 372.4 d 2 + 512.374 d 3 + 156.031 d 4 + 5.95955 d 5 - 0.0696905 d 6
8
1.37689 + 77.8604 d + 630.24 d 2 + 1317.36 d 3 + 790.7 d 4 + 98.4497 d 5

g d , a   

9  1. + 55.1925 d + 389.537 d 2 + 557.22 d 3 + 174.89 d 4 + 6.8188 d 5 - 0.0808317 d 6
Fracture
Design
91
Pressure fall-off
Dt D  Dt / te
Vte  Dt = Vi  2Ae S p  g Dt D ,a 2Ae C L te
wte  Dt
Vi

- 2 S p  2C L te g Dt D , a 
Ae
Fracture stiffness
pnet  S f w


pw   pC  S f Vi / Ae - 2S f S p  - 2S f CL te  g Dt D ,a 
Fracture
Design
92
pw  bN  mN  g Dt D , a 
Fracture Stiffness
(reciprocal compliance)
pnet  S f w
Pa/m
Table 5.5 Proportionality constant, Sf and suggested a for basic fracture geometries
Fracture
Design
93
PKN
KGD
Radial
a
4/5
2/3
8/9
Sf
2E '
h f
E'
x f
3E '
16 R f
Shlyapobersky assumption
No spurt-loss




Vi
pw   pC  S f
- 2S f S p  - 2S f CL te  g Dt D ,a 
Ae


Ae from intercept
bN
mN
pw
g
Fracture
Design
94
Nolte-Shlyapobersky
Leakoff
coefficient,
PKN a4/5
KGD a2/3
h f
x f
4 te E '
 mN 
2 te E '
 mN 
Radial a8/9
8R f
3 t e E '
 m N 
CL
Fracture
Extent
Fracture
Width
xf 
2 E Vi
h 2f bN  pC 
we 
Vi

x f hf
 2.830C L t e
Fluid
Efficiency
Fracture
Design
95
he 
we x f h f
Vi
xf 
E Vi
h f bN  pC 
we 
Vi

x f hf
 2.956C L t e
he 
Rf  3
we 
Vi: injected into one wing
Vi
2 
Rf

2
 2.754C L t e
we x f h f
Vi
3E Vi
8bN  pC 
he 
we R 2f
Vi

2